Max/Min power Transfer In String concept check

1. Oct 11, 2008

Andrusko

1. The problem statement, all variables and given/known data
We have a transverse sinusoidal wave being generated in a long horizontal string. The transverse displacement is described by:

$$y(x,t) = 0.01sin(27.3x - 240\pi t)$$

What is the displacement when the maximum and minimum power transfer occurs?

2. Relevant equations

$$P(x,t) = vA^{2}\mu\omega^{2}sin^{2}(kx - \omega t)$$

v = phase velocity
A = amplitude
mu = linear density
omega = angular frequency
k = angular wave number
x = element position
t = time

3. The attempt at a solution

I think I know the answer already, but I don't know how to mathematically write it down.

When a string element is at 0 displacement (relative to y axis) the element will be going at it's fastest speed and have it maximum tension, so the elastic potential and kinetic energy are at a max. So this will be the maximum power transfer.

When the string is at +-0.01m displacement, it's at its maximum amplitude and both kinetic energy and elastic energy are 0, so that's the minimum power transfer.

I think it's necessary to demonstrate this mathematically with the derivative of the power equation, but I'm just not sure how to do it because I've never done calculus with functions of two variables. I sort of understand that a partial derivative will be needed, but what do I differentiate with respect to and then how do I solve it for max and min? The still having x and t in the derivatives kinda throw me.

Or is it entirely unnecessary to demonstrate this mathematically?