Is there a general solution for 3d Wave equation?

Question

The general solution for 1D Wave equation is generally given as

f(kx-wt)+g(kx+wt)

Is there such general solution for the 3D Wave equation?

Answer 1

You ask:

The general solution for 1D Wave equation is generally given as

f(kx-wt)+g(kx+wt)

Is there such general solution for the 3D Wave equation

The answer is yes.

First note that general solution has a specific meaning when applied to differential equations:

https://www.hellovaia.com/explanations/math/calculus/general-solution-of-differential-equation/

"The general solution to a differential equation is a solution in its most general form. In other words, it does not take any initial conditions into account."

Second note that your 1D general solution is f(x-ct)+g(x+ct) in the time-space domain (vs. spatial frequency domain where it is your f(kx-wt)+g(kx+wt)).

Given that, the 3D general solution is f(r-ct)/r + g(r+ct)/r in the time-space domain which are concentric expanding and contracting spherical waves with amplitudes falling off as 1/r.