6.2: Real Solutions to the Wave Equation
- Page ID
- 34549
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
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.