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A transverse harmonic wave on a string i...

A transverse harmonic wave on a string is described by `y(x,t) = 3.0 sin (36 t + 0.018x + pi//4)` 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?

A

The wave is travelling from right to left.

B

The speed of the wave is 20 m/s.

C

Frequency of the wave is `5.7 Hz.`

D

The least distance between two successive crests in the wave is `2.5 cm.`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
Comparing `y = 3 sin (0.018 + 36 t + (pi)/(4))` with
` y =a sin (kx + omega l + phi)`
we get `k = 0.018 rad//cm, phi = (pi)/(4) rad, a =3 cm`
(a) Option (A) is correct as coefficient of x is +ve which indicates that wave propagates towards - X axis. (i.e. from right to left)
`implies ` Option (A) is correct.
(b) `v = (omega )/(k) = (36)/(0.018) =200 cm//s = 20 m//s`
`implies` Option (B) is correct.
(c) `omega = 2pif `
`implies f = (omega)/( 2pi ) = (36)/(2xx 3.14) =(36)/(6.28)`
`=5.732 Hz ~~ 5.7 Hz`
`implies `Option (C) is correct.
(d) `k = (2pi )/(lamda ) implies lamda = (2pi )/( k) = (2xx 3.14)/(0.018) = 348 cm = 3.48 m`
`therefore ` Distance between two consecutive crests is `lamda =3.48 m`
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Knowledge Check

  • The displacement of a string is given by y (x,t) = 0.06 sin ((2pi x )/(3)) cos (120 pit). 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 xx 10 ^(-2) kg .

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