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Overtones in a string (equation for wave)

  1. May 19, 2015 #1
    1. The problem statement, all variables and given/known data
    A string has the fundamental tone of
    ##s_1 = A_1\sin (\omega_1 t - k_1 x)##
    Determinate the wave equation for the first harmonic of the string if the sound intensity level of the harmonic is 20dB lower than the fundamental tone. ##\omega_1 = 1360/s## and ##k_1 = 4/m##.

    2. Relevant equations
    Sound intensity level
    ##L = 10\lg (\frac{I_1}{I_0} )##
    Sound intensity is proportional to the wave amplitude squared

    ##I \sim A^2##

    3. The attempt at a solution
    Sound intensity level
    ##-20 = 10\lg( \frac{I_2}{I_1}) \Longleftrightarrow I_2 = 0.01I_1##
    Amplitude relationship
    ##\frac{A_2}{A_1} = 0.1 \Longleftrightarrow A_2 = 0.1A_1##
    The first harmonic got double the frequency so
    ##s_2 = 0.1A_1 \sin(2 \omega_1 t - 2 k_1 x)##
    which is wrong. According to the answer key it should be
    ##s_2 = 0.05A_1\sin(2 \omega_1 t-2k_1 x)##
    Why is it 0.05?
     
  2. jcsd
  3. May 19, 2015 #2

    TSny

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  4. May 19, 2015 #3

    CWatters

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    As I recall the intensity of a wave is proportional to the square of its amplitude. But perhaps I'm rusty.
     
  5. May 19, 2015 #4
    Cheers!

    Always a bit tricky, luckily we got TSny to link us the formula :)
     
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