Question

An aluminum rod, 1.0 m long, is held lightly in the middle. One end is struck head-on with a rubber mallet so that a longitudinal pulse —a sound wave—travels down the rod. The fundamental frequency of the longitudinal vibration is 2.55 kHz. 

(a) Describe the locations of the displacement and pressure nodes and antinodes for the fundamental mode of vibration. 

(b) Calculate the speed of sound in aluminum from the information given in the problem. 

(c) The vibration of the rod produces a sound wave in air that can be heard. What is the wavelength of the sound wave in the air? Take the speed of sound in air to be 334 m/s. 

(d) Do the two ends of the rod vibrate longitudinally in phase or out of phase with each other? That is, at any given instant, do they move in the same direction or in opposite directions?

Answer 1

Length of the rod L = 1 m

Fundamental frequency of vibration f = 2.55 × 103 Hz

 

a)

The displacement antinodes are formed at the ends of the rod and displacement node at the center of the rod.

 

b)

The speed of sound in aluminum is given as

v = λf

 

The wavelength of wave, when the rod is like an open pipe

λ = 2L

 

Substituting in the above equation

v = 2Lf

   = 2(1m)(2.55 × 103 Hz)

   = 5.1 × 103 m/s

   = 5100 m/s

 

c)

The wavelength of the sound in air is

λair = vsound/f

        = (334 m/s)/2.55 × 103 Hz

        = 0/1309 m

       = 13.1 cm

 

d)

The longitudinal motion of the wave is symmetrical along the center. They move in opposite direction, so the two ends are in out of phase.