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Question

A 2 m long string fixed at both ends is set into vibrations in its first overtone. The wave speed on the string is 200 m s−1 and the amplitude is 0⋅5 cm. (a) Find the wavelength and the frequency. (b) Write the equation giving the displacement of different points as a function of time. Choose the X-axis along the string with the origin at one end and t = 0 at the instant when the point x = 50 cm has reached its maximum displacement.

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Solution

Given:
Length of the string (L) = 2.0 m
Wave speed on the string in its first overtone (v) = 200 m/s
Amplitude (A) = 0.5 cm

(a) Wavelength and frequency of the string when it is vibrating in its 1st overtone (n = 2):
L=nλ2
λ=L=2 mf=νλ=2002=100 Hz
(b) The stationary wave equation is given by:

y=2A cos2πxλsin2πvtλ =0.5cos2πx2sin2π×200 t2 =0.5 cmcosπm-1 xsin200πs-1 t

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