Question

The transverse displacement of a string (clamped at its both ends) is given by

Where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 ×10^–2 kg.

Answer the following:

(a) Does the function represent a travelling wave or a stationary wave?

(b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?

(c) Determine the tension in the string.

Answer 1

(a) The general equation representing a stationary wave is given by the displacement function:

y (x, t) = 2a sin kx cos ωt

This equation is similar to the given equation:

Hence, the given function represents a stationary wave.

(b) A wave travelling along the positive x-direction is given as:

The wave travelling along the negative x-direction is given as:

The superposition of these two waves yields:

The transverse displacement of the string is given as:

Comparing equations (i) and (ii), we have:

∴Wavelength, λ = 3 m

It is given that:

120π = 2πν

Frequency, ν = 60 Hz

Wave speed, v = νλ

= 60 × 3 = 180 m/s

(c) The velocity of a transverse wave travelling in a string is given by the relation:

Where,

Velocity of the transverse wave, v = 180 m/s

Mass of the string, m = 3.0 × 10^–2 kg

Length of the string, l = 1.5 m

Mass per unit length of the string,

Tension in the string = T

From equation (i), tension can be obtained as:

T = v²μ

= (180)² × 2 × 10^–2

= 648 N