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 $x=\frac{3λ}{4}$

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