Question

1. Two waves are traveling along a rope. The individual waves are de- scribed by hi(t, x) = h2(t, x) = 0.3 cos(8x – 4t) 0.3 cos(7.6x - 3.8t) (a) Write the superposition hi + h, in the form h(t, x) = Acos (į Aku - Awt) cos (Kx – wt) (b) What is the amplitude A of the combined wave? (c) What is the phase velocity of the combined wave? (d) What is the group velocity All of the combined wave?

Answer 1

Solution - Given that! @ h (t, x) = 0.3 Cos( 8x - 4t). hz (t, x) = 0.3 cos (7.6*- 3.8t) By theory of superposition : h (t, x) = halt, x) the (t, x) > h(t, x) = 0.3 cos (&x-4 t) +0.3 cos (7.6*- 3.8+) 7 h (dix) = 0.3{cos (8x-9t) + cas (7.6X-3.8 7 ) } r &x-at+7.6%-3.87 - ke(t, x) = 0.38 2 cos cos ( 8x-f t - 7.6x+3.8+) > h(t, x) - 0,6 cos (7.82-3.9+) 'Cos (0.2% - 0.10 h(t, x) = 0.6 cos( + (0.4)X - 1 (0.2)t) Cos (7.8x-3.94 about with hlt, x) = Acos (tok X-t swt) cas (kx-wit) A-0.6 DK=0.4, DW = 0.2 , w = 3.9 k=7.8 y comparing A-0.6 7

6 © A The value of amplitude is A=0.6 m. phase velocity phase velocity = 7.8 3.9 @ ok 0.2 2) phase velocity = £ mis Cloroup velocity Group velocity = Group velocity - I mis Thus the phase velocity & Group velocity Some for combined wave. are