four significant digits to get the right answer. Assume the speed of sound is 34

Question

four significant digits to get the right answer. Assume the speed of sound is 343 m/s. A string on a musical instrument is 1.20 m long and is tuned to that when unfingered it produces F below middle C, 175 Hz. How far from the end of the string must the string be held down to produce the A below middle C, 220 Hz? What is the wavelength on this string of this 220 Hz wave? What are the frequency and wavelength of the sound wave produced by this in air at 20 C? When standing at a certain distance from a jet on the ground on which all three engines are

Explanation / Answer

A) The speed of propagation of a wave in a string (v) is proportional to the square root of the tension of the string (T) and inversely proportional to the square root of the linear density () of the string.

V = sqrt(T / )

as your question shows there is nothing changed on the string; then the V must be a constant on both cases. Next equation, V = f * , Velocity equals to frequency times the wavelength.
on a violin string, its fundamental frequency wavelength is equal to two times of the string length. So that, on the case one, we can find the velocity_1 = 175 * 2 * 1.2 = 420m/s
Now, we can use this result to find the length of the C, its given frequency is 220 Hz

420 =220 *2 * Length_C
Length_C = 420 / 220*2 = 0.954 m

So that, the string length for playing the C is 95.4 cm
It is 1.2 -0.954 m from the end of the string.
Your finger should be place at 0.245m cm from the end of the string.

B) velocity, V =220 * 2 * 1.2 =528m/s

wavelength = V/f =528/220 = 2.4m

C) v = 343 m/s * SQRT (1 + T/273)= 493.88 m/s therefore wavelength = 2*L =2* 1.2 = 2.4 m

Frequency = velocity/wavelength=493.88/2.4 = 205.88 Hz

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