Question

Two waves, travelling along the same direction, are given by
y1(x, t) = asin (w1t − k1x)
and y2 (x, t) = asin (w2t − k2x)
Suppose that the values of w1 and k1 are respectively slightly greater than w2 and
k2 . i) Obtain an expression for the resultant wave due to their superposition. ii)
Explain the formation of wave packet

Answer 1

Superposition of two waves given by,
y = y_1+y_2 …………………………(1)
Put the values of y_1 and y_2 in equation (1), we get,
y = a sin (ω_1 t − k_1 x) + a sin(ω_2 t − k_2 x)
= 2 a [ sin ω_1 t . cos k_1 x - cos ω_1 t . sin k_1 x + sin ω_2 t . cos k_2 x - cos ω_2 t . sin k_2 x ]
This is the resultant wave equation.
Here resultant amplitude is 2a.
Answer: The resultant waves formed group wave, amplitude is 2a and produces beats.