In: Physics
A two dimensional transverse water waves spread in circular ripples a) show that the amplitude at distance r from the intial disturbance is proportional to sqrt of r b) the intensity of the above wave at distant r=2m is 10^-4 W. what is the intensity at 4m distance?
We use the equation derived for the transverse velocity in the 1D case:
−ωAcos(kx−ωt)
Consider one ripple, with the origin at the center. This equation can be applied along a cross section of the ripple (think of a ray emanating from the center, we apply the equation along that line). Since it's a circle, we'll use polar coordinates. Instead of x, the velocity now depends on r. We also take t to be 0 since we can start time whenever we want (at least I think this is the reason they do this). Now we integrate. It must be a double integral, since it's 2-dimensional.
Also, it doesn't depend on θ, so we get
12μω^2A^2 ∫0 to 2π ∫R-λ/2 to R+λ/2 cos^2(kr)rdrdθ
where we have used the fact that the kinetic energy is given by 1/2mv^2.
I'll skip the details of the integration
1/8μω^2A^2Rλ
solving for A, we would indeed get that it is proportional to 1/sqrt R.
The intensity will be halved at 4m distance