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Find sound wavelength from a vibrating string

  1. Aug 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Sound Wavelength From String
    During a concert a pianist hits a key that sets up a standing wave in a piano string that is vibrating in its fundamental mode. The string is 0.5 m long, has a mass density of 0.002 kg/m and is held under a tension of 120 N. What is the wavelength of the sound wave heard by the listener? The speed of sound in air is 343 m/s.
    v (sound) = 343 m/s
    L (length) = 0.5 m
    d (mass density) = 0.002 kg/m
    T (tension) = 120 N
    λ (wavelength) = unknown

    2. Relevant equations
    v = f*λ = ω/k

    f = frequency
    k = spring constant
    ω = angular frequency

    v = √(T/(m/L))

    T = force of tension
    m = mass
    L = length

    3. The attempt at a solution
    At first I attempted a solution using the second equation provided, however I am not quite sure what I would be solving for since the speed of sound in air is provided. Using the second equation I find the speed of the wave itself, but I am not sure how to apply the two speeds to finding the wavelength of the sound wave? Please help me with this problem!
     
    Last edited: Aug 16, 2012
  2. jcsd
  3. Aug 16, 2012 #2

    TSny

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    First find the frequency of vibration of the string.
     
  4. Aug 16, 2012 #3
    The fundametnal frequency is [tex] f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}}[/tex] They're giving you the info to find the frequency, they're giving you the velocity, so use that to find wavelength.
     
  5. Aug 16, 2012 #4

    CAF123

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    Gold Member

    Just to add:
    The frequency of vibration, [itex] f = \frac{v}{λ} = \frac{nv}{2L} [/itex]. The allowed wavelengths are: [itex] λ = \frac{2L}{n}, [/itex] where n is the mode of oscillation.

    Note also, in your listing of equations, you write that [itex] v = \frac{ω}{k} [/itex] and say that [itex] k [/itex] is the spring constant. This is not the case here, instead it is defined as the wavenumber, where [itex] k = \frac{2π}{λ}. [/itex] Substitution of this and [itex] ω = 2πf [/itex] gives back [itex] v = fλ. [/itex]
     
  6. Aug 16, 2012 #5
    Wow! Thank you so much. That made a lot more sense to me than when I first attempted the problem. Thanks :)
     
  7. Aug 16, 2012 #6
    Just to double check with you, the μ in this case is referring to the density given correct?
     
  8. Aug 16, 2012 #7
    yes, μ is the linear mass density, kg/m, which is what you're given as "d" actually.
     
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