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

##### Solution :-

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

Answered by Abhisek | 1 year ago

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