A transverse harmonic wave on a wire is expressed as:

y( x, t ) =3 sin ( 36t +0.018x +\( \dfrac{ π}{4}\) )

**(i)** Is it a stationary wave or a travelling one?

**(ii)** If it is a travelling wave, give the speed and direction of its propagation.

**(iii)** Find its frequency and amplitude.

**(iv)** Give the initial phase at the origin.

**(v) **Calculate the smallest distance between two adjacent crests in the wave.

[X and y are in cm and t in seconds. Assume the left to right direction as the positive direction of x]

Asked by Pragya Singh | 1 year ago | 140

Given,

y(x, t) =3 sin (36t +0.018x +\( \dfrac{ π}{4}\)) . . . . . . . . . . ( 1 )

**(i)** We know, the equation of a progressive wave travelling from right to left is:

y (x, t) = a sin (ωt + kx + Φ) . . . . . . . . . . . . ( 2 )

Comparing equation ( 1 ) to equation ( 2 ), we see that it represents a wave travelling from right to left and also we get:

a = 3 cm, ω = 36 rad/s , k = 0.018 cm and ϕ = \( \dfrac{ π}{4}\)

**(ii)**Therefore, the speed of propagation ,

v = \( \dfrac{ω}{k}\)=\( \dfrac{36}{0.018 }\) = 20 m/s

**(iii)** Amplitude of the wave, a = 3 cm

Frequency of the wave v =\(
\dfrac{ω}{2π}\)

= \(\dfrac{36}{2π}\) = 5.7 hz

**(iv)** Initial phase at the origin = \( \dfrac{ π}{4}\)

**(v)** the smallest distance between two adjacent crests in the wave,

λ = \( \dfrac{ 2π}{k}\) = \( \dfrac{ 2π}{0.018}\)= 349 cm

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