Waves
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Question : 8
Total: 27
A transverse harmonic wave on a string is described by y ( x , t ) = 3.0 s i n ( 36 t + 0.018 x +
)
where x and y are in cm and t in s. The positive direction of x is from left to right.
(a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation?
(b) What are its amplitude and frequency?
(c) What is the initial phase at the origin?
(d) What is the least distance between two successive crests in the wave?
where x and y are in cm and t in s. The positive direction of x is from left to right.
(a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation?
(b) What are its amplitude and frequency?
(c) What is the initial phase at the origin?
(d) What is the least distance between two successive crests in the wave?
Solution:
The equation of the form
y ( x , t ) = A sin (
( v t + x ) + ϕ )
represents a harmonic wave of amplitude A, wavelength l and travelling from right to left with a velocity v.
Now, the given equation for the transverse harmonic waveis
y ( x , t ) = 3.0 s i n ( 36 t + 0.018 x +
)
= 3.0 sin [ 0.018 (
t + x ) +
]
= 3.0 sin [ 0.018 ( 2000 t + x ) +
]
(a) Since the equation (i) and (ii) are of the same form, the given equation also represents a travellingwave propagating from right to left. Further, the coefficient of t gives the speed of the wave. Therefore,
v = 2000 cm s − 1 = 20 m s − 1
(b) Obviously, amplitude, A = 3.0 cm
Further,
= 0.018 λ =
c m
υ =
=
× 0.018 = 5.73 s − 1
(c) Initial phase at the origin,ϕ =
(d) Least distance between two successive crests in the wave is equal to wavelength. Therefore,
λ =
= 349.0 c m = 3.49 m
represents a harmonic wave of amplitude A, wavelength l and travelling from right to left with a velocity v.
Now, the given equation for the transverse harmonic waveis
(a) Since the equation (i) and (ii) are of the same form, the given equation also represents a travellingwave propagating from right to left. Further, the coefficient of t gives the speed of the wave. Therefore,
(b) Obviously, amplitude, A = 3.0 cm
Further,
(c) Initial phase at the origin,
(d) Least distance between two successive crests in the wave is equal to wavelength. Therefore,
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