Using d'Alemberts in a limited interval

Question

I am a bit confused using d'Alembert's formula for solving the one-dimensional wave equation, and more precisely when it involves the Dirac-function.

Some information before my question:

Let's say we have the most basic wave equation in a limited interval:

$u''_{tt}-\Delta u = 0, 0<x<2$, since 1D $\rightarrow \Delta u = u''_{xx}$.

With beginning value (and homogeneous Dirichlet conditions on the boundary):

$u'_{t}(x,0) = \delta (x-1) $

I am aware of the possibility of solving it with an initial guess of eigenfunctions to the Laplace operator and expanding into Fourier series, but for the sake of being able to visualize the solution analytically d'Alembert's gives an easier solution.

So, in order to solve this problem we need to expand our limited interval to $x \in R$, and since we have homogenous Dirichlet conditions we expand odd, so our beginning value is

$\tilde u'_{t}(x,0) = \delta (x-1) - \delta (x+1), -2<x<2 $.

Using d'Alembert's formula our solution would be the primitive function of our expanded beginning function:

$ \int_{x-t}^{x+t} (\delta (x-1)-\delta (x+1))dx$.

This solution would thus grant us four different $\theta$-functions (Heaviside step function).

My question is, does anything in the solution become affected of our expanding of the interval? I.e., since we are now looking at $-2<x<2$ instead of $0<x<2$.

I have been looking at some problems on this, and in the solution there seems to be some subtracted numbers, and my guess in this case would be a subtraction by 4 (length of the interval). I've seen this done more or less randomly with different intervals and so on, which is why I have no argument as to why it should be done and in what way it would be done.

Any help would be greatly appreciated.

Kind regards

Answer 1

D'almbert's solution to the wave equation is supposed to be used on infintely long strings. Also, the IVP you have written does not contain an equation. I assume you meant $u_{tt} = \Delta u = u_{xx}$.

If this was in regards to an infinite string, then without getting into the specifics, changing the interval you are considering should simply change the time at which the travelling wave reaches that position. As a result you must subtract or add this change where appropriate in the $f(x \pm ct)$ terms. That's the beauty of d'almbert, you just plug into the solution for your given x position an offset along the string.