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I am looking for a solution to a wave equation

$\frac{\partial^2 u}{\partial \tau^2} = \frac{\partial^2 u}{\partial \xi^2}$

in which $t_c\tau = t$, $L\xi = x$,

and $t_c = L/v_c$ is the characteristic time,

$L$ is the sample thickness,

and $v_c$ is the characteristic wave speed,

with an IC of

$\left [\frac{\partial u}{\partial \tau} \right]_{x,t=0} = \theta \left (x, t=0 \right)$

and a BC of

$\left [\frac{\partial u}{\partial \xi} \right]_{x=0,t} = \phi \left (x=0, t \right)$

I have tried the D' Alembert solution, but I get a function $u\left(\xi, \tau \right)$ that is a function of the integral of phi which I don't know since it is not analytic, and it also introduces two new unknowns, $f\left (\tau_0 \right)$ and $g\left (\tau_0 \right)$ and I'm actually trying to find $\frac{\partial u}{\partial \tau}$ and $\frac{\partial u}{\partial \xi}$ not u.

I haven't tried separation of variables, Sturm-Liouville or Fourier transform yet.

This system is similar to Cauchy-Riemann equations.

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1 Answer 1

up vote 2 down vote accepted

I'm going to ignore all that rescaling and just treat the wave equation in $\tau$ and $\xi$.

The general solution of that wave equation is

$$u(\tau,\xi)=f_+(\tau+\xi)+f_-(\tau-\xi)$$

with arbitrary functions $f_\pm$. Then

$$\frac{\partial u}{\partial \tau}=g_+(\tau+\xi)+g_-(\tau-\xi)$$

and

$$\frac{\partial u}{\partial \xi}=g_+(\tau+\xi)-g_-(\tau-\xi)\;,$$

with $g_\pm=f'_\pm$, which are arbitrary since the $f_\pm$ are arbitrary. Substituting your initial and boundary conditions (please don't use abbreviations like that; it's so much less effort for you to write it out than it is for all your readers to have to think about what it means) then yields (I'm not distinguishing between $x$ and $\xi$ or $t$ and $\tau$ here; you're going to have to add some scale factors)

$$g_+(\xi)+g_-(-\xi)=\theta(\xi)$$

and

$$g_+(\tau)-g_-(\tau)=\phi(\tau)$$

(where I've dropped the dummy zero argument on $\theta$ and $\phi$). Then solving for $g_\pm$ yields

$$g_+ = \frac{\theta+\phi}2$$

and

$$g_- = \frac{\theta-\phi}2\;,$$

so the derivatives you're interested in are

$$\frac{\partial u}{\partial \tau}=\frac{\theta(\tau+\xi)+\phi(\tau+\xi)+\theta(\tau-\xi)-\phi(\tau-\xi)}2$$

and

$$\frac{\partial u}{\partial \xi}=\frac{\theta(\tau+\xi)+\phi(\tau+\xi)-\theta(\tau-\xi)+\phi(\tau-\xi)}2\;.$$

Note that I used your conditions for all values of the arguments. That may not be what you had in mind, since you called them initial and boundary conditions. If you're only interested in one quadrant of the $(\tau,\xi)$ plane and intended the conditions only to apply to the boundaries of that quadrant, then the system is underspecified; you can have arbitrary waves coming in from infinity and being reflected at the boundary, and that freedom corresponds to the freedom of choosing the values at the remaining boundaries.

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Thanks for your response. You were correct that I am only looking for u in the 1st quadrant. But maybe I should back up. I am actually looking for a solution to a system of equations of only first derivatives. But there are two fields: theta(xi,tau) and phi(xi,tau). So the system of equations actually looks like this: dtheta/dtau = thetaphi and dphi/dxi = thetaphi. I just thought that by substitution of variables I could convert it to the wave equation which might be easier to solve. –  Mark Mikofski Jan 19 '12 at 19:21

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