How to solve the wave equation with Dirac delta function initial conditions?

FrattyMathMan
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Homework Statement


Solve the IVP for the wave equation:

Utt-Uxx=0 for t>0
U=0 for t=0
Ut=[dirac(x+1)-dirac(x-1)] for t=0


2. The attempt at a solution

By D' Almbert's solution:

1/2 integral [dirac(x+1)-dirac(x-1)] dx from (x-t) to (x+t)

I apologize for not using Latex- my browser does not seem to agree with the reference.

My issue is what am I to do with the dirac function inside the integral? Can anyone point me in the right direction? I've already read all common literature on the dirac delta function.

Thank you for any help!
 
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Any help? Seriously drawing a blank!

Am I supposed to just use the heavyside function?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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