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I have been studying the wave equation in $\mathbb{R}^n$ for the cases $n=1,2$ and $3$. In the three cases, working all over $\mathbb{R}^n$. That is:

$u_{tt}(x,t)=c^2 \Delta_{x} u(x,t)$ for $x \in \mathbb{R}^n$ and $t>0$

$u(x,0)=g(x)$ for $x \in \mathbb{R}^n$

$u_{t}(x,0)=h(x)$ for $x \in \mathbb{R}^n$

where $g$ and $h$ are given initial conditions.

The math part goes like this:

  1. Study the case $n=1$ and derive d'Alembert's formula for the solution.

  2. Applying the method of spherical means reduce the cases $n=2,3$ to the case $n=1$.

  3. Solve the case $n=3$ and obtain Kirchhoff's formula for the solution.

  4. Reduce the case $n=2$ to the case $n=3$ and obtain Poisson's formula for the solution.

For the cases $n=1$ and $n=2$ the solution $u$ at $(x,t)$ depends on the values of $g$ and $h$ over the entire, closed ball $B_{ct}(x)$. That is, the value of a wave that behaves according to the wave equation can be predicted at the point $x$ and at time $t$ knowing its values over the entire ball $B_{ct}(x)$ at time $0$.

However, for the case $n=3$ the solution $u$ at $(x,t)$ depends only on the values of $g$ and $h$ over the boundary of the ball $B_{ct}(x)$. That is, the value of a wave that behaves according to the wave equation can be predicted at the point $x$ end at time $t$ knowing only its values over $\partial B_{ct}(x)$ at time $0$.

I've read that the different behavior for the case $n=3$ has something to do with Huygens' principle. Anyway, I didn't get it.

My question is: is there a physical argument that allows one to understand, or even predict, why is it that the case $n=3$ depends only on the boundary of the ball around $x$ but the same doesn't hold for the cases $n=1,2$?

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  • 3
    $\begingroup$ math.stackexchange.com/q/8794 $\endgroup$ – lemon Jan 26 '15 at 9:23
  • $\begingroup$ @lemon made a great catch there. If you're interested, reading an introductory optics text may make the whole Huygens thing clearer. $\endgroup$ – Carl Witthoft Jan 26 '15 at 14:20
  • $\begingroup$ Should not $u(x,t)$ be $u_{xx}(x,t)$ in the first equation? $\endgroup$ – user45664 Jul 10 '16 at 21:27
  • $\begingroup$ @user45664 You are right!! $\endgroup$ – Pipicito Jul 15 '16 at 17:29
  • $\begingroup$ @pipicito See my question physics.stackexchange.com\q\267899. I am asking a question similar to yours but for n=1 only. Just received a helpful answer there. $\endgroup$ – user45664 Jul 17 '16 at 20:24

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