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I wonder if the wave equation $$u_{tt}-\Delta u=0$$ with initial conditions

$$u(x,0)=\delta, \ \ \ u_{t}(x,0)=0$$ makes sense, both mathematically and physically. Is this equation well-posed?

RobPratt
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MathGeek1024
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1 Answers1

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Broadly, yes it does make sense. Even though the Dirac delta function isn't a proper function in the standard sense, it is a form of generalised function that you can think of as a kind of impulse.

If you substitute the given initial conditions into d'Alembert's formula, you very quickly get the solution

$$u(x, t) = \frac{1}{2}\left(\delta(x-t) + \delta(x+t) \right)$$

meaning that the single delta impulse at $t = 0$ splits into two impulses of half size, one going left and the other going right at equal speeds.

ConMan
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  • Thank you ConMan. I was mainly interested in dimensions 3. So it would still make sense to think about the wave equation with Dirac delta as source? – MathGeek1024 Nov 10 '24 at 15:05