Wave Equation in Higher Dimensions
Summary
We treat the initial-value problem for the wave equation in $\mathbb{R}^n\times (0,\infty)$, for $n\geq 2$. Starting with the case $n=3$, using spherical means, we obtain Kirchoff’s solution formula for $u(x,t)$. We then employ Hadamard’s method of descent to obtain Poisson’s solution formula for $u(x,t)$ for the case $n=2$.
We generalize this framework to $\mathbb{R}^n$ for $n\geq 3$.
We then cover Huygens’ principle and highlight fundamental differences between the solutions and their domains of dependence in even vs. odd spatial dimensions.