- #1
Agent 47
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Homework Statement
Show that any function of the form
##z = f(x + at) + g(x - at)##
is a solution to the wave equation
##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}##
[Hint: Let u = x + at, v = x - at]
2. The attempt at a solution
My problem with this is not that I haven't been able to solve it. The book's solution is right here:
I began to have trouble when I decided not to use f'(u) and g'(v) instead I used ##\frac {\partial z} {\partial u}## and ##\frac {\partial z} {\partial v}##
When I did this I got
##\frac {\partial^2 z} {\partial x^2} = \frac {\partial^2 z} {\partial u^2} + \frac {2 \partial^2 z } {\partial u \partial v} + \frac {\partial^2 z} {\partial v^2}##
and
##\frac {\partial^2 z} {\partial t^2} = a^2 (\frac {\partial^2 z} {\partial u^2} - \frac {2 \partial^2 z } {\partial u \partial v} + \frac {\partial^2 z} {\partial v^2})##
And that does not fulfill the condition stated in the beginning.
So I have a two questions in the end:
1)What is the difference between writing f'(u) and ##\frac {\partial z} {\partial u}##?
2)Do ##\frac {2 \partial^2 z } {\partial u \partial v}## and ##- \frac {2 \partial^2 z } {\partial u \partial v}## both evaluate to 0? (That's the only way ##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}##) or did I do something wrong in my calculations?