Answer to Question #195545 in Mechanics | Relativity for Mohammad Balhas

Question

Two transverse sinusoidal waves combining in a string are described by the wave functions y1 = 0.02 sin⁡(2πx+πt) and y2 = 0.02 sin(2πx-πt), where x and y are in meters, and t is in seconds. If after superposition, a standing wave of 3 loops is formed, then the length of the string is:

2 m

1.5 m

0.75 m

0.5 m

1 m

Answer 1

$y_1=0.02\sin(2\pi x+\pi t)$

$y_2=0.02\sin(2\pi x-\pi t)$

Resultant wave, $y=y_1+y_2$

$y=0.04\sin(2\pi x)\cos(\pi t)$

Comparing this with standard equation,

$y=A\sin(kx)\cos(\omega t)$

$k=2\pi$

Also,

$k=2\pi\lambda$

$\lambda=1$

Number of loops, $n=3$

Length of string = $L$

Number of loops, $n=2L\lambda$

$L=\dfrac{n}{2\lambda}$

$L=\dfrac{3}{2}=1.5\space m$

Length of string = 1.5 m