Solving Forced Wave Equation with Causal Boundary Conditions

[nickthequick]

Hi,

I want to solve the forced wave equation

$$u_{tt}-c^2u_{xx} = f''(x)g(t)$$

(primes denote derivatives wrt x). The forcing I am interested in is

$$f(x,t)= e^{-t/T} (\alpha_o+\alpha_1 Tanh(-\frac{(x-x_o)}{L})$$.

I also am imposing causality, i.e. $$u =0$$ for $$t<0$$.

In the case where $$g(t) = \delta (t)$$ I know that the solution is

$$u = \frac{1}{2c} \left[ f'(x-c|t|) - f'(x+ct) \right]$$

My question is, can I build up the solutions to my particular type of time dependence through a Green's function type method from the case of impulse forcing?

My other approach would be via spectral methods but the inverse Fourier transform is quite complicated.


Any help is appreciated,

Nick

 

[Bill_K]

Change the independent variables from x, t to v = x+ct and w=x-ct. In these coordinates the left hand side is u_vw, and all you have to do is integrate both sides wrt v and then wrt w.

 

[nickthequick]

Excellent exploitation of the symmetry of the governing equation!

Thanks for pointing this out Bill_K

 

[nickthequick]

Using this method of characteristics, I can integrate the forcing twice (once wrt w and once wrt to v) but I end up finding a divergent solution. Am I just taking the anti-derivatives of the forcing function twice or are these definite integrals that depend on the geometry of (v,w) space?

PS This is the equation I end up finding in (x,t) space

[tex]u(x,y)= 2\omega\Delta \mathcal{E} e^{\frac{-c_g^2t}{cL}}\left\{\sum_n (-1)^{n+1} \frac{e^{\frac{2x}{L}(n+1)}}{n+1 -\alpha}\left(c_g^2\frac{1-\alpha}{n+1-\alpha} + c_g(c_g-2c)\frac{\alpha}{n+\alpha}\right)\right.[\tex]$$\left.+2c(c_g-2c)\left(2+4\sum_n (-1)^{n+1}\frac{\alpha+1}{\alpha+1+n} e^{\frac{2x}{L}(n+1)} - Tanh{\frac{x}{L}}\right)\right\}$$[/tex]