MCQ
Two waves ${y_1} = {A_1}\sin (\omega t - {\beta _1})$, ${y_2} = {A_2}\sin (\omega t - {\beta _2})$ Superimpose to form a resultant wave whose amplitude is
  • $\sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos ({\beta _1} - {\beta _2})} $
  • B
    $\sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\sin ({\beta _1} - {\beta _2})} $
  • C
    ${A_1} + {A_2}$
  • D
    $|{A_1} + {A_2}|$

Answer

Correct option: A.
$\sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos ({\beta _1} - {\beta _2})} $
a
(a) Phase difference between the two waves is

$\phi = (\omega t - {\beta _2}) - (\omega t - {\beta _1}) = ({\beta _1} - {\beta _2})$

Resultant amplitude $A = \sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos ({\beta _1} - {\beta _2})} $

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