Group and Phase Velocity in the Example of Water Waves

Abstract

As a preparation for quantum mechanics we shall in this chapter consider the propagation of waves. As the basic wave form we take a plane, scalar wave of infinite extent and sinusoidal profile. If we choose the x-axis in the direction of propagation, the wave function becomes

$$ \psi \left( {x,t} \right) = {\psi _0}\sin \left( {kx - \omega t} \right)\,\,,\,\,\,{\text{or}}\,\,\psi \left( {x,t} \right) = {\psi _0}\cos \left( {kx,\omega t} \right). $$

(12.1)

The modulus of the wave number k and the angular frequency ω are determined by the wavelength λ and the wave period τ:

$$ \left| k \right| = \frac{{2\pi }}{\lambda },\,\,\,\,\omega = \frac{{2\pi }}{\tau }. $$

(12.2)

If one follows the maximum elevation of the wave with time, this moves with the velocity

$$ {v_{\text{p}}} = \frac{\omega }{k}. $$

(12.3)

The velocity v _p is called the phase velocity. In nature there are no infinite plane waves. Any wave motion occurring in nature has a beginning and an end, both with regard to space and time. One speaks in this case of a wave train or a wave packet. Following Fourier we can represent a wave packet as a superposition of plane waves. If we again consider only propagation in the x-direction and take a wave packet that is an even function of x at time t = 0, then we can write

$$ \psi \left( {x,t} \right) = \int\limits_{ - \infty }^\infty {a\left( k \right)\cos \left( {kx - \omega t} \right)dk.} $$

(12.4)

The function a(k) determines the form of the wave packet and also decides in which direction it moves. In many fields of mechanics, acoustics, optics and quantum mechanics the principle of superposition of waves is valid: the linear superposition (12.4), as well as its individual components, satisfy the relevant wave equation. The superposition principle, however, is not valid for all wave motions. For example, it is not valid for water waves which have breaking crests.