Question

A violin string vibrates, when struck, as a standing wave with a frequency of 260Hz. When you place your finger on the same string so that its length is reduced to 2/3 of its original length, what is its new vibration frequency? What is the new 3rd harmonic?

Answer 1

new vibration frequency is 3/2 times the original frequency, and the new 3rd harmonic frequency is 9/2 times the original frequency

in 3rd harmonic 260hz = 780 hertz ~384.349305 kilometers

Step-by-step explanation:

wave speed equation

Assuming the original length of the string is L and its vibration frequency is 260 Hz, the speed of sound in the string can be calculated as follows:

v = fλ

where v is the speed of sound, f is the vibration frequency, and λ is the wavelength of the standing wave.

For the fundamental frequency (1st harmonic), the wavelength is twice the length of the string:

λ1 = 2L

So, we have:

v = f1 λ1

f1 = v/λ1 = v/(2L)

For the new length of the string (2/3 L), the wavelength of the fundamental frequency is:

λ'1 = 2/3 L

The new vibration frequency (f') can be calculated as follows:

v = f' λ'1

f' = v/λ'1 = v/(2/3 L) = 3v/(2L)

So, the new vibration frequency is:

f' = 3f1

Substituting f1 = v/(2L) into the above equation gives:

f' = 3v/(2L)

Now, we can calculate the new 3rd harmonic frequency (f'3) using the formula:

f'3 = 3f'

Substituting f' = 3v/(2L) into the above equation gives:

f'3 = 9v/(2L)

Therefore, the new vibration frequency is 3/2 times the original frequency, and the new 3rd harmonic frequency is 9/2 times the original frequency.

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