Question

A transverse harmonic wave on a string is described by

y(x, t) = 3.0 sin (36 t + 0.018 x + $\frac{π}{4}$)

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 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 ?

Answer 1

Yes; Speed = 20 m/s, Direction = Right to left

3 cm; 5.73 Hz

$\frac{\pi}{4}$

3.49 m

** Explanation: **

The equation of a progressive wave travelling from right to left is given by the displacement function:

y (x, t) = a sin (ωt + kx + Φ) … (i)

The given equation is:

$y(x,t)=3.0 sin(36t+0.0118x+\frac{\pi}{4})....(ii)$

On comparing both the equations, we find that equation (ii) represents a travelling wave, propagating from right to left.

Now, using equations (i) and (ii), we can write:

ω = 36 rad/s and k = 0.018 m–1

We know that:

$v=\frac{ω }{2\pi}$ and $λ=\frac{2\pi}{k}$

Also

v=vλ

$∴ v=(\frac{ω}{2\pi})×(\frac{2\pi}{k})=\frac{ω}{k}$

$=\frac{36}{0.018}=2000\,cm/s=20\,m/s$

Hence, the speed of the given travelling wave is 20 m/s.

Amplitude of the given wave, a = 3 cm

Frequency of the given wave:

$v=\frac{ω}{2\pi}=\frac{36}{2×3.14}=5.73\,Hz$

On comparing equations (i) and (ii), we find that the initial phase angle, $ϕ=\frac{\pi}{4}$

The distance between two successive crests or troughs is equal to the wavelength of the wave.

Wavelength is given by the relation:

$k=\frac{2\pi}{λ}$

$∴ λ\frac{2\pi}{k}=\frac{2×3.14}{0.018}=348.89\,cm=3.49\,m$