Question

A transverse harmonic wave on a string is described by y(x,t) = 3.0 sin (36t + 0.018x + π/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 a 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) It is travelling wave which is propagating from right to left. Comparing the given equation with,

y(x,t) = γ sin(ωt + kx + ϕ)

we get γ = 3 cm

 ω = 36 rad s^-1

k = 2π/λ = 0.018 cm^-1

and ϕ = π/4

The speed of the wave,

v = λv = {2πv}/{2π/λ} = ω/k = 36/0.018 = 2000 cm s^-1

= 20 ms^-1

(b) Amplitude of the wave,

γ = 3 cm

and frequency, v = ω/2π = {36}/{2 x 3.14} = 5.73 Hz

(c) Initial phase

ϕ = π/4 rad

(d) Least distance between two successive crest = wavelength

= λ = 2π/k = {2 x 3.14}/{0.018} cm = 3.49 x 10² cm = 3.49 m