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Determing wavelength of sound wave from steel string

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known data

    A 120 cm-long steel string with a linear density of 1.2 g/m is under 100 N tension. It is plucked and vibrates at its fundamental frequency.

    What is the wavelength of the sound wave that reaches your ear in a 20 [tex]\circ[/tex]C room?

    2. Relevant equations

    Fundamental Frequency

    f1 = [tex]\frac{v}{2L}[/tex]

    Fundamental Frequency of a stretched string

    f1 = [tex]\frac{1}{2L}[/tex][tex]\sqrt{\frac{T_s}{\mu}}[/tex]

    Wavelengths of standing wave modes

    [tex]\lambda[/tex]m = [tex]\frac{2L}{m}[/tex]

    [tex]\lambda[/tex]m = [tex]\frac{v}{f_m}[/tex]

    3. The attempt at a solution

    I solved fundamental frequency as 143.3 then used

    [tex]\lambda[/tex]m = [tex]\frac{v}{f_m}[/tex] to find the wavelength.

    I also tried solving for fundamental frequency using

    f1 = [tex]\frac{1}{2L}[/tex][tex]\sqrt{\frac{T_s}{\mu}}[/tex]

    All of my answers have been incorrect. Please help!
     
  2. jcsd
  3. Feb 7, 2010 #2
    Speed of wave in string: v = [tex]\sqrt{\frac{T_s}{\mu}}[/tex]
    Ts = 100N​
    [tex]\mu[/tex] = 0.0012kg/m​
    v = 288.7​

    Fundamental Frequency: F0 = [tex]\frac{v}{\lambda_0}[/tex]
    [tex]\lambda[/tex]0 = 2L = 2*1.2​
    [tex]\lambda[/tex]0 = 2.4​

    f0 = [tex]\frac{\sqrt{\frac{T_s}{\mu}}}{2L}[/tex]

    In the air:

    [tex]\lambda[/tex] = [tex]\frac{v}{f_0}[/tex]

    [tex]\lambda[/tex] = [tex]\frac{v}{\frac{\sqrt{\frac{T_s}{\mu}}}{2L}}[/tex]

    [tex]\lambda[/tex] = [tex]\frac{v2L}{\frac{\sqrt{T_s}}{\mu}}[/tex]

    [tex]\lambda[/tex] = [tex]\frac{344m/s * 2.4}{288.7}[/tex] = 2.86
     
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