A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36 t + 0.018 x + 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? (d) What is the least distance between two successive crests in the wave?

Question

Philoid · Accepted Answer

Given:

Wave function, y = 3.0 sin (36 t + 0.018 x + π/4) …(1)

(a) The equation of a travelling wave is given by,

y(x,t) = a× sin(wt + kx + θ) …(2)

The given equation (1) is in the form of equation (2),

By comparing equation (1) and equation (2), we get that,

k = 0.018 m^-1

w = 36 rad/sec

We know that,

v = w/2π …(3)

Where,

v = frequency

w = angular velocity

λ = 2π/k …(4)

velocity, V = vλ …(5)

By combing equation (3) and (4) we get,

v = w/k

v = 36rads^-1/0.018m^-1

v = 20ms^-1

The velocity of the wave is 20 m/s.

(b) Amplitude a, of the given wave can be found by comparing equation 1 and 2.

a = 3cm = 0.03 m

Frequency, v = w/2π

v = 35rad s^-1/(2× 3.14)

v = 573 Hz

(c) Initial phase angle can be found by comparing equation (1) and (2),

θ = π/4

(d) The distance between consecutive crests or troughs is the wavelength of the wave, it is given by,

λ = 2π/k

λ = 2π/0.018m^-1

λ = 3.49 m