6.2: Real Solutions to the Wave Equation

We first consider real solutions to the wave equation. One family of solutions are travelling waves of the form $$f(x,t) = f_0 \, \cos\!\big(kx - \omega t + \phi\big),\quad\mathrm{where}\;\, \left|\frac{\omega}{k}\right| = v. \label{realsol}$$ By direct substitution, we can verify that this satisfies the PDE. We call $f_0$ the amplitude of the wave, $\phi$ the phase, $\omega$ the (angular) frequency, and $k$ the wavenumber. By convention, $\omega$ is taken to be a positive real number. However, $k$ can be either positive or negative, and its sign determines the direction of propagation of the wave; the magnitude of the wavenumber is inversely related to the wavelength $\lambda$ by $\lambda = 2\pi/|k|$.

As $t$ increases, the wave moves to the right if $k$ is positive, whereas it moves to the left if $k$ is negative. Here’s one way to reason out why this is the case. Consider introducing a small change in time, $\delta t$, into the function $\cos(kx - \omega t + \phi)$. If, together with this time shift, we change $x$ by $\delta x = (\omega/k)\, \delta t$, then the change in the $kx$ term and the change in the $\omega t$ term cancel, leaving the value of the cosine unchanged:

Figure $\PageIndex{1}$

This implies that the wave shifts by $\delta x = (\omega/k)\, \delta t$ during the time interval $\delta t$. Hence, the wave velocity is $$\textrm{velocity} = \frac{\delta x}{\delta t} = \frac{(\omega/k)\,\delta t}{\delta t} = \frac{\omega}{k}.$$ As previously noted, $\omega$ is conventionally taken to be a positive real number. Hence, positive $k$ implies that the wave is right-moving (positive velocity), and negative $k$ implies the wave is left-moving (negative velocity). Moreover, the wave speed is the absolute value of the velocity, which is precisely equal to the constant $v$: $$\textrm{speed}\; = \, \left|\frac{\delta x}{\delta t}\right| = \frac{\omega}{\left|k\right|} = v.$$

Standing waves

Suppose we have two traveling wave solutions, with equal amplitude and frequency, moving in opposite directions: $$f(x,t) = f_0 \, \cos(kx - \omega t + \phi_1) + f_0 \cos(-kx - \omega t + \phi_2).$$ Here, we denote $k = \omega/c$. Such a superposition is also a solution to the wave equation, called a standing wave. It can be re-written in a variable-separated form (i.e., as the product of a function of $x$ and a function of $t$): $$f(x,t) = 2f_0 \, \cos\big[kx + (\phi_1-\phi_2)/2\big]\, \cos\big[\omega t - (\phi_1+\phi_2)/2\big]. \label{standingsol}$$ This can be proven using the trigonometric addition formulas, but the proof is tedious.