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 πt)

where x and y are in m and t is s. The length of the string is 1.5 m and its mass is 3.0 x 10-2 kg.

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 direction. What are the wavelength, frequency and speed of each wave?

(c) Determine the tension in the string.

Answer 1

(a) The function does not represent a travelling wave. It represents a stationary wave.

(b) Using the relation,

2 sin A cos B = sin(A + B) + sin(A - B)

y(x,t) = 0.3 x 2 sin(2x/3 x) cos(120 πt)

We have,

Comparing the equation with,

We get, γ = 0.3 m

2π/λ = 2π/3 ⇒ λ = 3 cm

and 2π/λ v = 120 π

⇒ v = 60 λ

= 60 x 3 = 180 ms^-1

Also frequency,

ν = v/λ = 180/3 = 60 Hz

(c) Tension in the string,

T = mν²

Here, m = mass/length

= {3 x 10^-2}/{1.5} = 2 x 10^-2 kg m^-1

Hence, T = 2 x 10^-2 x (180)²

= 648 N