Note: This is not really a strict mathematical discussion, but a quasi-math.
My post about Fourier thing got a nice interesting comment about a week ago, which reminded me that I have not settled the theme yet, and that I suspended the topic even before mentioning Fourier transform. Actually the post was not really about Fourier, but a complaint about a sound engineering textbook that discusses wave equations a lot. So, let me get back to the main trail.
The 1D wave equation of my interest looked like
And I was about to take Fourier transform of this equation, with respect to . So let us just do that. The left hand side becomes
(here denotes Fourier transform of ), and F.T. of the right hand side is
Here let me rely on a wishful thinking again that vanishes when tends to . Then Fourier transform of 1st derivative of a function is
(the term vanishes). Likewise, F.T. of the 2nd derivative is
So F.T. of the right hand side of the wave equation becomes
Since both sides should be equal, we get another differential equation in frequency domain
This differential equation is actually only about , and can be solved fairly easily. Its solution is given by:
where and are arbitrary functions of , independent of .
Then inverse-transform of this will give me the solution to the original wave equation. Computing the inverse gives:
Well here we note the and came up again, that were the sources of change of variable in the earlier post, this time in a natural way by a simple computation.
Okay, though the integral above might look scary, it can reach back to the definition of inverse Fourier transform by the same change of variables:
By this, the integral becomes (no complaints this time):
Since the 2 terms are just definition of the inverse-transform of and respectively, letting them be and finally we get
Now I got a bit happier than before… But wait a second,