🚀 go-pugleaf

In article <1992Nov18.105130.17567@husc15.harvard.edu> blom@husc15.harvard.edu writes:
>I wrote a simple computer program to simulate waves in a cable fixed at both
>ends as if that cable were a number of springs joined together (320 springs, to
>be exact).  I expected to see motion like that predicted by classical wave
>mechanics, obeying the wave equation
>
>d2y    2 d2y
>--- = v  ---
>dt2      dx2
>
>from which we find that the shape of a travelling wave pulse does not change
>over time.  It occurs to me, though, that this should only work for analytic
>functions.  What happens to places on a pulse where the wave comes to a point? 
>(i.e. where the function has no definable derivative)  

Well, this is one reason why people invented distributions and the
notion of "distributional derivatives".  Using these concepts the wave
equation still makes sense at nasty points such as you describe.  

If you're not up to learning distribution theory, there's a cheap way
out, since your equation is easy to solve exactly.  The general solution
is a sum of left- and right-moving waves, i.e. f(x-vt) and g(x+vt).
Here f and g are ANY function and need not be smooth or even continuous.
(In fact, they can be any distributions -- e.g., nasty things like the
Dirac delta "function".)  

So - the pulses never change shape; there is no dispersion in this
equation.  Of course, simulating the equation on the computer may
introduce dispersion if your program isn't well-written.