Determing wavelength of sound wave from steel string

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LBRRIT2390
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Homework Statement



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?

Homework 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]

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!
 

Answers and Replies

  • #2
LBRRIT2390
29
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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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