Question 112: Waves

Question

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.

Accepted Answer

y
1
= A
1
sin k(x $$-$$ vt)
y
1
= 12 sin 6.28 (x $$-$$ vt)
y
2
= 5 sin 6.28 (x $$-$$ vt + 3.5)
$$\Delta \phi  = {{2\pi } \over \lambda }(\Delta x)$$
$$ = K(\Delta x)$$
$$ = 6.28 	imes 3.5 = {7 \over 2} 	imes 2\pi  = 7\pi $$
$${A_{net}} = \sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \phi } $$
$${A_{net}} = \sqrt {{{(12)}^2} + {{(5)}^2} + 2(12)(5)\cos (7\pi )} $$
$$ = \sqrt {144 + 25 - 120} $$

Waves

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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