A digestible introduction to the wave equation with working code
github.com
github.com
And a good example of why the 'exact' solution is often not used, because it's so slow
you can boil the whole thing down to a convolution matrix type trick which a modern gpu will eat for breakfast.
i should find some time and do one of these... the quality of 'impressive webgl tech demo' seems to be about 15 years out of date in general...