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.

Full Set of Lecture Notes

The notes for this lecture are available here (39 pages).