Question

a sinusoidal transverse progressive wave of amplitude 50mm, wavelength 200mm, is travelling along the positive direction of the x- axis with a velocity of 4m/s. At a given instant there is a crest of the wave at the origin.

Determine the displacement, velocity and direction of the motion of a particle 120mm to the right of the origin:

a) at a given instant

b) 1/50 s later

Answer 1

We know that v = f , is wavelength = 200 mm = 0.2 m

Velocity v = 4 m/sec , so frequency f = 4/0.2 = 20 Hz

Angular frequency = 2f = 40

Wave number k = 1/ = 1/0.2 = 5

Now,

If wave is traveling in positive x axis then general equation of wave is given by :

y = Acos(t - kx + ) here is initial phase

Putting the values at t= 0 , x = 0 ( y is crest i.e. y = +A)

A = A cos(0 + 0 + )

cos = 1 , = 0

So wave equation is

y = (50 mm)cos(40t - 5x)

Particle velocity = dy/dt = - (2 m)sin(40t - 5x)

(a) at given instant

At x = 0.12m and t = 0 sec

Displacement y = (50mm)cos(0 - 5*0.12) = 48 mm

Velocity V = - (2)sin(0-5*0.12) = 1.76 m/sec

Here particle velocity is positive it means it is moving upward.

(b)

At time t = 1/50 sec

Displacement y = (50mm)cos(40/50 - 5*0.12) = -47.05 mm

Velocity V = - (2)sin(40/50 - 5*0.12) = 2.13 m/sec

Here velocity is again positive so particle are moving upward.