Standing Wave Energy: Find Total Mech. Energy

[atavistic]

Homework Statement

A standing wave is maintained in a homogenous string of cross-sectional area a and density $$\rho$$ . It is formed by the superposition of two waves traveling in opposite directions given by the equations

y1 = Asin(wt-kx)
y2 = 2Asin(wt +kx)

Find the total mechanical energy confined between the section corresponding to the adjacent antinodes.2. The attempt at a solution

The wave is given by y = Asin(wt-kx) + 2Asin(wt +kx)

KE of a small element is 1/2 u dx v^2 , where u = mass per unit length.

I find v by differentiating y wrt t.

Then I integrate with proper limits. But the integration looks outrageous looking at the simple answer. Is there any shortcut ? Also using above method we only get the KE, what about the PE?

 

[tiny-tim]

Hi atavistic!

But the integration looks outrageous …

Show us!

 

[atavistic]

Well can you tell me what I did is right or not and the thing about PE.