One of the 51.0-cm-long strings of an ordinary guitar is tuned to produce the no

Question

One of the 51.0-cm-long strings of an ordinary guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode.

A. Find the speed of transverse waves on this string.

B. If the tension in this string is increased by 1.7 %, what will be the new fundamental frequency of the string?

C. If the speed of sound in the surrounding air is 344 m/s, find the frequency of the sound wave produced in the air by the vibration of the B3 string.

D. If the speed of sound in the surrounding air is 344 m/s, find the wavelength of the sound wave produced in the air by the vibration of the B3 string.

E. How do your answers in Part C and Part D compare to the frequency and wavelength of the standing wave on the string?

Explanation / Answer

A)

Wave speed is given by v = sqrt(T/) where T is tension and is the linear mass density which is equal to mass/length fn = n*v/(2L) where n is the mode, v is the wave speed and l is the length

2L*fn/n = v thus v = 2*0.51 m * 245Hz/1

v = 249.9 m/s
B)

2L*fn/n = v =sqrt(T1/), here T1 is1.017T, so new velocity is 249.9 *1.017 = 254.148 m/s

2L*fn/n =254.148

New Fundamental frequency is 254.148/(2*0.51) = 249.16 Hz

New Fundamental frequency is 249.16 Hz.

C) Frequency in air too is 245 Hz

D)

Wavelength = v/f = 344/245 = 1.40 m

E) Wavelength on string is 2L = 1.02 m

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