Calculating kinetic energy density of wavepacket in deep water waves

[twinklestar28]

Assume that an orca can generate a wavepacket which is half a sine wave. Show that the kinetic energy density in this wave can be written

U= ρgAλ/(32(π^2))

2. Homework Equations

mass per unit length=∫ρ dxdy
KE=1/2mv^2

3. The Attempt at a Solution

I'm confused on what to use for dx and dy. Would dx and dy be the displacements :

displacement in x = Ae^ky sin(wt-kx)
displacement in y = Ae^ky cos(wt-kx)

This however gives me complicated answers so I think my integral is completely wrong. Also for the velocity which they have given as vp= √gk/2 and vg=1/2vp which one do i use?

 

[jbsteele]

So far I have: U=1/2 ∫ρ (Ae^ky sin(wt-kx))^2 + (Ae^ky cos(wt-kx))^2 dxdy = 1/2 ρ A^2 e^2k^2 ∫sin^2(wt-kx) + cos^2(wt-kx) dxdy = 1/2 ρ A^2 e^2k^2 But I'm not sure if this is correct. Any help would be much appreciated. Thank you!