Determing wavelength of sound wave from steel string

In summary, the problem involves finding the wavelength of a sound wave that reaches the ear in a 20 \circC room. The solution involves solving for the fundamental frequency using the formulas for fundamental frequency and speed of wave in a string, and then using that value to calculate the wavelength in the air. The final result is a wavelength of 2.86m.
  • #1
LBRRIT2390
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0

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!
 
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  • #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
 

1. How is the wavelength of a sound wave on a steel string determined?

The wavelength of a sound wave on a steel string is determined by the length of the string, the tension applied to the string, and the mass per unit length of the string. This can be calculated using the equation: λ = 2L/n, where λ is the wavelength, L is the length of the string, and n is the harmonic number.

2. What is the relationship between wavelength and frequency in a sound wave on a steel string?

The relationship between wavelength and frequency in a sound wave on a steel string is inversely proportional. This means that as the wavelength increases, the frequency decreases, and vice versa. This relationship can be described by the equation: f = v/λ, where f is the frequency, v is the speed of sound, and λ is the wavelength.

3. How does the tension on the steel string affect the wavelength of the sound wave?

The tension on the steel string directly affects the wavelength of the sound wave. As the tension increases, the wavelength decreases, resulting in a higher frequency sound. This is because higher tension in the string causes it to vibrate at a higher frequency, producing shorter wavelengths.

4. Can the wavelength of a sound wave on a steel string be changed?

Yes, the wavelength of a sound wave on a steel string can be changed by altering the length, tension, or mass per unit length of the string. Shortening the string or increasing the tension will result in a shorter wavelength and a higher frequency sound, while lengthening the string or decreasing tension will result in a longer wavelength and a lower frequency sound.

5. How is the speed of sound on a steel string determined?

The speed of sound on a steel string is determined by the tension and mass per unit length of the string. It can be calculated using the equation: v = √(T/μ), where v is the speed of sound, T is the tension, and μ is the mass per unit length. As tension and mass per unit length increase, so does the speed of sound on the string.

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