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]$$\left.+2c(c_g-2c)\left(2+4\sum_n (-1)^{n+1}\frac{\alpha+1}{\alpha+1+n} e^{\frac{2x}{L}(n+1)} - Tanh{\frac{x}{L}}\right)\right\}$$

 

[PJC01]

Hello Nick,

Thank you for sharing your problem and providing some background information. It seems like you are trying to solve a forced wave equation with a specific type of forcing and causal boundary conditions. It is certainly a challenging problem, but there are a few approaches you can take to solve it.

One approach is to use a Green's function method, as you suggested. This involves using the solution for the case of impulse forcing as a building block to construct the solution for your specific forcing function. This can be a useful approach, but it may be complicated and require some mathematical manipulation to get the desired solution.

Another approach is to use spectral methods, as you mentioned. This involves using the Fourier transform to simplify the equation and then using inverse Fourier transform to obtain the solution. It is important to carefully consider the boundary conditions and how they affect the solution in this method.

Both of these approaches have their advantages and disadvantages, so it may be helpful to try both and see which one gives you a more manageable solution. Additionally, it may be worth looking into numerical methods such as finite difference or finite element methods to solve the equation.

I hope this helps and good luck with your research!