Notes on standing and traveling waves, and phase, (group,) and “material” velocity standtravel_wiki

Present the continuous “wave” first as a series of discrete oscillators at different locations, each of whose phase has a definite relationship to the phase of the others. Show animations of standing waves and traveling waves. Point out that standing waves can be viewed as in-phase oscillators with amplitudes that depend on position and how the mathematical form for the standing wave, \(\psi \left( x,t \right)=A\cos kx\cos \omega t\), makes this clear. Point out the separation of the space and time variables in different functions. Ask about alternative forms, e.g. \[\psi \left( x,t \right)=A\sin kx\cos \left(\omega t+\varphi \right).\]

Present the traveling wave animation, where the phase of equal-amplitude oscillators varies linearly with position: \[\psi \left( x,t \right)=A\cos \left( kx-\omega t+\varphi \right).\] This wave travels, and the combination of \(kx-\omega t\) in the cosine function makes it travel.

Students at this level are usually not able to identify features of wave motion from the mathematical form, and they need lots of practice with the language of waves.

Phase velocity: focus on a particular phase in the traveling wave and derive the phase velocity of the wave. If \[d\left( phase \right)=d\left( kx\pm \omega t \right)=0,\] then \(kdx\pm \omega dt=0\), and the phase velocity, \(v_{ph}\equiv \frac{dx}{dt}=\mp \frac{\omega }{k}\). Then follows a discussion about the direction of travel, the signs of $\omega$ and k, and so on. The message should be that the mathematics tells you the answer. If asked to perform an experiment to measure the speed of a wave, students usually invoke the wavelength, frequency and the formula they have learned: $v=\lambda f$. If pressed, some will come up with a correct experiment: focus on some feature (usually a maximum) and time how long it takes to move a certain distance. Few can translate that into a mathematical model: “pick a point of constant phase ($d\varphi=0$), and calculate the velocity of that point of constant phase”. They just don't know how to connect the mathematical manipulations to an experiment, real or gedanken. They need training in the language.

Material velocity: This is word I invented for \(\frac{\partial \psi \left( x,t \right)}{\partial t}\), the rate of change of $\psi$ at a particular position. For waves in a rope, it gives the sped of the rope at a point, hence “material”, but of course, $\psi$ can be an abstract quantity. There is an important distinction between the two velocities. The material velocity is relevant to the calculation of energy density, while the phase velocity (in non-dispersive systems) is related to information flow. A brief mention of group velocity is helpful here (ask the question of how to define the velocity of a superposition of waves if each member of the superposition has a different phase velocity) as food for thought for later discussion.

The discussion of standing waves as linear combinations of traveling waves and vice versa is in the next lecture.