6.2: Real Solutions to the Wave Equation
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We first consider real solutions to the wave equation. One family of solutions are travelling waves of the form f(x,t)=f0cos(kx−ωt+ϕ),where|ωk|=v. By direct substitution, we can verify that this satisfies the PDE. We call f0 the amplitude of the wave, ϕ the phase, ω the (angular) frequency, and k the wavenumber. By convention, ω 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 λ by λ=2π/|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, δt, into the function cos(kx−ωt+ϕ). If, together with this time shift, we change x by δx=(ω/k)δt, then the change in the kx term and the change in the ωt term cancel, leaving the value of the cosine unchanged:

This implies that the wave shifts by δx=(ω/k)δt during the time interval δt. Hence, the wave velocity is velocity=δxδt=(ω/k)δtδt=ωk. As previously noted, ω 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: speed=|δxδt|=ω|k|=v.
Standing waves
Suppose we have two traveling wave solutions, with equal amplitude and frequency, moving in opposite directions: f(x,t)=f0cos(kx−ωt+ϕ1)+f0cos(−kx−ωt+ϕ2). Here, we denote k=ω/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)=2f0cos[kx+(ϕ1−ϕ2)/2]cos[ωt−(ϕ1+ϕ2)/2]. This can be proven using the trigonometric addition formulas, but the proof is tedious.