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Sound Waves and Interference Problem - Check My Work?

  1. Sep 12, 2011 #1
    Okay, I have this problem worked out, but I have no clue if my answers are right. Could someone please check my work and give me some confidence or show me where I may have messed up.

    1. The problem statement, all variables and given/known data
    Two loudspeakers, one of them generates sound with 400 W power the other with 100 W. The sound is generated isotropically (the same in all directions) from each loudspeaker. A listener is seated 20m from the 100 W loudspeaker and 40 m from the other loudspeaker. A signal generator drives the two speakers in phase with the same frequency. The frequency is swept through the audible range from 20-20,000 Hz.

    a) What is the ratio of the intensities of the two sounds (I(for 400W)/I(for 100W)) at the listener's position?

    b) What are the two highest frequencies at which the listener will hear a maximum signal because of constructive interference?

    c) What are the two highest frequencies at which the listener will hear a minimum signal because of destructive interference?

    d) Assuming a single frequency of sound (f= 5000Hz) from the 400 W loudspeaker. What is the displacement amplitude of the sound waves that arrives at the listener's position from the 400 W loudspeaker only?

    2. Relevant equations
    I = Ps/(4πr2)
    I = 0.5ρvω2sm2
    ΔL/λ = 0, 1, 2... (fully constructive interference)
    ΔL/λ = 0.5, 1.5, 2.5... (fully destructive interference)
    v = λf
    vair = 343 m/s
    ρair = 1.21 kg/m3

    3. The attempt at a solution
    a) I1 = Ps/(4πr2) = 400/(4π402) = 1.99*10-2 W/m2
    I2 = 100/(4π202) = 1.99*10-3 W/m2
    I1/I2 = 10

    b) ΔL/λ = ΔL/(v/f) = fΔL/v = f(40-20)/343 = 0, 1, 2...
    f = (0, 1, 2...)*17.15 Hz
    fmaxf = 20,000 Hz
    Looking for two highest frequencies: (n)17.15 Hz ≤ 20,000 Hz
    n ≤ 1166.18
    nmax = 1165, 1166
    fmax = 19,979.75 Hz, 19,996.9 Hz

    c)f = (0.5, 1.5, 2.5...)*17.15 Hz
    (n)17.15 Hz ≤ 20,000 Hz
    n ≤ 1166.18
    nmax = 1164.5, 1165.5
    fmax = 19,971.18 Hz, 19,988.33 Hz

    d)ω = 2πf = 2π(5000) = 10000π
    I = 0.5ρvω2sm2
    sm = √(2I/(ρvω2))
    sm = 3.18*10-7 m
     
  2. jcsd
  3. Sep 13, 2011 #2

    Redbelly98

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    Homework Helper

    For (a), try recalculating I2. I get a different number.

    (b), (c) and (d) look good.
     
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