List of solutions to the wave equation in three dimensions

Question

Wikipedia describes plane waves and $e^{i(\omega t\pm kr)}$ as well as a general integral formula as solutions to the wave equation. As far as I understand, all solutions can be constructed from these by finite and infinite sums. But: are there other "simple" closed form functions known that solve the wave equation?

Answer 1

Suppose the general solution for the wave equation in three dimensions as the following

$$y(x)=c_1e^{i(\omega t\pm kr)}+c_2e^{-i(\omega t\pm kr)}$$

with $c_1,c_2\in\mathbb{C}$. From hereon we can construct some other functions, which are closely related to the exponential

$$\begin{align} &y_{1,2}(x)~=~ce^{\pm i(\omega t\pm kr)}\tag{I}\\ &y_{3,4}(x)~=~c\lambda^{\pm i(\omega t\pm kr)}\tag{II}\\ &y_{5,6}(x)~=~c_1\sin(\omega t\pm kr)+c_2\cos(\omega t\pm kr)\tag{III}\\ &y_{5,6}(x)~=~c_1\sin(\lambda_1(\omega t\pm kr))+c_2\cos(\lambda_2(\omega t\pm kr)\tag{IV}\\ &y_{7,8}(x)~=~c_1\sinh(i(\omega t\pm kr))+c_2\cosh(i(\omega t\pm kr))\tag{V}\\ &y_{9,10}(x)~=~c_1\sinh(\lambda_1i(\omega t\pm kr))+c_2\cosh(\lambda_2i(\omega t\pm kr)\tag{VI}\\ \end{align}$$

We can go even further and say the Generalized Hypergeometric Function $_0F_0(;;i(\omega t\pm kr))$ is a solution of the equation since it is just a more general way to write down the exponential.

The main problem with your question is that you can in fact construct many other closed functions out of the exponential. But hence they are just a construction out of the general solution mathematician agreed on just considering the general solution - or in this case the fundamental set of solutions - as the one solution to the differential equation so that they have not to write down a list with about $10$ entries every single time they solve an equation.

Howsoever I hope that you are satisfied with my collocation of possible functions which can be constructed.