Two waves, each having a frequency of 100 Hz and a wavelength of 2.0 cm, are travelling in the same direction on a string. What is the phase difference between the waves (a) if the second wave was produced 0.015 s later than the first one at the same instant but the first one was produced a distance 4.0 cm behind the second one? (c) If each of the waves has an amplitude second one? (c) If each of the waves has an amplitude of 2.0 mm, what would be the amplitude of the resultant waves in part (a) and (b)?

Sol. f = 100 Hz, λ = 2 cm = 2 * 10 base 2 m ∴ wave speed, v = f λ = 2 m/s a) in 0.015 sec 1^ST wave has travelled x = 0.015 * 2 = 0.03 m = path diffn ∴ corresponding phase difference, ∅ = 2π*/λ = {2π/ (2 * 10^2)} * 0.03 = 3π. b) Path different x = 4 cm = 0.04 m ⇒ ∅ = (π/λ)x = {(2π/2 * 10^2) * 0.04} = 4 π. C) The waves have same frequency, same wavelength and same amplitude Let, Y base 1 = r sin wt y base 2 = r sin (wt + ∅) ⇒ Y = Y base 1 + y base 2 = [r sin wt + (wt + ∅) = 2r sin (wt + ∅/2) cos (∅/2) ∴ resultant amplitude = 2r cos ∅/2 So when ∅ = 3π, r = 2 * 10^ 3 m R base res = 2 * (2 * 10^3) cos (3π/2) = 0 Again, when ∅/2π, R base res = 2 * (2 * 10^3) cos (4π/2) = 4mm.