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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 place, (b) if the two waves were produced 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 of 2.0 mm, what would be the amplitudes of the resultant waves in part (a) and (b)?
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.
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