Question

The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin (2π/3)x ( cos (120 π t) 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 ×10¯²kg. Answer the following : (a) Does the function represent a travelling wave or a stationary wave? (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ? (c) Determine the tension in the string.

Answer 1

Given, the length of the string is 1.5 m and the mass of the string is 3× 10 −2  kg.

a)

The general equation of a travelling wave is,

y( x,t )=Asinω( x±vt )

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The general equation of a stationary wave is,

y( x,t )=Asinωtsin nπx L

Here, the amplitude of the wave is A, it’s angular velocity is ω, it’s position is x and it’s velocity is v.

The given equation of the transverse displacement of the string is,

y( x,t )=0.06sin( 2 3 πx )cos( 120πt )

As the given equation is similar to the standard standing wave equation, so the given wave is a stationary wave.

b)

The general equation of a wave travelling along the +direction of the x axis is,

y 1 =rsin 2π λ ( vt−x )

Here, the amplitude of the wave is r, it’s wavelength is λ, it’s position is x and it’s velocity is v.

The stationary wave which is superimposed by a reflective stationary wave traveling in the opposite direction is given by the equation,

y 2 =−rsin 2π λ ( vt+x )

Here, the amplitude of the wave is r, it’s wavelength is λ, it’s position is x and it’s velocity is v.

The stationary wave that is formed by the superposition of two waves has the following equation:

y= y 1 + y 2

Substituting the values in the above equation, we get:

y=rsin 2π λ ( vt−x )−rsin 2π λ ( vt+x ) =r( sin 2π λ ( vt−x )−sin 2π λ ( vt+x ) ) =−r( sin 2π λ ( vt+x )−sin 2π λ ( vt−x ) ) =−2rsin 2π λ xcos 2π λ ( vt ) (1)

The provided equation of the transverse displacement of the string is,

y( x,t )=0.06sin( 2 3 πx )cos( 120πt )(2)

Comparing equation (1) and equation (2) to find the value of wavelength λ,we get:

2π λ = 2π 3 λ=3 m

Comparing equation (1) and equation (2) to find the value of velocity v, we get:

2π λ v=120π v=60( 3 ) =180 m/s

Comparing equation (1) and equation (2) to find the value of the frequency of the wave f,we get:

f= v λ = 180 3 =60 Hz

Thus, both the waves have the same wavelength, frequency and speed.

c)

The formula to calculate the velocity of a transverse wave is,

v= T M l v 2 = Tl M T= v 2 M l

Substituting the values in the above equation, we get:

T= ( 180 ) 2 ( 3× 10 −2 ) 1.5 = ( 180 ) 2 ( 2× 10 −2 ) =648 N

Thus, the tension in the string is 648 N.