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) \(\frac{λ}{2}\), 

(d) \(\frac{3λ}{4}\)

Answer 1

Equation for a travelling harmonic wave is given as: 

y (x, t) = 2.0 cos 2π (10t – 0.0080x + 0.35) 

= 2.0 cos (20πt – 0.016πx + 0.70 π) 

Where, 

Propagation constant, k = 0.0160 π 

Amplitude, a = 2 cm 

Angular frequency, ω= 20 π rad/s 

Phase difference is given by the relation: 

\(Φ=ky=\frac{2\pi}{λ}\)

For x = 4 m = 400 cm 

Φ = 0.016 π × 400 = 6.4 π rad

For 0.5 m = 50 cm 

Φ = 0.016 π × 50 = 0.8 π rad

For \(x=\frac{λ}{2}\)

\(ϕ=\frac{2\pi}{λ}×\frac{λ}{2}=\pi\,rad\)

For \(x=\frac{3λ}{4}\)

\(ϕ=\frac{2\pi}{λ}×\frac{3λ}{2}=\,1.5\,\pi\,rad\)