(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a traveling wave u(x,t) =f(x - at) where f is a given function of one variable.

(a) If it is a solution of the wave equation, show that the speed must be [tex]a = \pm c[/tex] (unless f is a linear function).

(b) If it is a solution of the diffusion equation, find f and show that the speed a is arbitrary.

2. Relevant equations

Wave: u(x,t) = f(x+ct) + g(x-ct) and [tex]u_{tt} = c^2u_{xx}[/tex]

Diffusion: [tex]u_{t} = ku_{xx}[/tex] and [tex]u(x,t) = \frac{1}{\sqrt{4kt\pi}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy[/tex]

3. The attempt at a solution

(a) The wave equation is [tex] u_{tt} = c^2u_{xx}[/tex] with the general solutions u(x,t) = f(x + ct) + g(x - ct), so for the solution to be u(x,t) = f(x-at), [tex]a = \pm c[/tex]. If f is linear, then [tex]u_{tt} =u_{xx}=0[/tex], so it doesn't matter what a equals.

(b) [tex]u(x,t) = f(x-at) = \frac{1}{\sqrt{4kt\pi}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy[/tex] I don't know where to go from here.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# PDE: a traveling wave solution to the diffusion equation

**Physics Forums | Science Articles, Homework Help, Discussion**