Find sound wavelength from a vibrating string

[kchurchi]

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!

 

[TSny]

First find the frequency of vibration of the string.

 

[AdkinsJr]

The fundametnal frequency is $$f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$$ They're giving you the info to find the frequency, they're giving you the velocity, so use that to find wavelength.

 

[CAF123]

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

Note also, in your listing of equations, you write that $v = \frac{ω}{k}$ and say that $k$ is the spring constant. This is not the case here, instead it is defined as the wavenumber, where $k = \frac{2π}{λ}.$ Substitution of this and $ω = 2πf$ gives back $v = fλ.$

 

[kchurchi]

Wow! Thank you so much. That made a lot more sense to me than when I first attempted the problem. Thanks :)

 

[kchurchi]

Just to double check with you, the μ in this case is referring to the density given correct?

 

[AdkinsJr]

yes, μ is the linear mass density, kg/m, which is what you're given as "d" actually.