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<script type="text/javascript">
/*<![CDATA[*/
/*
ASCIIMathML.js
==============
This file contains JavaScript functions to convert ASCII math notation
to Presentation MathML. The conversion is done while the (X)HTML page
loads, and should work with Firefox/Mozilla/Netscape 7+ and Internet
Explorer 6+MathPlayer (http://www.dessci.com/en/products/mathplayer/).
Just add the next line to your (X)HTML page with this file in the same folder:
This is a convenient and inexpensive solution for authoring MathML.
Version 1.4.7 Dec 15, 2005, (c) Peter Jipsen http://www.chapman.edu/~jipsen
Latest version at http://www.chapman.edu/~jipsen/mathml/ASCIIMathML.js
For changes see http://www.chapman.edu/~jipsen/mathml/asciimathchanges.txt
If you use it on a webpage, please send the URL to jipsen@chapman.edu
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or (at
your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
General Public License (at http://www.gnu.org/copyleft/gpl.html)
for more details.
*/
var checkForMathML = true; // check if browser can display MathML
var notifyIfNoMathML = true; // display note if no MathML capability
var alertIfNoMathML = false; // show alert box if no MathML capability
var mathcolor = ""; // change it to "" (to inherit) or any other color
var mathfontfamily = "serif"; // change to "" to inherit (works in IE)
// or another family (e.g. "arial")
var displaystyle = true; // puts limits above and below large operators
var showasciiformulaonhover = true; // helps students learn ASCIIMath
var decimalsign = "."; // change to "," if you like, beware of `(1,2)`!
var AMdelimiter1 = "`", AMescape1 = "\\\\`"; // can use other characters
var AMdelimiter2 = "$", AMescape2 = "\\\\\\$", AMdelimiter2regexp = "\\$";
var doubleblankmathdelimiter = false; // if true, x+1 is equal to `x+1`
// for IE this works only in <!-- -->
//var separatetokens;// has been removed (email me if this is a problem)
var isIE = document.createElementNS==null;
if (document.getElementById==null)
alert("This webpage requires a recent browser such as\
\nMozilla/Netscape 7+ or Internet Explorer 6+MathPlayer")
// all further global variables start with "AM"
function AMcreateElementXHTML(t) {
if (isIE) return document.createElement(t);
else return document.createElementNS("http://www.w3.org/1999/xhtml",t);
}
function AMnoMathMLNote() {
var nd = AMcreateElementXHTML("h3");
nd.setAttribute("align","center")
nd.appendChild(AMcreateElementXHTML("p"));
nd.appendChild(document.createTextNode("To view the "));
var an = AMcreateElementXHTML("a");
an.appendChild(document.createTextNode("ASCIIMathML"));
an.setAttribute("href","http://www.chapman.edu/~jipsen/asciimath.html");
nd.appendChild(an);
nd.appendChild(document.createTextNode(" notation use Internet Explorer 6+"));
an = AMcreateElementXHTML("a");
an.appendChild(document.createTextNode("MathPlayer"));
an.setAttribute("href","http://www.dessci.com/en/products/mathplayer/download.htm");
nd.appendChild(an);
nd.appendChild(document.createTextNode(" or Netscape/Mozilla/Firefox"));
nd.appendChild(AMcreateElementXHTML("p"));
return nd;
}
function AMisMathMLavailable() {
if (navigator.appName.slice(0,8)=="Netscape")
if (navigator.appVersion.slice(0,1)>="5") return null;
else return AMnoMathMLNote();
else if (navigator.appName.slice(0,9)=="Microsoft")
try {
var ActiveX = new ActiveXObject("MathPlayer.Factory.1");
return null;
} catch (e) {
return AMnoMathMLNote();
}
else return AMnoMathMLNote();
}
// character lists for Mozilla/Netscape fonts
var AMcal = [0xEF35,0x212C,0xEF36,0xEF37,0x2130,0x2131,0xEF38,0x210B,0x2110,0xEF39,0xEF3A,0x2112,0x2133,0xEF3B,0xEF3C,0xEF3D,0xEF3E,0x211B,0xEF3F,0xEF40,0xEF41,0xEF42,0xEF43,0xEF44,0xEF45,0xEF46];
var AMfrk = [0xEF5D,0xEF5E,0x212D,0xEF5F,0xEF60,0xEF61,0xEF62,0x210C,0x2111,0xEF63,0xEF64,0xEF65,0xEF66,0xEF67,0xEF68,0xEF69,0xEF6A,0x211C,0xEF6B,0xEF6C,0xEF6D,0xEF6E,0xEF6F,0xEF70,0xEF71,0x2128];
var AMbbb = [0xEF8C,0xEF8D,0x2102,0xEF8E,0xEF8F,0xEF90,0xEF91,0x210D,0xEF92,0xEF93,0xEF94,0xEF95,0xEF96,0x2115,0xEF97,0x2119,0x211A,0x211D,0xEF98,0xEF99,0xEF9A,0xEF9B,0xEF9C,0xEF9D,0xEF9E,0x2124];
var CONST = 0, UNARY = 1, BINARY = 2, INFIX = 3, LEFTBRACKET = 4,
RIGHTBRACKET = 5, SPACE = 6, UNDEROVER = 7, DEFINITION = 8,
LEFTRIGHT = 9, TEXT = 10; // token types
var AMsqrt = {input:"sqrt", tag:"msqrt", output:"sqrt", tex:null, ttype:UNARY},
AMroot = {input:"root", tag:"mroot", output:"root", tex:null, ttype:BINARY},
AMfrac = {input:"frac", tag:"mfrac", output:"/", tex:null, ttype:BINARY},
AMdiv = {input:"/", tag:"mfrac", output:"/", tex:null, ttype:INFIX},
AMover = {input:"stackrel", tag:"mover", output:"stackrel", tex:null, ttype:BINARY},
AMsub = {input:"_", tag:"msub", output:"_", tex:null, ttype:INFIX},
AMsup = {input:"^", tag:"msup", output:"^", tex:null, ttype:INFIX},
AMtext = {input:"text", tag:"mtext", output:"text", tex:null, ttype:TEXT},
AMmbox = {input:"mbox", tag:"mtext", output:"mbox", tex:null, ttype:TEXT},
AMquote = {input:"\"", tag:"mtext", output:"mbox", tex:null, ttype:TEXT};
var AMsymbols = [
//some greek symbols
{input:"alpha", tag:"mi", output:"\u03B1", tex:null, ttype:CONST},
{input:"beta", tag:"mi", output:"\u03B2", tex:null, ttype:CONST},
{input:"chi", tag:"mi", output:"\u03C7", tex:null, ttype:CONST},
{input:"delta", tag:"mi", output:"\u03B4", tex:null, ttype:CONST},
{input:"Delta", tag:"mo", output:"\u0394", tex:null, ttype:CONST},
{input:"epsi", tag:"mi", output:"\u03B5", tex:"epsilon", ttype:CONST},
{input:"varepsilon", tag:"mi", output:"\u025B", tex:null, ttype:CONST},
{input:"eta", tag:"mi", output:"\u03B7", tex:null, ttype:CONST},
{input:"gamma", tag:"mi", output:"\u03B3", tex:null, ttype:CONST},
{input:"Gamma", tag:"mo", output:"\u0393", tex:null, ttype:CONST},
{input:"iota", tag:"mi", output:"\u03B9", tex:null, ttype:CONST},
{input:"kappa", tag:"mi", output:"\u03BA", tex:null, ttype:CONST},
{input:"lambda", tag:"mi", output:"\u03BB", tex:null, ttype:CONST},
{input:"Lambda", tag:"mo", output:"\u039B", tex:null, ttype:CONST},
{input:"mu", tag:"mi", output:"\u03BC", tex:null, ttype:CONST},
{input:"nu", tag:"mi", output:"\u03BD", tex:null, ttype:CONST},
{input:"omega", tag:"mi", output:"\u03C9", tex:null, ttype:CONST},
{input:"Omega", tag:"mo", output:"\u03A9", tex:null, ttype:CONST},
{input:"phi", tag:"mi", output:"\u03C6", tex:null, ttype:CONST},
{input:"varphi", tag:"mi", output:"\u03D5", tex:null, ttype:CONST},
{input:"Phi", tag:"mo", output:"\u03A6", tex:null, ttype:CONST},
{input:"pi", tag:"mi", output:"\u03C0", tex:null, ttype:CONST},
{input:"Pi", tag:"mo", output:"\u03A0", tex:null, ttype:CONST},
{input:"psi", tag:"mi", output:"\u03C8", tex:null, ttype:CONST},
{input:"Psi", tag:"mi", output:"\u03A8", tex:null, ttype:CONST},
{input:"rho", tag:"mi", output:"\u03C1", tex:null, ttype:CONST},
{input:"sigma", tag:"mi", output:"\u03C3", tex:null, ttype:CONST},
{input:"Sigma", tag:"mo", output:"\u03A3", tex:null, ttype:CONST},
{input:"tau", tag:"mi", output:"\u03C4", tex:null, ttype:CONST},
{input:"theta", tag:"mi", output:"\u03B8", tex:null, ttype:CONST},
{input:"vartheta", tag:"mi", output:"\u03D1", tex:null, ttype:CONST},
{input:"Theta", tag:"mo", output:"\u0398", tex:null, ttype:CONST},
{input:"upsilon", tag:"mi", output:"\u03C5", tex:null, ttype:CONST},
{input:"xi", tag:"mi", output:"\u03BE", tex:null, ttype:CONST},
{input:"Xi", tag:"mo", output:"\u039E", tex:null, ttype:CONST},
{input:"zeta", tag:"mi", output:"\u03B6", tex:null, ttype:CONST},
//binary operation symbols
{input:"*", tag:"mo", output:"\u22C5", tex:"cdot", ttype:CONST},
{input:"**", tag:"mo", output:"\u22C6", tex:"star", ttype:CONST},
{input:"//", tag:"mo", output:"/", tex:null, ttype:CONST},
{input:"\\\\", tag:"mo", output:"\\", tex:"backslash", ttype:CONST},
{input:"setminus", tag:"mo", output:"\\", tex:null, ttype:CONST},
{input:"xx", tag:"mo", output:"\u00D7", tex:"times", ttype:CONST},
{input:"-:", tag:"mo", output:"\u00F7", tex:"divide", ttype:CONST},
{input:"@", tag:"mo", output:"\u2218", tex:"circ", ttype:CONST},
{input:"o+", tag:"mo", output:"\u2295", tex:"oplus", ttype:CONST},
{input:"ox", tag:"mo", output:"\u2297", tex:"otimes", ttype:CONST},
{input:"o.", tag:"mo", output:"\u2299", tex:"odot", ttype:CONST},
{input:"sum", tag:"mo", output:"\u2211", tex:null, ttype:UNDEROVER},
{input:"prod", tag:"mo", output:"\u220F", tex:null, ttype:UNDEROVER},
{input:"^^", tag:"mo", output:"\u2227", tex:"wedge", ttype:CONST},
{input:"^^^", tag:"mo", output:"\u22C0", tex:"bigwedge", ttype:UNDEROVER},
{input:"vv", tag:"mo", output:"\u2228", tex:"vee", ttype:CONST},
{input:"vvv", tag:"mo", output:"\u22C1", tex:"bigvee", ttype:UNDEROVER},
{input:"nn", tag:"mo", output:"\u2229", tex:"cap", ttype:CONST},
{input:"nnn", tag:"mo", output:"\u22C2", tex:"bigcap", ttype:UNDEROVER},
{input:"uu", tag:"mo", output:"\u222A", tex:"cup", ttype:CONST},
{input:"uuu", tag:"mo", output:"\u22C3", tex:"bigcup", ttype:UNDEROVER},
//binary relation symbols
{input:"!=", tag:"mo", output:"\u2260", tex:"ne", ttype:CONST},
{input:":=", tag:"mo", output:":=", tex:null, ttype:CONST},
{input:"lt", tag:"mo", output:"<", tex:null, ttype:CONST},
{input:"<=", tag:"mo", output:"\u2264", tex:"le", ttype:CONST},
{input:"lt=", tag:"mo", output:"\u2264", tex:"leq", ttype:CONST},
{input:">=", tag:"mo", output:"\u2265", tex:"ge", ttype:CONST},
{input:"geq", tag:"mo", output:"\u2265", tex:null, ttype:CONST},
{input:"-<", tag:"mo", output:"\u227A", tex:"prec", ttype:CONST},
{input:"-lt", tag:"mo", output:"\u227A", tex:null, ttype:CONST},
{input:">-", tag:"mo", output:"\u227B", tex:"succ", ttype:CONST},
{input:"-<=", tag:"mo", output:"\u2AAF", tex:"preceq", ttype:CONST},
{input:">-=", tag:"mo", output:"\u2AB0", tex:"succeq", ttype:CONST},
{input:"in", tag:"mo", output:"\u2208", tex:null, ttype:CONST},
{input:"!in", tag:"mo", output:"\u2209", tex:"notin", ttype:CONST},
{input:"sub", tag:"mo", output:"\u2282", tex:"subset", ttype:CONST},
{input:"sup", tag:"mo", output:"\u2283", tex:"supset", ttype:CONST},
{input:"sube", tag:"mo", output:"\u2286", tex:"subseteq", ttype:CONST},
{input:"supe", tag:"mo", output:"\u2287", tex:"supseteq", ttype:CONST},
{input:"-=", tag:"mo", output:"\u2261", tex:"equiv", ttype:CONST},
{input:"~=", tag:"mo", output:"\u2245", tex:"cong", ttype:CONST},
{input:"~~", tag:"mo", output:"\u2248", tex:"approx", ttype:CONST},
{input:"prop", tag:"mo", output:"\u221D", tex:"propto", ttype:CONST},
//logical symbols
{input:"and", tag:"mtext", output:"and", tex:null, ttype:SPACE},
{input:"or", tag:"mtext", output:"or", tex:null, ttype:SPACE},
{input:"not", tag:"mo", output:"\u00AC", tex:"neg", ttype:CONST},
{input:"=>", tag:"mo", output:"\u21D2", tex:"implies", ttype:CONST},
{input:"if", tag:"mo", output:"if", tex:null, ttype:SPACE},
{input:"<=>", tag:"mo", output:"\u21D4", tex:"iff", ttype:CONST},
{input:"AA", tag:"mo", output:"\u2200", tex:"forall", ttype:CONST},
{input:"EE", tag:"mo", output:"\u2203", tex:"exists", ttype:CONST},
{input:"_|_", tag:"mo", output:"\u22A5", tex:"bot", ttype:CONST},
{input:"TT", tag:"mo", output:"\u22A4", tex:"top", ttype:CONST},
{input:"|--", tag:"mo", output:"\u22A2", tex:"vdash", ttype:CONST},
{input:"|==", tag:"mo", output:"\u22A8", tex:"models", ttype:CONST},
//grouping brackets
{input:"(", tag:"mo", output:"(", tex:null, ttype:LEFTBRACKET},
{input:")", tag:"mo", output:")", tex:null, ttype:RIGHTBRACKET},
{input:"[", tag:"mo", output:"[", tex:null, ttype:LEFTBRACKET},
{input:"]", tag:"mo", output:"]", tex:null, ttype:RIGHTBRACKET},
{input:"{", tag:"mo", output:"{", tex:null, ttype:LEFTBRACKET},
{input:"}", tag:"mo", output:"}", tex:null, ttype:RIGHTBRACKET},
{input:"|", tag:"mo", output:"|", tex:null, ttype:LEFTRIGHT},
//{input:"||", tag:"mo", output:"||", tex:null, ttype:LEFTRIGHT},
{input:"(:", tag:"mo", output:"\u2329", tex:"langle", ttype:LEFTBRACKET},
{input:":)", tag:"mo", output:"\u232A", tex:"rangle", ttype:RIGHTBRACKET},
{input:"<<", tag:"mo", output:"\u2329", tex:null, ttype:LEFTBRACKET},
{input:">>", tag:"mo", output:"\u232A", tex:null, ttype:RIGHTBRACKET},
{input:"{:", tag:"mo", output:"{:", tex:null, ttype:LEFTBRACKET, invisible:true},
{input:":}", tag:"mo", output:":}", tex:null, ttype:RIGHTBRACKET, invisible:true},
//miscellaneous symbols
{input:"int", tag:"mo", output:"\u222B", tex:null, ttype:CONST},
{input:"dx", tag:"mi", output:"{:d x:}", tex:null, ttype:DEFINITION},
{input:"dy", tag:"mi", output:"{:d y:}", tex:null, ttype:DEFINITION},
{input:"dz", tag:"mi", output:"{:d z:}", tex:null, ttype:DEFINITION},
{input:"dt", tag:"mi", output:"{:d t:}", tex:null, ttype:DEFINITION},
{input:"oint", tag:"mo", output:"\u222E", tex:null, ttype:CONST},
{input:"del", tag:"mo", output:"\u2202", tex:"partial", ttype:CONST},
{input:"grad", tag:"mo", output:"\u2207", tex:"nabla", ttype:CONST},
{input:"+-", tag:"mo", output:"\u00B1", tex:"pm", ttype:CONST},
{input:"O/", tag:"mo", output:"\u2205", tex:"emptyset", ttype:CONST},
{input:"oo", tag:"mo", output:"\u221E", tex:"infty", ttype:CONST},
{input:"aleph", tag:"mo", output:"\u2135", tex:null, ttype:CONST},
{input:"...", tag:"mo", output:"...", tex:"ldots", ttype:CONST},
{input:":.", tag:"mo", output:"\u2234", tex:"therefore", ttype:CONST},
{input:"/_", tag:"mo", output:"\u2220", tex:"angle", ttype:CONST},
{input:"\\ ", tag:"mo", output:"\u00A0", tex:null, ttype:CONST},
{input:"quad", tag:"mo", output:"\u00A0\u00A0", tex:null, ttype:CONST},
{input:"qquad", tag:"mo", output:"\u00A0\u00A0\u00A0\u00A0", tex:null, ttype:CONST},
{input:"cdots", tag:"mo", output:"\u22EF", tex:null, ttype:CONST},
{input:"vdots", tag:"mo", output:"\u22EE", tex:null, ttype:CONST},
{input:"ddots", tag:"mo", output:"\u22F1", tex:null, ttype:CONST},
{input:"diamond", tag:"mo", output:"\u22C4", tex:null, ttype:CONST},
{input:"square", tag:"mo", output:"\u25A1", tex:null, ttype:CONST},
{input:"|__", tag:"mo", output:"\u230A", tex:"lfloor", ttype:CONST},
{input:"__|", tag:"mo", output:"\u230B", tex:"rfloor", ttype:CONST},
{input:"|~", tag:"mo", output:"\u2308", tex:"lceiling", ttype:CONST},
{input:"~|", tag:"mo", output:"\u2309", tex:"rceiling", ttype:CONST},
{input:"CC", tag:"mo", output:"\u2102", tex:null, ttype:CONST},
{input:"NN", tag:"mo", output:"\u2115", tex:null, ttype:CONST},
{input:"QQ", tag:"mo", output:"\u211A", tex:null, ttype:CONST},
{input:"RR", tag:"mo", output:"\u211D", tex:null, ttype:CONST},
{input:"ZZ", tag:"mo", output:"\u2124", tex:null, ttype:CONST},
{input:"f", tag:"mi", output:"f", tex:null, ttype:UNARY, func:true},
{input:"g", tag:"mi", output:"g", tex:null, ttype:UNARY, func:true},
//standard functions
{input:"lim", tag:"mo", output:"lim", tex:null, ttype:UNDEROVER},
{input:"Lim", tag:"mo", output:"Lim", tex:null, ttype:UNDEROVER},
{input:"sin", tag:"mo", output:"sin", tex:null, ttype:UNARY, func:true},
{input:"cos", tag:"mo", output:"cos", tex:null, ttype:UNARY, func:true},
{input:"tan", tag:"mo", output:"tan", tex:null, ttype:UNARY, func:true},
{input:"sinh", tag:"mo", output:"sinh", tex:null, ttype:UNARY, func:true},
{input:"cosh", tag:"mo", output:"cosh", tex:null, ttype:UNARY, func:true},
{input:"tanh", tag:"mo", output:"tanh", tex:null, ttype:UNARY, func:true},
{input:"cot", tag:"mo", output:"cot", tex:null, ttype:UNARY, func:true},
{input:"sec", tag:"mo", output:"sec", tex:null, ttype:UNARY, func:true},
{input:"csc", tag:"mo", output:"csc", tex:null, ttype:UNARY, func:true},
{input:"log", tag:"mo", output:"log", tex:null, ttype:UNARY, func:true},
{input:"ln", tag:"mo", output:"ln", tex:null, ttype:UNARY, func:true},
{input:"det", tag:"mo", output:"det", tex:null, ttype:UNARY, func:true},
{input:"dim", tag:"mo", output:"dim", tex:null, ttype:CONST},
{input:"mod", tag:"mo", output:"mod", tex:null, ttype:CONST},
{input:"gcd", tag:"mo", output:"gcd", tex:null, ttype:UNARY, func:true},
{input:"lcm", tag:"mo", output:"lcm", tex:null, ttype:UNARY, func:true},
{input:"lub", tag:"mo", output:"lub", tex:null, ttype:CONST},
{input:"glb", tag:"mo", output:"glb", tex:null, ttype:CONST},
{input:"min", tag:"mo", output:"min", tex:null, ttype:UNDEROVER},
{input:"max", tag:"mo", output:"max", tex:null, ttype:UNDEROVER},
//arrows
{input:"uarr", tag:"mo", output:"\u2191", tex:"uparrow", ttype:CONST},
{input:"darr", tag:"mo", output:"\u2193", tex:"downarrow", ttype:CONST},
{input:"rarr", tag:"mo", output:"\u2192", tex:"rightarrow", ttype:CONST},
{input:"->", tag:"mo", output:"\u2192", tex:"to", ttype:CONST},
{input:"|->", tag:"mo", output:"\u21A6", tex:"mapsto", ttype:CONST},
{input:"larr", tag:"mo", output:"\u2190", tex:"leftarrow", ttype:CONST},
{input:"harr", tag:"mo", output:"\u2194", tex:"leftrightarrow", ttype:CONST},
{input:"rArr", tag:"mo", output:"\u21D2", tex:"Rightarrow", ttype:CONST},
{input:"lArr", tag:"mo", output:"\u21D0", tex:"Leftarrow", ttype:CONST},
{input:"hArr", tag:"mo", output:"\u21D4", tex:"Leftrightarrow", ttype:CONST},
//commands with argument
AMsqrt, AMroot, AMfrac, AMdiv, AMover, AMsub, AMsup,
{input:"hat", tag:"mover", output:"\u005E", tex:null, ttype:UNARY, acc:true},
{input:"bar", tag:"mover", output:"\u00AF", tex:"overline", ttype:UNARY, acc:true},
{input:"vec", tag:"mover", output:"\u2192", tex:null, ttype:UNARY, acc:true},
{input:"dot", tag:"mover", output:".", tex:null, ttype:UNARY, acc:true},
{input:"ddot", tag:"mover", output:"..", tex:null, ttype:UNARY, acc:true},
{input:"ul", tag:"munder", output:"\u0332", tex:"underline", ttype:UNARY, acc:true},
AMtext, AMmbox, AMquote,
{input:"bb", tag:"mstyle", atname:"fontweight", atval:"bold", output:"bb", tex:null, ttype:UNARY},
{input:"mathbf", tag:"mstyle", atname:"fontweight", atval:"bold", output:"mathbf", tex:null, ttype:UNARY},
{input:"sf", tag:"mstyle", atname:"fontfamily", atval:"sans-serif", output:"sf", tex:null, ttype:UNARY},
{input:"mathsf", tag:"mstyle", atname:"fontfamily", atval:"sans-serif", output:"mathsf", tex:null, ttype:UNARY},
{input:"bbb", tag:"mstyle", atname:"mathvariant", atval:"double-struck", output:"bbb", tex:null, ttype:UNARY, codes:AMbbb},
{input:"mathbb", tag:"mstyle", atname:"mathvariant", atval:"double-struck", output:"mathbb", tex:null, ttype:UNARY, codes:AMbbb},
{input:"cc", tag:"mstyle", atname:"mathvariant", atval:"script", output:"cc", tex:null, ttype:UNARY, codes:AMcal},
{input:"mathcal", tag:"mstyle", atname:"mathvariant", atval:"script", output:"mathcal", tex:null, ttype:UNARY, codes:AMcal},
{input:"tt", tag:"mstyle", atname:"fontfamily", atval:"monospace", output:"tt", tex:null, ttype:UNARY},
{input:"mathtt", tag:"mstyle", atname:"fontfamily", atval:"monospace", output:"mathtt", tex:null, ttype:UNARY},
{input:"fr", tag:"mstyle", atname:"mathvariant", atval:"fraktur", output:"fr", tex:null, ttype:UNARY, codes:AMfrk},
{input:"mathfrak", tag:"mstyle", atname:"mathvariant", atval:"fraktur", output:"mathfrak", tex:null, ttype:UNARY, codes:AMfrk}
];
function compareNames(s1,s2) {
if (s1.input > s2.input) return 1
else return -1;
}
var AMnames = []; //list of input symbols
function AMinitSymbols() {
var texsymbols = [], i;
for (i=0; i<AMsymbols.length; i++)
if (AMsymbols[i].tex)
texsymbols[texsymbols.length] = {input:AMsymbols[i].tex,
tag:AMsymbols[i].tag, output:AMsymbols[i].output, ttype:AMsymbols[i].ttype};
AMsymbols = AMsymbols.concat(texsymbols);
AMsymbols.sort(compareNames);
for (i=0; i<AMsymbols.length; i++) AMnames[i] = AMsymbols[i].input;
}
var AMmathml = "http://www.w3.org/1998/Math/MathML";
function AMcreateElementMathML(t) {
if (isIE) return document.createElement("m:"+t);
else return document.createElementNS(AMmathml,t);
}
function AMcreateMmlNode(t,frag) {
// var node = AMcreateElementMathML(name);
if (isIE) var node = document.createElement("m:"+t);
else var node = document.createElementNS(AMmathml,t);
node.appendChild(frag);
return node;
}
function newcommand(oldstr,newstr) {
AMsymbols = AMsymbols.concat([{input:oldstr, tag:"mo", output:newstr,
tex:null, ttype:DEFINITION}]);
}
function AMremoveCharsAndBlanks(str,n) {
//remove n characters and any following blanks
var st;
if (str.charAt(n)=="\\" && str.charAt(n+1)!="\\" && str.charAt(n+1)!=" ")
st = str.slice(n+1);
else st = str.slice(n);
for (var i=0; i<st.length && st.charCodeAt(i)<=32; i=i+1);
return st.slice(i);
}
function AMposition(arr, str, n) {
// return position >=n where str appears or would be inserted
// assumes arr is sorted
if (n==0) {
var h,m;
n = -1;
h = arr.length;
while (n+1<h) {
m = (n+h) >> 1;
if (arr[m]<str) n = m; else h = m;
}
return h;
} else
for (var i=n; i<arr.length && arr[i]<str; i++);
return i; // i=arr.length || arr[i]>=str
}
function AMgetSymbol(str) {
//return maximal initial substring of str that appears in names
//return null if there is none
var k = 0; //new pos
var j = 0; //old pos
var mk; //match pos
var st;
var tagst;
var match = "";
var more = true;
for (var i=1; i<=str.length && more; i++) {
st = str.slice(0,i); //initial substring of length i
j = k;
k = AMposition(AMnames, st, j);
if (k<AMnames.length && str.slice(0,AMnames[k].length)==AMnames[k]){
match = AMnames[k];
mk = k;
i = match.length;
}
more = k<AMnames.length && str.slice(0,AMnames[k].length)>=AMnames[k];
}
AMpreviousSymbol=AMcurrentSymbol;
if (match!=""){
AMcurrentSymbol=AMsymbols[mk].ttype;
return AMsymbols[mk];
}
// if str[0] is a digit or - return maxsubstring of digits.digits
AMcurrentSymbol=CONST;
k = 1;
st = str.slice(0,1);
var integ = true;
while ("0"<=st && st<="9" && k<=str.length) {
st = str.slice(k,k+1);
k++;
}
if (st == decimalsign) {
st = str.slice(k,k+1);
if ("0"<=st && st<="9") {
integ = false;
k++;
while ("0"<=st && st<="9" && k<=str.length) {
st = str.slice(k,k+1);
k++;
}
}
}
if ((integ && k>1) || k>2) {
st = str.slice(0,k-1);
tagst = "mn";
} else {
k = 2;
st = str.slice(0,1); //take 1 character
tagst = (("A">st || st>"Z") && ("a">st || st>"z")?"mo":"mi");
}
if (st=="-" && AMpreviousSymbol==INFIX) {
AMcurrentSymbol = INFIX; //trick "/" into recognizing "-" on second parse
return {input:st, tag:tagst, output:st, ttype:UNARY, func:true};
}
return {input:st, tag:tagst, output:st, ttype:CONST};
}
function AMremoveBrackets(node) {
var st;
if (node.nodeName=="mrow") {
st = node.firstChild.firstChild.nodeValue;
if (st=="(" || st=="[" || st=="{") node.removeChild(node.firstChild);
}
if (node.nodeName=="mrow") {
st = node.lastChild.firstChild.nodeValue;
if (st==")" || st=="]" || st=="}") node.removeChild(node.lastChild);
}
}
/*Parsing ASCII math expressions with the following grammar
v ::= [A-Za-z] | greek letters | numbers | other constant symbols
u ::= sqrt | text | bb | other unary symbols for font commands
b ::= frac | root | stackrel binary symbols
l ::= ( | [ | { | (: | {: left brackets
r ::= ) | ] | } | :) | :} right brackets
S ::= v | lEr | uS | bSS Simple expression
I ::= S_S | S^S | S_S^S | S Intermediate expression
E ::= IE | I/I Expression
Each terminal symbol is translated into a corresponding mathml node.*/
var AMnestingDepth,AMpreviousSymbol,AMcurrentSymbol;
function AMparseSexpr(str) { //parses str and returns [node,tailstr]
var symbol, node, result, i, st,// rightvert = false,
newFrag = document.createDocumentFragment();
str = AMremoveCharsAndBlanks(str,0);
symbol = AMgetSymbol(str); //either a token or a bracket or empty
if (symbol == null || symbol.ttype == RIGHTBRACKET && AMnestingDepth > 0) {
return [null,str];
}
if (symbol.ttype == DEFINITION) {
str = symbol.output+AMremoveCharsAndBlanks(str,symbol.input.length);
symbol = AMgetSymbol(str);
}
switch (symbol.ttype) {
case UNDEROVER:
case CONST:
str = AMremoveCharsAndBlanks(str,symbol.input.length);
return [AMcreateMmlNode(symbol.tag, //its a constant
document.createTextNode(symbol.output)),str];
case LEFTBRACKET: //read (expr+)
AMnestingDepth++;
str = AMremoveCharsAndBlanks(str,symbol.input.length);
result = AMparseExpr(str,true);
AMnestingDepth--;
if (typeof symbol.invisible == "boolean" && symbol.invisible)
node = AMcreateMmlNode("mrow",result[0]);
else {
node = AMcreateMmlNode("mo",document.createTextNode(symbol.output));
node = AMcreateMmlNode("mrow",node);
node.appendChild(result[0]);
}
return [node,result[1]];
case TEXT:
if (symbol!=AMquote) str = AMremoveCharsAndBlanks(str,symbol.input.length);
if (str.charAt(0)=="{") i=str.indexOf("}");
else if (str.charAt(0)=="(") i=str.indexOf(")");
else if (str.charAt(0)=="[") i=str.indexOf("]");
else if (symbol==AMquote) i=str.slice(1).indexOf("\"")+1;
else i = 0;
if (i==-1) i = str.length;
st = str.slice(1,i);
if (st.charAt(0) == " ") {
node = AMcreateElementMathML("mspace");
node.setAttribute("width","1ex");
newFrag.appendChild(node);
}
newFrag.appendChild(
AMcreateMmlNode(symbol.tag,document.createTextNode(st)));
if (st.charAt(st.length-1) == " ") {
node = AMcreateElementMathML("mspace");
node.setAttribute("width","1ex");
newFrag.appendChild(node);
}
str = AMremoveCharsAndBlanks(str,i+1);
return [AMcreateMmlNode("mrow",newFrag),str];
case UNARY:
str = AMremoveCharsAndBlanks(str,symbol.input.length);
result = AMparseSexpr(str);
if (result[0]==null) return [AMcreateMmlNode(symbol.tag,
document.createTextNode(symbol.output)),str];
if (typeof symbol.func == "boolean" && symbol.func) { // functions hack
st = str.charAt(0);
if (st=="^" || st=="_" || st=="/" || st=="|" || st==",") {
return [AMcreateMmlNode(symbol.tag,
document.createTextNode(symbol.output)),str];
} else {
node = AMcreateMmlNode("mrow",
AMcreateMmlNode(symbol.tag,document.createTextNode(symbol.output)));
node.appendChild(result[0]);
return [node,result[1]];
}
}
AMremoveBrackets(result[0]);
if (symbol.input == "sqrt") { // sqrt
return [AMcreateMmlNode(symbol.tag,result[0]),result[1]];
} else if (typeof symbol.acc == "boolean" && symbol.acc) { // accent
node = AMcreateMmlNode(symbol.tag,result[0]);
node.appendChild(AMcreateMmlNode("mo",document.createTextNode(symbol.output)));
return [node,result[1]];
} else { // font change command
if (!isIE && typeof symbol.codes != "undefined") {
for (i=0; i<result[0].childNodes.length; i++)
if (result[0].childNodes[i].nodeName=="mi" || result[0].nodeName=="mi") {
st = (result[0].nodeName=="mi"?result[0].firstChild.nodeValue:
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</head>
<body class="article">
<div id="header">
<h1>Gfer: Reference manual</h1>
<span id="author">`` P. A. Giuliano Albo, L. Pitre, F. Sparasci</span><br />
<span id="revdate">Istituto Nazionale di Ricerca Metrologica (INRiM) and Conservatoire National des Arts et Métiers (Le-Cnam)</span>
<div id="toc">
<div id="toctitle">Table of Contents</div>
<noscript><p><b>JavaScript must be enabled in your browser to display the table of contents.</b></p></noscript>
</div>
</div>
<div id="content">
<div class="sect1">
<h2 id="_introduction">Introduction</h2>
<div class="sectionbody">
<div class="paragraph"><p>The code of this library is stored in the file Gfer.f08 in the form of
Fortran 2008 source. This file can be compiled using
<a href="https://gcc.gnu.org/wiki/GFortranBinaries">GFortran</a> free compiler
or other alternatives. For Linux operating systems, it is usually
simpler to install the compiler from the official distribution
repository. The code has been designed to not require further
dependency and can be compiled using the command line reported in the
source code.</p></div>
<div class="paragraph"><p>Alternatively, the library is distributed in the form of compiled
shared library Gfer.dll and Gfer.so for Windows™ 64 bit and Linux
64 bit operative systems. A 32 bit version is available only for
Windows™.</p></div>
<div class="paragraph"><p>The name of the functions, exposed by the library, are reported in the
<em>List of function</em> session. It has been chosen to use the "C" format
for the compiled code so that the functions can be called by almost
all the development environment and software for numerical analysis.</p></div>
<div class="paragraph"><p>All the quantities are represented using <em>double</em> precision floating
point with a length of 8 bytes while integer indexes, when necessary,
are implemented using signed <em>integer</em> of 4 bytes. Arrays are
<em>pointers</em> to <em>double</em>.</p></div>
<div class="paragraph"><p><strong>All values are passed by reference</strong> in the form of <em>pointers</em>.</p></div>
<div class="paragraph"><p>For a sake of testing, a LabView™ 16 bit library has been
implemented to link Gfer functions in a form of a set of virtual
instruments. LabView™ Library is contained in the Gfer.llb file. A
Maple™ 16 bit version is already available if requested.</p></div>
<div class="paragraph"><p>It has been chosen to leave to the end user the freedom to provide the
properties needed to calculate the corrections.</p></div>
<div class="paragraph"><p>Sometimes corrections depend on mechanical dimensions of the
resonator, ducts, tubes and so on. These quantities are expressed in
millimeters while thermodynamic quantities are expressed in SI
fundamental units. Internally, the conversion from millimeters to
meters is applied when necessary.</p></div>
<div class="paragraph"><p>An implementation for the Bessel function, optimized for the
corrections applied to radial modes, has been included in the library
and are called internally when necessary. In this way, Gfer library
doesn’t depend from external math libraries.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_installation">Installation</h2>
<div class="sectionbody">
<div class="paragraph"><p>Please check the terms of use described in the license before
installing this software.</p></div>
<div class="paragraph"><p>Gfer library is not provided with an automatic installer because its
installation depends on the choices of the end-user. For LabView™,
the shared library, being Gfer.dll or Gfer.so, should be saved in the
same folder of the Gfer.llb file. Other development platforms and
software for numerical analysis might request to save libraries in
determined places. Please refer to the manual of the software
connecting to the Gfer libraries to get the necessary information.</p></div>
<div class="paragraph"><p>For 32 bits platforms (only Windows™), it is possible use
Gfer_32bit.dll. To this end, rename the file to Gfer.dll before using.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_acoustic_modes">Acoustic modes</h2>
<div class="sectionbody">
<div class="paragraph"><p>The following functions will consider only radial acoustic modes with
<em>n</em><11. Modes are labeled as (0,<em>n</em>) with <em>n</em>=2,3,… 10.</p></div>
<div class="sect3">
<h4 id="_ideal_frequencies">Ideal frequencies</h4>
<div class="paragraph"><p><code>f_id(n, w, req)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>w</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m/s</p></td>
<td align="left" valign="top"><p class="table">Speed of sound</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the mode</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, M. R. Moldover and L. Pitre, Designing quasi-spherical
resonators for acoustic thermometry, Metrologia 41 (2004) 295–304;
<a href="https://doi.org/10.1088/0026-1394/41/4/011">https://doi.org/10.1088/0026-1394/41/4/011</a>
</p>
</li>
</ol></div>
</div>
<div class="sect2">
<h3 id="_boundary_layer">Boundary layer</h3>
<div class="sect3">
<h4 id="_thermal_penetration_length_d_t">Thermal penetration length: d_t</h4>
<div class="paragraph"><p><code>d_t(K, rho, cp, f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>rho</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kg/m<sup>3</sup></p></td>
<td align="left" valign="top"><p class="table">Density of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>cp</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">J/(kg K)</p></td>
<td align="left" valign="top"><p class="table">Constant pressure specific heat capacity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
M. R. Moldover, J. B. Mehl and M. Greenspan, Gas-filled spherical
resonators: Theory and experiment, J. Acoust. Soc. Am. 79 (2},
February 1986; <a href="https://doi.org/10.1121/1.393566">https://doi.org/10.1121/1.393566</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_viscous_penetration_length_d_v">Viscous penetration length: d_v</h4>
<div class="paragraph"><p><code>d_v(Nu, rho, f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>Nu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Pa s</p></td>
<td align="left" valign="top"><p class="table">Viscosity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>rho</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kg/m<sup>3</sup></p></td>
<td align="left" valign="top"><p class="table">Density of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Viscous penetration length</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
M. R. Moldover, J. B. Mehl and M. Greenspan, Gas-filled spherical
resonators: Theory and experiment, J. Acoust. Soc. Am. 79 (2},
February 1986; <a href="https://doi.org/10.1121/1.393566">https://doi.org/10.1121/1.393566</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_thermal_accommodation_length_i_t">Thermal accommodation length: I_t</h4>
<div class="paragraph"><p><code>I_t(T, p, cv, K, m, h)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>T</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">K</p></td>
<td align="left" valign="top"><p class="table">Temperature of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>p</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kPa</p></td>
<td align="left" valign="top"><p class="table">Pressure of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>cv</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">J/(kg K)</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>m</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kg/mol</p></td>
<td align="left" valign="top"><p class="table">Molar mass of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>h</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Accommodation coefficient</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal accommodation length</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
G. Benedetto, R. M. Gavioso, R. Spagnolo, P. Marcarino and A.
Merlone, Acoustic measurements of the thermodynamic temperature
between the triple point of mercury and 380 K, Metrologia 41, 2004,
74–98; <a href="https://doi.org/10.1088/0026-1394/47/4/005">https://doi.org/10.1088/0026-1394/47/4/005</a>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect2">
<h3 id="_frequency_shift">Frequency shift</h3>
<div class="sect3">
<h4 id="_thermal_boundary_layer_frequency_shift">Thermal boundary layer frequency shift</h4>
<div class="paragraph"><p><code>df_th(g, dt, dt_Cu, I_t, K, K_Cu, Req, f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>I_t</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal accommodation length</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Shift of the resonant frequency</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
L. Pitre, M. R. Moldover and W. L. Tew, Acoustic thermometry: new results from
273 K to 77 K and progress towards 4 K, Metrologia 43 (2006)
142–162; <a href="https://doi.org/10.1088/0026-1394/43/1/020">https://doi.org/10.1088/0026-1394/43/1/020</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_separated_contributions">Separated contributions</h4>
<div class="paragraph"><p>Next three functions reproduce the three terms used to calculate
df_th. Sometimes researchers prefer to keep the contributions
separated to investigate them.</p></div>
<div class="sect4">
<h5 id="_frequency_shift_by_em_d_t_em">Frequency shift by <em>d_t</em></h5>
<div class="paragraph"><p><code>df_dth(g, Req, d_t)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>d_t</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table">Shift of the resonant frequency</p></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="sect4">
<h5 id="_frequency_shift_by_em_i_t_em">Frequency shift by <em>I_t</em></h5>
<div class="paragraph"><p><code>df_Ith(g, Req, I_t)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>I_t</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal accommodation length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table">Shift of the resonant frequency</p></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="sect4">
<h5 id="_frequency_shift_by_shell_gas_coupling">Frequency shift by shell-gas coupling</h5>
<div class="paragraph"><p><code>df_CuTh(g, Req, dt_Cu, K, K_Cu)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table">Shift of the resonant frequency</p></td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div class="sect3">
<h4 id="_ducts">Ducts</h4>
<div class="paragraph"><p><code>df_ducts(r, L, N, rho, cp,w, g, K, Nu, Req, f, n, df_f, dg_g)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>r</em></p></td>
<td align="left" valign="top"><p class="table">Array[Real(8)]</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the tube sections starting from the resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>L</em></p></td>
<td align="left" valign="top"><p class="table">Array(Real(8)]</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Length of the tube sections starting from the resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>N</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Number of sections loaded in the arrays</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>rho</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kg/m<sup>3</sup></p></td>
<td align="left" valign="top"><p class="table">Density of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>cp</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">J/(kg K)</p></td>
<td align="left" valign="top"><p class="table">Constant pressure specific heat capacity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>w</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m/s</p></td>
<td align="left" valign="top"><p class="table">speed of sound of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Nu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Ps s</p></td>
<td align="left" valign="top"><p class="table">Viscosity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">frequency of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>df_f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> the relative frequency shift caused by ducts</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dg_f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dg/f</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> the relative half-width increasing of the acoustic mode caused by ducts</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, M. R. Moldover and L. Pitre, Designing quasi-spherical
resonators for acoustic thermometry, Metrologia 41 (2004) 295–304;
<a href="https://doi.org/10.1088/0026-1394/41/4/011">https://doi.org/10.1088/0026-1394/41/4/011</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_microphones">Microphones</h4>
<div class="paragraph"><p><code>df_mic(r, Req, rho w)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>r</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">radius of the microphone</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>rho</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kg/m<sup>3</sup></p></td>
<td align="left" valign="top"><p class="table">Density of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>w</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m/s</p></td>
<td align="left" valign="top"><p class="table">Speed of sound in the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table">Frequency shift of the acoustic mode</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, M. R. Moldover and L. Pitre, Designing quasi-spherical
resonators for acoustic thermometry, Metrologia 41 (2004) 295–304;
<a href="https://doi.org/10.1088/0026-1394/41/4/011">https://doi.org/10.1088/0026-1394/41/4/011</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_radial_modes_eigenvalues_correction_for_triaxial_ellipsoids">Radial modes eigenvalues correction for triaxial-Ellipsoids</h4>
<div class="paragraph"><p><code>O2dZ2AC(Eps, n)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>Eps</em></p></td>
<td align="left" valign="top"><p class="table">Array[Real(8)]</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Values of <em>epsilon_1</em> and <em>epsilon_2</em> encapsulated in an array</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dz<sup>2</sup>/z<sup>2</sup></p></td>
<td align="left" valign="top"><p class="table">Shift of the square of the eigenvalue <em>dz</em><sup>2</sup>/<em>z</em><sup>2</sup></p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, Acoustic Eigenvalues of a Quasispherical
Resonator: Second Order Shape Perturbation
Theory for Arbitrary Modes, J. Res. Natl. Inst. Stand. Technol. 112,
163-173 (2007); <a href="https://doi.org/10.6028/jres.112.013">https://doi.org/10.6028/jres.112.013</a>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect2">
<h3 id="_half_width_of_the_acoustic_modes">Half-width of the acoustic modes</h3>
<div class="sect3">
<h4 id="_half_width_of_radial_acoustic_modes">Half-width of radial acoustic modes</h4>
<div class="paragraph"><p><code>g_t(T, p,g, dt, dt_Cu, K, K_Cu, Req, f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>T</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">K</p></td>
<td align="left" valign="top"><p class="table">Temperature of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>p</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kPa</p></td>
<td align="left" valign="top"><p class="table">Pressure of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Increment of the half-with for thermal boundary layer</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
K. A. Gillis, I. I. Shinder and M. R. Moldover, Phys. Rev. E
70 021201 (2004); <a href="https://doi.org/10.1103/PhysRevE.70.021201">https://doi.org/10.1103/PhysRevE.70.021201</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_separated_contributions_2">Separated contributions</h4>
<div class="paragraph"><p>Next three functions calculate the separated contributions included in g_t.</p></div>
<div class="sect4">
<h5 id="_half_width_increase_by_thermal_boundary_layer_first_order">Half-width increase by thermal boundary layer (first order)</h5>
<div class="paragraph"><p><code>g_th(g, Req, d_th)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dg/f</p></td>
<td align="left" valign="top"><p class="table">Half-with for thermal boundary layer</p></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="sect4">
<h5 id="_half_width_increase_by_thermal_boundary_layer_second_order">Half-width increase by thermal boundary layer (second order)</h5>
<div class="paragraph"><p><code>g_thO2(g, Req, d_th)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dg/f</p></td>
<td align="left" valign="top"><p class="table">Half-with for thermal boundary layer</p></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="sect4">
<h5 id="_half_width_increase_by_shell_gas_coupling">Half-width increase by shell-gas coupling</h5>
<div class="paragraph"><p><code>g_Cu(g, Req, dt_Cu, K, K_Cu)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>K_Cu</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dg/f</p></td>
<td align="left" valign="top"><p class="table">Half-with for thermal boundary layer</p></td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div class="sect3">
<h4 id="_half_width_by_bulk_attenuation">Half-width by bulk attenuation</h4>
<div class="paragraph"><p><code>g_b(T, p, g, dv, dt, f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>T</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">K</p></td>
<td align="left" valign="top"><p class="table">Temperature of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>p</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">kPa</p></td>
<td align="left" valign="top"><p class="table">Pressure of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>g</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">cp/cv</p></td>
<td align="left" valign="top"><p class="table">Specific heat capacities ratio</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dv</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Viscous penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dt</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Thermal penetration length of the gas</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the acoustic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Increment of the half-with</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
M. R. Moldover, J. B. Mehl and M. Greenspan, Gas-filled spherical
resonators: Theory and experiment, J. Acoust. Soc. Am. 79 (2},
February 1986; <a href="https://doi.org/10.1121/1.393566">https://doi.org/10.1121/1.393566</a>
</p>
</li>
</ol></div>
</div>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_electromagnetic_modes">Electromagnetic modes</h2>
<div class="sectionbody">
<div class="sect2">
<h3 id="_skin_effect">Skin effect</h3>
<div class="sect3">
<h4 id="_depth_penetration_of_electromagnetic_waves">Depth penetration of electromagnetic waves</h4>
<div class="paragraph"><p><code>d_sk(mu_r, s, f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>mu_r</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Relative magnetic permeability of copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>s</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">S / m</p></td>
<td align="left" valign="top"><p class="table">Electric conductivity of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the electromagnetic mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Penetration length in the copper</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
E. F. May, L. Pitre, J.B. Mehl, M. R. Moldover and J. W. Schmidt,
Quasi-spherical cavity resonators for metrology based on the
relative dielectric permittivity of
gases, Rev. Sci. Instrum., Vol. 75, No. 10, October 2004;
<a href="https://doi.org/10.1063/1.1791831">https://doi.org/10.1063/1.1791831</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_te_frequency_and_half_width_perturbation">TE frequency and half-width perturbation</h4>
<div class="paragraph"><p><code>df_skTE(d_sk, R_Eq, df_f dg_f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>d_sk</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Penetration length in the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>df_f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> shift of the TE mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dg_f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dg/f</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> half-width of the TE mode</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
E. F. May, L. Pitre, J.B. Mehl, M. R. Moldover and J. W. Schmidt,
Quasi-spherical cavity resonators for metrology based on the
relative dielectric permittivity of
gases, Rev. Sci. Instrum., Vol. 75, No. 10, October 2004;
<a href="https://doi.org/10.1063/1.1791831">https://doi.org/10.1063/1.1791831</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_tm1n_frequency_and_half_width_perturbation">TM1n frequency and half-width perturbation</h4>
<div class="paragraph"><p><code>df_skTM1n(d_sk, R_Eq, n, df_f dg_f)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>d_sk</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">m</p></td>
<td align="left" valign="top"><p class="table">Penetration length in the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>df_f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> relative shift of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>dg_f</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dg/f</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> relative increase of the half-width of the TM1n mode</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
E. F. May, L. Pitre, J.B. Mehl, M. R. Moldover and J. W. Schmidt,
Quasi-spherical cavity resonators for metrology based on the
relative dielectric permittivity of
gases, Rev. Sci. Instrum., Vol. 75, No. 10, October 2004;
<a href="https://doi.org/10.1063/1.1791831">https://doi.org/10.1063/1.1791831</a>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect2">
<h3 id="_ducts_2">Ducts</h3>
<div class="sect3">
<h4 id="_te_modes_frequency_perturbation">TE modes frequency perturbation</h4>
<div class="paragraph"><p><code>df_dcTE(r, R_Eq)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>r</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the duct</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table">Shift of the TE modes due to a duct</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
R. J. Underwood, J. B. Mehl, L. Pitre, G. Edwards, G. Sutton
and M. de Podesta, Waveguide effects on quasispherical microwave
cavity resonators, Meas. Sci. Technol. 21 (2010) 075103;
<a href="https://doi.org/10.1088/0957-0233/21/7/075103">https://doi.org/10.1088/0957-0233/21/7/075103</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_tm1n_modes_frequency_perturbation">TM1n modes frequency perturbation</h4>
<div class="paragraph"><p><code>df_dcTM1n(r, R_Eq, n)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>r</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the duct</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>Req</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">mm</p></td>
<td align="left" valign="top"><p class="table">Radius of the equivalent sphere</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">df/f</p></td>
<td align="left" valign="top"><p class="table">Shift of the TM1n modes due to a duct</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
R. J. Underwood, J. B. Mehl, L. Pitre, G. Edwards, G. Sutton
and M. de Podesta, Waveguide effects on quasispherical microwave
cavity resonators, Meas. Sci. Technol. 21 (2010) 075103;
<a href="https://doi.org/10.1088/0957-0233/21/7/075103">https://doi.org/10.1088/0957-0233/21/7/075103</a>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect2">
<h3 id="_epsilon_calculations">Epsilon calculations</h3>
<div class="sect3">
<h4 id="_using_tm1n_first_order_approximation_model">Using TM1n first order approximation model</h4>
<div class="paragraph"><p><code>Eps_TM1n(f1, f2, f3, n, ep1, ep2)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>f1</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the first component of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f2</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the second component of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f3</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the third component of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>ep1</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">1-Rx/Rz</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> value of <em>epsilon_1</em> of the ellipsoidal resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>ep2</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">1-Ry/Rz</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> value of <em>epsilon_2</em> of the ellipsoidal resonator</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, Second-order electromagnetic
eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009)
554–559; <a href="https://doi.org/10.1088/0026-1394/46/5/020">https://doi.org/10.1088/0026-1394/46/5/020</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_using_te1n_first_order_approximation_model">Using TE1n first order approximation model</h4>
<div class="paragraph"><p><code>Eps_TE1n(f1, f2, f3, n, ep1, ep2)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>f1</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the first component of the TE1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f2</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the second component of the TE1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>f3</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">Hz</p></td>
<td align="left" valign="top"><p class="table">Frequency of the third component of the TE1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>ep1</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">1-Rx/Rz</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> value of <em>epsilon_1</em> of the ellipsoidal resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>ep2</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">1-Ry/Rz</p></td>
<td align="left" valign="top"><p class="table"><em>Return</em> value of <em>epsilon_2</em> of the ellipsoidal resonator</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, Second-order electromagnetic
eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009)
554–559; <a href="https://doi.org/10.1088/0026-1394/46/5/020">https://doi.org/10.1088/0026-1394/46/5/020</a>
</p>
</li>
</ol></div>
</div>
</div>
<div class="sect2">
<h3 id="_second_order_shape_perturbations_applied_to_the_mean_value">Second order shape perturbations (applied to the mean value)</h3>
<div class="sect3">
<h4 id="_tm1n_perturbed_eigen_values">TM1n perturbed eigen values</h4>
<div class="paragraph"><p><code>dzm2_zTM1n( ep1, ep2, n)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>ep1</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Value of <em>epsilon_1</em> of the ellipsoidal resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>ep2</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Value of <em>epsilon_2</em> of the ellipsoidal resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the TM1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dz<sup>2</sup>/z<sup>2</sup></p></td>
<td align="left" valign="top"><p class="table">Shift of the square of the eigenvalue <em>dz</em><sup>2</sup>/z<sup>2</sup> for the mode TE1n</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, Second-order electromagnetic
eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009)
554–559; <a href="https://doi.org/10.1088/0026-1394/46/5/020">https://doi.org/10.1088/0026-1394/46/5/020</a>
</p>
</li>
</ol></div>
</div>
<div class="sect3">
<h4 id="_te1n_perturbed_eigen_values">TE1n perturbed eigen values</h4>
<div class="paragraph"><p><code>dzm2_zTE1n( ep1, ep2, n)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>ep1</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Value of <em>epsilon_1</em> of the ellipsoidal resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>ep2</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Value of <em>epsilon_2</em> of the ellipsoidal resonator</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>n</em></p></td>
<td align="left" valign="top"><p class="table">Integer(4)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Index of the TE1n mode</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">dz<sup>2</sup>/z<sup>2</sup></p></td>
<td align="left" valign="top"><p class="table">Shift of the square of the eigenvalue <em>dz</em><sup>2</sup>/z<sup>2</sup> for the mode TE1n</p></td>
</tr>
</tbody>
</table>
</div>
<div class="paragraph"><p><strong>Reference:</strong></p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
J. B. Mehl, Second-order electromagnetic
eigenfrequencies of a triaxial ellipsoid, Metrologia 46 (2009)
554–559; <a href="https://doi.org/10.1088/0026-1394/46/5/020">https://doi.org/10.1088/0026-1394/46/5/020</a>
</p>
</li>
</ol></div>
</div>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_copper_properties">Copper properties</h2>
<div class="sectionbody">
<div class="sect3">
<h4 id="_constant_pressure_specific_heat_capacity">Constant pressure specific heat capacity</h4>
<div class="paragraph"><p><code>cp_cu(T)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>T</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">K</p></td>
<td align="left" valign="top"><p class="table">Temperature of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">J/(kg K)</p></td>
<td align="left" valign="top"><p class="table">Constant pressure specific heat capacity of the copper</p></td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="sect3">
<h4 id="_thermal_conductivity">Thermal conductivity</h4>
<div class="paragraph"><p><code>k_cu(T, RRR)</code></p></div>
<div class="tableblock">
<table rules="all"
width="100%"
frame="border"
cellspacing="0" cellpadding="4">
<col width="16%" />
<col width="16%" />
<col width="16%" />
<col width="50%" />
<tbody>
<tr>
<td align="left" valign="top"><p class="table"><em>T</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">K</p></td>
<td align="left" valign="top"><p class="table">Temperature of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table"><em>RRR</em></p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">none</p></td>
<td align="left" valign="top"><p class="table">Residual resistance ratio of the copper</p></td>
</tr>
<tr>
<td align="left" valign="top"><p class="table">Return</p></td>
<td align="left" valign="top"><p class="table">Real(8)</p></td>
<td align="left" valign="top"><p class="table">W/(m K)</p></td>
<td align="left" valign="top"><p class="table">Thermal conductivity of the copper</p></td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_acknowledgement">Acknowledgement</h2>
<div class="sectionbody">
<div class="paragraph"><p>This project (18SIB02-RMG1) has received funding from the EMPIR programme
co-financed by the Participating States and from the European Union’s
Horizon 2020 research and innovation programme.</p></div>
<div class="paragraph"><p><span class="image">
<img alt="EMPIR_logo.jpg" width="300"
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</span></p></div>
</div>
</div>
<div class="sect1">
<h2 id="_license">License</h2>
<div class="sectionbody">
<div class="openblock">
<div class="content">
<div class="paragraph"><p>Copyright (c) 2022 Le-Cnam/INRiM.</p></div>
<div class="paragraph"><p>MIT-like license with commercial restrictions.</p></div>
<div class="paragraph"><p>Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
“Software”), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute and sublicense copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:</p></div>
<div class="olist arabic"><ol class="arabic">
<li>
<p>
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
</p>
</li>
<li>
<p>
The Software, or part of it, is not included or used in commercial
applications without prior written agreement from Le-Cnam/INRiM.
</p>
</li>
<li>
<p>
Except as contained in this notice, the name of Le-Cnam/INRiM shall
not be used in advertising or otherwise to promote the sale, use or
other dealings in this Software without prior written authorization
from Le-Cnam/INRiM.
</p>
</li>
</ol></div>
<div class="paragraph"><p>THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.</p></div>
</div></div>
</div>
</div>
</div>
<div id="footnotes"><hr /></div>
<div id="footer">
<div id="footer-text">
Last updated
2023-03-28 08:08:01 CEST
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