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. | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-15 Waves



Q.11:- 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 equation represents a stationary wave.
(b) A wave travelling along the positive x-direction is given as:
y1 = a sin (ωt kx)
The wave travelling along the positive x-direction is given as:
y2 = a sin (ωt + kx)
The supposition of these two waves yields:
y = y1 + y2 = a sin (ωt – kx) – a sin (ωt + kx)
a sin (ωt) cos (kx) – a sin (kx) cos (ωt) –  a sin (ωt) cos (kx) – a sin (kx) cos (ωt)
= 2 a sin (kx) cos (ωt)

∴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:
v = √T/µ              ….(i)
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, µ = m/l
= 3.0 × 1.5 = 10-2
= 2 × 10-2 kg m-1

Tension in the string = T
From equation (i), tension can be obtained as:
T = v2μ
= (180)2 × 2 × 10–2
= 648 N