Question

Derive dynamic equation of uniformly progressive wave.

Answer 1

Wave phenomena are ubiquitous in nature. Examples include water waves, sound waves, electro-

magnetic waves (radio waves, light, X-rays, gamma rays etc.), the waves that in quantum mechanics

are found to be an alternative (and often better) description of particles, etc. Some features are

common for most waves, e.g. that they in cases of small amplitude can be well approximated by

a simple trigonometric wave function (Section 4.1) Other features differ. In some cases, all waves

travel with the same speed (e.g. sound waves or light in vacuum) whereas in other cases, the speed

depends strongly on the wave length (e.g. water waves or quantum mechanical particle waves). In

most cases, one can start from basic physical principles and from these derive partial differential

equations (PDEs) that govern the waves. In Section 4.2 we will do this for transverse waves on a

tight string, and for Maxwell’s equations describing electromagnetic waves. In both of these cases,

we obtain linear PDEs that can quite easily be solved numerically. In other cases, such as water

waves, discussed in Section 4.3, the full governing equations are too complex to give here, and we

need to restrict ourselves to a number of general observations. In still other cases, such as the

Schrödinger equation for quantum wave functions, a quite simple set of PDEs are well known and

extremely accurate (often said to describe all of chemistry!) but these are prohibitively difficult to

solve in all but the simplest special cases. We note in Section 4.4 that some important nonlinear

wave equations can be formulated as systems of first order PDEs. Not only are these systems

usually very well suited for numerical solution, they also allow a quite simple analysis regarding

various features, such as types of waves they support and their speeds. In some cases, discussed

in Section 4.5, we find some closed-form analytic solutions. We arrive in Section 4.6 to Hamilton’s

equations. These are fundamental in many applications, such as mechanical and dynamical sys-

tems, and the study of chaotic motions. I