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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 -

(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 beobtained as:

T = v2μ

= (180)2 × 2 × 10–2

= 648 N

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