In Walter Strauss, the author derives the solution of wave equation in 3 dimension and observes the following point
- The solution involves and integration over the surface of sphere (Krichoff formula) which implies that only surface affect the solution not the points inside it (and is called Huygen's principle).
In the next section he attempts to prove the solution in 2 dimension by ignoring the third coordinate i.e. $z$-coordinate. The result is we manage to simplify the surface integral into a regular integral (i.e. having dxdy instead of dS) and limits changes from the surface of a sphere to a disk. Due to this the author claimed that Huygen's principle is violated in the 2 dimension case because the solution is influenced by the points inside the disk.
One can see the acutall calculation here https://en.wikipedia.org/wiki/Wave_equation#Scalar_wave_equation_in_general_dimension_and_Kirchhoff's_formulae
Now here is my doubt.
In the two dimension case, there is a step where we change the surface integral into double integral (see the section Scalar wave equation in two space dimensions in the above link). My doubt is why I can't do the same step in the 3 dimension case. This step for me (uptill now) doesn't seem to utilize the fact that solution is not dependent of $z$-variable.