# Introductory acoustics

## 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).

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.

Thanks very much!!

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haruspex
Homework Helper
Gold Member
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?

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
Homework Helper
Gold Member
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