Question

A 4.70-m-long string that is fixed at one end and attached to a long string of negligible mass at the other end is vibrating in its fifth harmonic, which has a frequency of 428 Hz. The amplitude of the motion at each antinode is 2.82 cm.

(a) What is the wavelength of this wave?
?₅ =  m

(b) What is the wave number?
k₅ =  m^?1

(c) What is the angular frequency?
?₅ =  s^?1

(d) Write the wave function for this standing wave. (Use the following as necessary: x, and t.)
y₅(x, t) =

Answer 1

When String that is fixed with one end, length of string is given by:

L = n*lambda/4, where n = 1, 3, 5, ------

lambda = wavelength of wave

lambda = 4*L/n

n = fifth harmonic wave = 5

lambda = 4*4.70/5 = 3.76 m

Part B

wave number, k = 2*pi/lambda = 2*pi/3.76 = 1.67 m^-1

Part C

Angular frequency is given by:

w = 2*pi*f

f = 428 Hz

w = 2*pi*428 = 856*pi rad/sec = 2689.20 rad/sec

Part D.

Wave function for standing wave in nth harmonic is given by:

y₅(x, t) = A*sin kx* cos wt

A = Amplitude = 2.82 cm = 0.0282 m

k = 1.67 m^-1

w = 856*pi s^-1

y₅(x, t) = 0.0282*(sin 1.67*x)*(cos 856*pi*t)

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