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.
Solution
(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,
Therefore,
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 =
...(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
=
= 2 × 10-2 kg m-1
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:
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 - 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)
Therefore,
Wavelength, λ = 3 m
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 =
...(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
=
= 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
From equation (i), tension can be obtained as
T = v2μ
= (180)2 × 2 × 10–2
= 648 N
