"Two waves are simultaneously passing through a string and their equations are : y1 = A1 sin k(x $-$ vt), y2 = A2 sin k(x $-$ vt + x0). Given amplitudes A1 = 12 mm and A2 = 5 mm, x0 = 3.5 cm and wave number k = 6.28 cm$-$1. The amplitude of resulting wave will be ................ mm."

Question

Accepted Answer

"y₁ = A₁ sin k(x $-$ vt)

y₁ = 12 sin 6.28 (x $-$ vt)

y₂ = 5 sin 6.28 (x $-$ vt + 3.5)

$\Delta \phi = {{2\pi } \over \lambda }(\Delta x)$

$= K(\Delta x)$

$= 6.28 \times 3.5 = {7 \over 2} \times 2\pi = 7\pi$

${A_{net}} = \sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \phi }$

${A_{net}} = \sqrt {{{(12)}^2} + {{(5)}^2} + 2(12)(5)\cos (7\pi )}$

$= \sqrt {144 + 25 - 120}$"

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Two waves are simultaneously passing through a string and their equations are : y1 = A1 sin k(x $-$ vt), y2 = A2 sin k(x $-$ vt + x0). Given amplitudes A1 = 12 mm and A2 = 5 mm, x0 = 3.5 cm and wave number k = 6.28 cm$-$1. The amplitude of resulting wave will be ................ mm.

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