Question 73: Waves

Question

Two waves are simultaneously passing through a string and their equations are :
y
1
= A
1
sin k(x $$-$$ vt), y
2
= A
2
sin k(x $$-$$ vt + x
0
). Given amplitudes A
1
= 12 mm and A
2
= 5 mm, x
0
= 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 : y 1 = A 1 sin k(x $- vt), y 2 = A 2 sin k(x - vt + x 0 ). Given amplitudes A 1 = 12 mm and A 2 = 5 mm, x 0 = 3.5 cm and wave number k = 6.28 cm -$1 . The amplitude of resulting wave will be ................ mm.

Integer / Numeric Answer Type