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): (∂2∂x2−1v2∂2∂t2)Re[f(x,t)]=Re[(∂2∂x2−1v2∂2∂t2)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(kx−ωt)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(kx−ωt)}=Re{|A|eiarg[A]ei(kx−ωt)}=|A|Re{eiarg[A]ei(kx−ωt)}=|A|cos(kx−ωt+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(kx−ωt+ϕ1)+|A|ei(−kx−ωt+ϕ2)=|A|(ei(kx+ϕ1)+e−i(kx−ϕ2))e−iωt=|A|(ei[kx+(ϕ1−ϕ2)/2]+e−i[kx+(ϕ1−ϕ2)/2])ei(ϕ1+ϕ2)/2e−iωt=2|A|cos[kx+(ϕ1−ϕ2)/2]e−i[ωt−(ϕ1+ϕ2)/2] Taking the real part yields Eq. (6.2.5), without the need for tedious manipulations of trigonometric formulas.