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.

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