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Find sound wavelength from a vibrating string

  • Thread starter kchurchi
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  • #1
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Homework Statement


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

Homework 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

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:

Answers and Replies

  • #2
TSny
Homework Helper
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First find the frequency of vibration of the string.
 
  • #3
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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.
 
  • #4
CAF123
Gold Member
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88
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]
 
  • #5
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Wow! Thank you so much. That made a lot more sense to me than when I first attempted the problem. Thanks :)
 
  • #6
39
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Just to double check with you, the μ in this case is referring to the density given correct?
 
  • #7
150
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yes, μ is the linear mass density, kg/m, which is what you're given as "d" actually.
 

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