Waves

Question : 11

Total: 27

The transverse displacement of a string (clamped at its two ends) is given by
y(x,t)=0.06‌sin(

2π

x)‌cos(120πt)
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 are the wavelength, frequency, and speed of each wave?
(c) Determine the tension in the string.

Solution:

Here, the equation for transverse displacement is given by
y(x,t)=0.06‌sin(

2π

x)‌cos(120πt)...(i)
(a) The displacement, which involves harmonic functions of x and t separately, represents a stationary wave and the displacement, which is harmonic function of the form (vt±x), represents a travelling wave.
Hence, the equation given above represents a stationary wave.
(b) When a wave pulse
y′=A‌sin‌

2π

(vt−x)
travelling along X-axis is superimposed by the reflected pulse
y′′=−A‌sin‌

2π

(vt−x)
from the other end, a stationary wave is formed and is given by
y=y′+y′′ =−2A‌sin‌

2π

‌x‌cos‌

2π

‌v‌t...(ii)
Comparing the equations (i) and (ii), we have

2π

or λ=3m
Also,

2π

v=120π or v=60λ =60×3=180ms–1
Now, frequency, υ=

180

=60Hz
(c) Velocity of transverse wave in a string is given by v=√

Here, µ=

3.0×10−2

1.5

=2×10−2kgm−1
Also, v=180ms−1
∴T=v2µ=(180)2×2×10−2=648N

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