Question

In: Physics

A two dimensional transverse water waves spread in circular ripples a) show that the amplitude at...

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?

Solutions

Expert Solution

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


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