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:

\(k=\frac{2\pi}{λ}\)

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