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Physics LibreTexts

6.3: Complex Solutions to the Wave Equation

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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): (2x21v22t2)Re[f(x,t)]=Re[(2x21v22t2)f(x,t)]=0. There exists a nice set of complex solutions to the wave equation, called complex travelling waves, which take the form f(x,t)=Aei(kxωt)where|ωk|=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: Re{Aei(kxωt)}=Re{|A|eiarg[A]ei(kxωt)}=|A|Re{eiarg[A]ei(kxωt)}=|A|cos(kxωt+arg[A]) Comparing this to Eq. (6.2.1), we see that |A| serves as the amplitude of the real wave, while arg(A) serves as the phase factor ϕ. 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: f(x,t)=|A|ei(kxωt+ϕ1)+|A|ei(kxωt+ϕ2)=|A|(ei(kx+ϕ1)+ei(kxϕ2))eiωt=|A|(ei[kx+(ϕ1ϕ2)/2]+ei[kx+(ϕ1ϕ2)/2])ei(ϕ1+ϕ2)/2eiωt=2|A|cos[kx+(ϕ1ϕ2)/2]ei[ωt(ϕ1+ϕ2)/2] Taking the real part yields Eq. (6.2.5), without the need for tedious manipulations of trigonometric formulas.


This page titled 6.3: Complex Solutions to the Wave Equation is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform.

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