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

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SUMMARY

The discussion focuses on solving the initial value problem (IVP) for the wave equation Utt - Uxx = 0 with Dirac delta function initial conditions. The solution approach utilizes D'Alembert's formula, specifically integrating the expression 1/2 ∫ [dirac(x+1) - dirac(x-1)] dx from (x-t) to (x+t). The main challenge highlighted is the treatment of the Dirac delta functions within the integral, prompting inquiries about the application of the Heaviside function in this context.

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  • Understanding of wave equations and their properties
  • Familiarity with D'Alembert's solution for wave equations
  • Knowledge of the Dirac delta function and its properties
  • Basic concepts of the Heaviside step function
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Mathematics students, physicists, and engineers dealing with wave equations and initial value problems, particularly those interested in advanced concepts involving Dirac delta functions.

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?
 

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