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Introductory acoustics

  1. Oct 8, 2017 #1
    1. The problem statement, all variables and given/known data
    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).

    New Bitmap Image (3).jpg

    2. Relevant equations
    For Q4
    :
    B015988463_271-361.jpg


    3. 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,

    31afc7e1-17c7-495e-9f92-248600d71349.jpg

    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.
    a.jpg

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

    Thanks very much!!
     
  2. jcsd
  3. Oct 8, 2017 #2

    haruspex

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    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?
     
  4. Oct 8, 2017 #3
    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.
     
  5. Oct 8, 2017 #4

    haruspex

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    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.
     
  6. Oct 8, 2017 #5

    haruspex

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    I had not read Q3. Now that I have I realise 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 vr in the equation is the component of velocity normal to the surface element dS. Or maybe there is a shortcut.
     
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