A wave y = a cos (kx + omega t) superimposes | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) stationary wave is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.Given the first w

Vedclass · Accepted Answer

$ - a \cos (kx - \omega t)$

Vedclass · Answer

(C) stationary wave is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.Given the first wave is $y_1 = a \cos (kx + \omega t)$.For a stationary wave to have a node at $x = 0$,the resultant displacement $y = y_1 + y_2$ must be zero at $x = 0$ for all $t$.So,$y_1(0, t) + y_2(0, t) = 0$.$a \cos (\omega t) + y_2(0, t) = 0$,which implies $y_2(0, t) = -a \cos (\omega t)$.Since the waves must travel in opposite directions,if the first wave is traveling in the negative $x$-direction (as indicated by $kx + \omega t$),the second wave must travel in the positive $x$-direction,taking the form $y_2 = a' \cos (kx - \omega t + \phi)$.At $x = 0$,$y_2(0, t) = a' \cos (- \omega t + \phi) = a' \cos (\omega t - \phi)$.Comparing this with $-a \cos (\omega t)$,we get $a' = a$ and $\phi = \pi$.Thus,$y_2 = a \cos (kx - \omega t + \pi) = -a \cos (kx - \omega t)$.

$A$ wave $y = a \cos (kx + \omega t)$ superimposes on another wave to form a stationary wave having a node at $x = 0$. What is the equation of the other wave?

Similar Questions

$A$ stationary wave is formed with $3$ nodes along the length of a string of $90 \ cm$. The wavelength of the wave is: (in $cm$)

$A$ wave $y = a \cos(kx - \omega t)$ superposes on another wave to form a stationary wave having a node at $x = 0$. What is the equation of the other wave?

$A$ string of mass $0.1 \ kg \ m^{-1}$ has length $0.9 \ m$. It is fixed at both ends and stretched such that it has a tension of $40 \ N$. The string vibrates in three segments with amplitude $0.3 \ cm$. The amplitude (maximum) of the particle velocity is (in $m/s$):

When a wave propagating on a string gets reflected from a fixed rigid support,what is the change in its phase?

Two travelling waves produce a standing wave represented by the equation,
${y} = 1.0 \, \text{mm} \cos(1.57 \, \text{cm}^{-1} x) \sin(78.5 \, \text{s}^{-1} t)$
The node closest to the origin in the region ${x} > 0$ will be at ${x} = \dots \, \text{cm}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module