6.3: Complex Solutions to the Wave Equation

It is much easier to deal with the wave equation if we promote it into a complex PDE by letting \(f(x,t)\) take on complex values. However, \(x\) and \(t\) will remain real. We will also take the wave speed \(v\) to be real, for now.

From any complex solution to the wave equation, we can take the real part to get a solution to the real PDE, thanks to linearity (see Section 4.1): \[\left(\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right) \mathrm{Re}\left[f(x,t)\right] = \mathrm{Re} \left[ \left(\frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right) f(x,t)\right] = 0.\] There exists a nice set of complex solutions to the wave equation, called complex travelling waves, which take the form \[f(x,t) = A \, e^{i(kx - \omega t)} \quad\mathrm{where}\;\; \left|\frac{\omega}{k}\right| = v.\] It can be verified by direct substitution that this satisfies the PDE. The complex constant \(A\) is called the complex amplitude of the wave. Consider what happens if we take the real part of the above solution: \[\begin{align}\mathrm{Re}\Big\{A \, e^{i(kx - \omega t)}\Big\} &= \mathrm{Re}\Big\{ |A|\, e^{i\mathrm{arg}[A]} \; e^{i(kx - \omega t)}\Big\} \\ &= \big|A\big|\; \mathrm{Re}\Big\{ e^{i\mathrm{arg}[A]} \, e^{i(kx - \omega t)}\Big\} \\ &= \big|A\big|\; \cos\big(kx - \omega t + \mathrm{arg}[A]\big)\end{align}\] Comparing this to Eq. (6.2.1), we see that \(|A|\) serves as the amplitude of the real wave, while \(\mathrm{arg}(A)\) serves as the phase factor \(\phi\). Mathematically, the complex solution is more succinct than the real solution: a single complex parameter \(A\) combines the roles of two parameters in the real solution.

The complex representation also makes wave superpositions easier to handle. As an example, consider the superposition of two counter-propagating waves of equal amplitude and frequency, with arbitrary phases. Using complex traveling waves, we can calculate the superposition with a few lines of algebra: \[\begin{align}f(x,t) &= \displaystyle \big|A\big| \, e^{i(kx - \omega t + \phi_1)} + \big|A\big| \, e^{i(-kx - \omega t + \phi_2)} \\ &= \displaystyle \big|A\big|\, \left(e^{i(kx + \phi_1)} + e^{-i(kx - \phi_2)}\right)\, e^{-i\omega t} \\ &= \displaystyle \big|A\big|\, \left(e^{i[kx + (\phi_1-\phi_2)/2]} + e^{-i[kx + (\phi_1 - \phi_2)/2]}\right)\, e^{i(\phi_1 + \phi_2)/2} \,e^{-i\omega t} \\ &= \displaystyle 2\big|A\big|\, \cos\left[kx + (\phi_1-\phi_2)/2\right] \,e^{-i[\omega t -(\phi_1+\phi_2)/2]}\end{align}\] Taking the real part yields Eq. (6.2.5), without the need for tedious manipulations of trigonometric formulas.