Question

For the travelling harmonic wave y(x, t) = 2.0 cos 2π (10t – 0.0080 x + 0.35) where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (a) 4 m, (b) 0.5 m, (c) λ /2, (d) 3λ /4

Answer 1

The given equation of the travelling harmonic wave is,

y( x,t )=2.0cos2π( 10t−0.0080x+0.35 )

Here, the time taken is t and the distance covered is x.

Modifying the given equation we get:

y( x,t )=2.0cos( 2π( 0.0080 )( 10 0.0080 t−x )+0.7π )

On comparing the given equation with the standard equation of a travelling harmonic wave, we get:

2π λ =2π( 0.0080 ) λ= 1 0.0080  cm =125 cm

a)

The phase angle between the oscillatory motions of two points separated by a distance Δx is given by the equation,

Δϕ= 2π λ Δx

Substituting the values in the above equation, we get:

Δϕ= 2π 125 ( 4 m× 100 cm 1 m ) =6.4π rad

Thus, the phase angle is 6.4π rad.

b)

The phase angle between the oscillatory motions of two points separated by a distance Δx is given by the equation,

Δϕ= 2π λ Δx

Substituting the values in the above equation, we get:

Δϕ= 2π 125 ( 0.5 m× 100 cm 1 m ) =0.8π rad

Thus, the phase angle is 0.8π rad.

c)

The phase angle between the oscillatory motions of two points separated by a distance Δx is given by the equation,

Δϕ= 2π λ Δx

Substituting the values in the above equation, we get:

Δϕ= 2π λ ( λ 2 ) =π

Thus, the phase angle is π rad.

d)

The phase angle between the oscillatory motions of two points separated by a distance Δx is given by the equation,

Δϕ= 2π λ Δx

Substituting the values in the above equation, we get:

Δϕ= 2π λ ( 3λ 4 ) = 3π 2 =π+ π 2

The phase angle between the two points is advanced by π. Therefore, the phase angle is π 2  rad.