Question

For the travelling harmonic wave

y(x,t) = 2.0 cos 2π (10t + 0.0080x + 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

Comparing the equation given in the question, with the standard equation of travelling wave:

we get, λ = 2 cm

v/λ = 10 Hz

1/λ = 0.008, and ϕ₀ = 0.35 rad

Hence, λ = 1/0.008 = 125 cm

The phase difference between oscillatory motion of two points separated by a distance Δx is given by

Δϕ = 2π/λ Δx   (in magnitude)

(a) Here, Δx = 4 m = 400 cm

Therefore, Δϕ = 2π/125 x 400 = 6.4 π rad

(b) Here, Δx = 0.5 m = 50 cm

Therefore, Δϕ = 2π/125 x 50 = 0.8 π rad

(c) Here, Δx = λ/2 = 125/2 cm

Therefore, Δϕ = 2π/125 x 125/2 = π rad

(d) Here, Δx = 3λ/2 = {3 x 125}/{4} cm

Therefore, Δϕ = 2π/125 x {3 x 125}/{4}

= 3/2 π rad.