Traveling Waves in Infinite Continuous Medium

Consider solutions of the wave equation, (6.1), in an infinite medium. Such a medium does not possess any spatial boundaries, and so is not subject to boundary constraints. Hence, there is no particular reason why a wave of definite wavelength should have stationary nodes or anti-nodes. In other words, Equation (6.2) may not be the only permissible type of wave solution in an infinite medium. What other kind of solution could we have? Suppose that

$\displaystyle \psi(x,t) = A\,\cos(\omega\,t-k\,x-\phi),$ (6.7)

where $A>0$, $k>0$, $\omega>0$, and $\phi$ are constants. This solution is interpreted as a wave of amplitude $A$, wavenumber $k$, wavelength $\lambda=2\pi/k$, angular frequency $\omega $, frequency (in hertz) $f=\omega/2\pi$, period $T=1/f$, and phase angle $\phi$. It can be seen that $\psi(x+\lambda,t)=\psi(x,t)$, and $\psi(x,t+T)=\psi(x,t)$, for all $x$ and $t$. In other words, the wave is periodic in space with period $\lambda $, and periodic in time with period $T$. A wave maximum corresponds to a point at which $\cos(\omega\,t-k\,x-\phi)=1$. It follows, from the well-known properties of the cosine function, that the various wave maxima are located at

$\displaystyle \omega\,t-k\,x-\phi = n\,2\pi,$ (6.8)

where $n$ is an integer. Differentiating the previous expression with respect to $t$, and rearranging, the equation of motion of a particular maximum becomes

$\displaystyle \frac{dx}{dt}=\frac{\omega}{k}.$ (6.9)

We conclude that the wave maximum in question propagates along the $x$-axis at the velocity

$\displaystyle v_p = \frac{\omega}{k}.$ (6.10)

It can be shown that the other wave maxima (as well as the wave minima and the wave zeros) also propagate along the $x$-axis at the same velocity. In fact, the whole wave pattern propagates in the positive $x$-direction without changing shape. The characteristic propagation velocity $v_p$ is known as the phase velocity, because it is the velocity of constant phase points on the wave disturbance (i.e., points that satisfy $\omega\,t-k\,x-\phi = {\rm constant}$). For obvious reasons, the type of wave solution specified in Equation (6.7) is called a traveling wave.

Substitution of Equation (6.7) into the wave equation, (6.1), yields the familiar dispersion relation

$\displaystyle \omega = k\,v.$ (6.11)

We conclude that the traveling wave solution (6.7) satisfies the wave equation provided

$\displaystyle v_p = \frac{\omega}{k} = v;$ (6.12)

that is, provided the phase velocity of the wave takes the fixed value $v$. It follows that the constant $v^{\,2}$, that appears in the wave equation, (6.1), can be interpreted as the square of the velocity with which traveling waves propagate through the medium in question. Hence, from the discussions in Sections 4.3, 5.2, and 5.3, transverse waves propagate along strings of tension $T$ and mass per unit length $\rho$ at the phase velocity $\sqrt{T/\rho}$, longitudinal waves propagate along thin elastic rods of Young's modulus $Y$ and mass density $\rho$ at the phase velocity $\sqrt{Y/\rho}$, and sound waves propagate through ideal gases of pressure $p$, mass density $\rho$, and ratio of specific heats $\gamma$, at the phase velocity $\sqrt{\gamma\,p/\rho}$.

Table: 6.1 Calculated and measured longitudinal wave speeds in thin rods made up of common metals. Sources: Haynes and Lide 2011c, Wikipedia contributors 2018.
Metal $Y \,({\rm N\,m}^{-2})$ $\rho \,({\rm kg\,m}^{-3})$ $\sqrt{Y/\rho}\, ({\rm m\,s}^{-1})$ $v\,({\rm m\,s}^{-1})$
Aluminum $7.0\times 10^{10}$ $2.7\times 10^3$ $5100$ $5000$
Nickel $2.0\times 10^{11}$ $8.9\times 10^3$ $4700$ $4900$
Zinc $1.1\times 10^{11}$ $7.1\times 10^3$ $3900$ $3900$
Copper $1.2\times 10^{11}$ $8.9\times 10^3$ $3600$ $3800$
Silver $8.3\times 10^{10}$ $1.1\times 10^4$ $2800$ $2700$
Tin $5.0\times 10^{10}$ $7.4\times 10^3$ $2600$ $2700$
Lead $1.6\times 10^{10}$ $1.1\times 10^4$ $1100$ $1100$

Table 6.1 displays calculated and measured longitudinal wave speeds in thin rods made up of various common metals. It can be seen that the agreement between the two is excellent.

An ideal gas of mass $m$ and molecular weight $M$ satisfies the ideal gas equation of state,

$\displaystyle p\,V = \frac{m}{M}\,R\,T,$ (6.13)

where $p$ is the pressure, $V$ the volume, $R=8.3145\,{\rm J\,mol}^{-1}\,{\rm K}^{-1}$ the molar ideal gas constant, and $T$ the absolute temperature (Reif 2008). Because the ratio $m/V$ is equal to the density, $\rho$, the expression for the sound speed, $v=\sqrt{\gamma\,p/\rho}$, yields

$\displaystyle v= \left(\frac{\gamma\,R\,T}{M}\right)^{1/2}.$ (6.14)

We conclude that the speed of sound in an ideal gas is independent of the pressure or the density, proportional to the square root of the absolute temperature, and inversely proportional to the square root of the molecular mass. Incidentally, the root-mean-square molecular speed in an ideal gas in thermal equilibrium is $v_{\rm rms}= \sqrt{3\,R\,T/M}$ (Reif 2008). Hence, the speed of sound in an ideal gas is of order, but slightly less than (because the maximum possible value of $\gamma$ is $5/3$), the mean molecular speed.

A comparison between Equations (6.6) and (6.11) reveals that standing waves and traveling waves in a given medium satisfy the same dispersion relation. However, because traveling waves in infinite media are not subject to boundary constraints, there is no restriction on the possible wavenumbers, or wavelengths, of such waves. Hence, any traveling wave solution whose wavenumber, $k$, and angular frequency, $\omega $, are related according to the dispersion relation (6.11) is a valid solution of the wave equation. In other words, any traveling wave solution whose wavelength, $\lambda=2\pi/k$, and frequency, $f=\omega/2\pi$, are related according to

$\displaystyle v = f\,\lambda$ (6.15)

is a valid solution. We conclude that relatively high-frequency traveling waves propagating through a given medium possess relatively short wavelengths, and vice versa.

Consider the alternative wave solution

$\displaystyle \psi(x,t) = A\,\cos(\omega\,t+k\,x-\phi),$ (6.16)

where $A>0$, $k>0$, $\omega>0$, and $\phi$ are constants. As before, this solution is interpreted as a wave of amplitude $A$, wavenumber $k$, angular frequency $\omega $, and phase angle $\phi$. However, the wave maxima are now located at

$\displaystyle \omega\,t+k\,x-\phi = n\,2\pi,$ (6.17)

where $n$ is an integer, and have equations of motion of the form

$\displaystyle \frac{dx}{dt} =-\frac{ \omega}{k}.$ (6.18)

Equation (6.16), thus, represents a traveling wave that propagates in the minus $x$-direction at the phase velocity $v_p = \omega/k$. Moreover, substitution of Equation (6.16) into the wave equation, (6.1), again yields the dispersion relation (6.11), which implies that $v_p=v$. It follows that traveling wave solutions to the wave equation, (6.1), can propagate in either the positive or the negative $x$-direction, as long as they always travel at the fixed speed $v$.