First Versus Second Order Differential Equations for Waves

     A wave on a string is characterized by a second order wave equation which follows from Newton’s second law. The equation for a photon is also a second order wave equation which follows from Maxwell’s classical electromagnetic equations. Both Newton’s second law and Maxwell’s equations are linked to force, but the wave moves at a constant velocity and is described by v=f wavelength or  v=E/p. This is a linear equation and constant velocity suggests no force in Newtonian mechanics. Nevertheless some kind of action/reaction or internal force seems to be associated with a wave. 

   In this note we suggest that either a free particle with rest mass or a classical wave (light etc) move at a constant velocity and so one should think in terms of a linear equation. Secondly, we suggest that a particle or classical wave represents force i.e. the particle is not simply the effect of a cause called an external force. Newton already stated this by stating that for every action there is an equal and opposite reaction. Thus we argue that a particle (or a wave like a photon etc) is linked directly to force and so more than constant velocity motion is described. As a result, we suggest that linear equations are needed to describe E (energy) and p (momentum), but these should convey the idea of force. Furthermore, these ideas should be linked to special relativity because p and E depend on the constant velocity frame in which they are observed.