Question

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

Answer 1

(a)

The general equation of a wave travelling from right to left is,

y( x,t )=Asin( kx+ωt+ϕ )(1)

Here, the amplitude of the wave is A, its angular velocity is ω, its position is x and its phase angle is ϕ.

The provided equation of a transverse harmonic wave on the string is,

y( x,t )=3.0sin( 36t+0.018x+ π 4 )(2)

Compare equation (1) with equation (2) to find the amplitude of the wave.

A=3.0 cm

Compare equation (1) with equation (2) to find the angular velocity of the wave.

ω=36  rad/s

Compare equation (1) with equation (2) to find the displacement of the wave.

k=0.018  cm −1

Compare equation (1) with equation (2) to find the phase angle of the wave.

ϕ= π 4

The equation to find the value of the speed of wave propagation is,

v= ω k

Substituting the values in the above equation, we get:

v= 36  rad/s 0.018× 10 −2   m −1 1  cm −1 =2000( 10 −2 ) m/s =20 m/s

Thus, the speed of the travelling wave is 20 m/s moving from right to left.

(b)

The amplitude of the wave is,

A=3.0 cm =3.00× 1 m 100 cm =0.03 m

The formula to determine the frequency of the wave is,

v= ω 2π

Substituting the value in the above equation, we get:

v= 36 2π =5.7 Hz

Thus, the amplitude of the wave is 0.03 m and the frequency of the wave is 5.7 Hz.

(c)

The phase angle is,

ϕ= π 4

Thus, the phase angle of the wave is π 4 .

(d)

The equation for the least distance between two successive crests in the wave is,

λ= 2π k

Substituting the value in the above equation, we get:

λ= 2π 0.018 =349 cm× 1 m 100 cm =3.5 m

Thus, the wavelength of the wave is 3.5 m.