Introductory Acoustics Homework Help

[davon806]

Homework Statement

Hi there,
I am a beginner in acoustics and there are severals problems I am currently struggling while I was reading the first chapter of "Theory of Vortex Sound" (available on Google books).

Homework Equations

For Q4[/B]:

The Attempt at a Solution

For Q2 and 3, as underlined in the first picture, if the velocity and pressure doesn't go like 1/r , the integral will diverge as r-> inf. However, there are no 1/r terms in U₀cos(wt)

For Q4,

I am not sure how to proceed, since the variable y is contained in two terms within the dirac delta,I don't know how to eliminate it.

BTW, I googled sth that might be helpful for Q4.

(From P.115-116 of https://books.google.co.uk/books?id...WAhVKDxoKHe_nDiQQ6AEIMjAC#v=onepage&q&f=false )

Thanks very much!

 

[haruspex]

For Q2, I feel the discussion about large r that follows eqn 1.8.4 is not relevant. You have no idea what p is.
Note that in the eqn. the left side involves p and v. A substitution is then made to eliminate v in favour of p.
In Q2 you are given information about v but not p. What alternative step does that suggest?

 

[davon806]

For Q2, I feel the discussion about large r that follows eqn 1.8.4 is not relevant. You have no idea what p is.
Note that in the eqn. the left side involves p and v. A substitution is then made to eliminate v in favour of p.
In Q2 you are given information about v but not p. What alternative step does that suggest?

Since v = U₀cos(wt) and using the substitution v = p/ρ₀c₀ , rearranging we can solve for p. Then I use eq 1.8.4 with the integrand
pv = U₀ ^2 cos^2 (ωt) ρ₀ c₀ .But then the integrand doesn't go like 1/r^2 ,when I integrate over the surface it gives a 4πr^2 term.

In addition, I am not sure on the difference between Q2 and Q3. I guess the surface area for Q3 is like 4πr^2,but have no idea on Q2.

 

[haruspex]

But then the integrand doesn't go like 1/r^2

You don't need it to. As I wrote, we are not here considering large r. This is the radius R of the compact sphere. We are told it makes small oscillations, so to a first approximation its area is always 4πR².

 

[haruspex]

I am not sure on the difference between Q2 and Q3

I had not read Q3. Now that I have I realize that is the question I have been leading you to answer.
Q2 says translational oscillations, i.e. the sphere is oscillating side-to-side.
You could try to figure out the correct integral for that, remembering that the v_r in the equation is the component of velocity normal to the surface element dS. Or maybe there is a shortcut.