A question on "finite speed propagation". Anwers provided but i don't understand it.

**panda** · November 2nd 2008, 04:42 PM

Hi,
I have to prove the finite speed propagation property for the solutions of the wave equation of the real line of equation(2).
I've attached the image of the equation 2 and the solution. But I don't understand what I have to prove to show finite speed propagation property.

Thanks in advance

**ThePerfectHacker** · November 2nd 2008, 05:12 PM

I assume you are solving the equation,

$\left\{ \begin{array}{c}\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u }{\partial x^2} \\ -\infty < x,t < \infty \\ u(x,0) = f(x) \text{ and }u_t(x,0)=g(x) \end{array} \right.$

D'Alembert formula gives us,
$u(x,t) = \frac{1}{2}\left[ f(x-t) + f(x+t) \right] + \int_{x-t}^{x+t} g(\mu) d\mu$

Now pick a point $(x,t)$ and mark off the interval $[x-t,x+t]$ (assuming $t>0$ ). This is the interval of dependence. It is showing us that $u$ , the solution to the wave equation, is completely determined by its values on this interval. This is because there are two summands in D'Alembers formula. The first summand simply depends on the starting and ending values of this interval. The second summand just depends on the values of this entire interval. Therefore, the value for position of the wave at $(x,t)$ is completely determined in this interval. It does not depend on anything else which is going on far away. Therefore, the waves propagate according to this model. (We do not have the same situation with the heat equation, the heat is propagated instanteneously).

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