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Intensity of a sound wave problem

  1. Mar 26, 2014 #1
    1. The problem statement, all variables and given/known data

    Two loudspeakers are placed beside each other and produce sound of the same intensity at the position of a listener. One speaker produces a low note of 40 Hz and the other produces a high note of 2560 Hz. What is the ratio of the maximum displacements of the speakers vibrating cones?

    2. Relevant equations

    1) [itex]I[/itex]=P/4piR^2
    2) I=1/2BωkA^2
    3) I=1/2[itex]\sqrt{ρB}[/itex]ω^2A^2

    3. The attempt at a solution
    I understand that I am supposed to use equation 3 for both frequencies and set them equal to each other to come up with a ratio. I just don't understand how equation 3 is derived from equation 1, or how equation 2 is derived from equation 1 for that matter. If I could see how to manipulate these equations I would understand this type of problem much better. (The only equation I am given on my equation sheet is the first one).
  2. jcsd
  3. Mar 26, 2014 #2

    Simon Bridge

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    Well what do each of the terms in the equations mean?
  4. Mar 28, 2014 #3
    Well I is intensity of the wave, p is the pressure, r is radius, ω is angular frequency, k is 2pi/[itex]\lambda[/itex], A is amplitude, and ρ is density. Right?
  5. Mar 28, 2014 #4

    Simon Bridge

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    This is incomplete, and you have not been consistent in your notation.

    Taken in order:
    - Intensity of the sound wave - good;
    - there is no "p" in your equations. Do you mean "P" here?
    ##\qquad##... sound is a pressure wave, so there are lots of pressures all over the place so which pressure does P refer to? Or is that P for "power"?
    - there is no "r" in your equations, do you mean "R"? What is R the radius of?
    - ##\small{\omega}## = angular frequency of the wave
    ##\qquad \small{\omega = vt = 2\pi f}## where v is the wave-speed and f is the frequency of the wave. ##\small{k(x-vt)=kx-\omega t}##
    - ##\small{k=2\pi/\lambda}## good, it's called the wave number.
    - What is A the amplitude of
    ##\qquad##- if "the sound wave" then is it a pressure or a displacement or something else?
    - there are lots of different kinds of density - what is ##\small{\rho}## the density of?
    - what is B? You missed it out.

    If you don't know what the terms refer to then you won't be able to understand the equations.
    I think you need to check your ideas about what sound intensity means:
    ... the intensity of the sound is the rate that energy is delivered to the listeners location per unit area.
    Energy rate = energy per unit time = Power, so ##I=P/A## i.e. is power per unit area.
    Revisit the equation list in post #1 with that in mind.
    Last edited: Mar 28, 2014
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