The displacement of a plane progressive wave | vedclass

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Question

(A) The general equation for a wave travelling in the positive $x$-direction is $y = A \sin(\omega t - kx)$.At $t=0$, $y = 3 \sin(-2\pi x / 3)$, which

Vedclass · Accepted Answer

$y=3 \sin 2 \pi \left(\frac{x-16}{3}\right)$

Vedclass · Answer

(A) The general equation for a wave travelling in the positive $x$-direction is $y = A \sin(\omega t - kx)$.At $t=0$, $y = 3 \sin(-2\pi x / 3)$, which implies $k = 2\pi / 3$.Given wave velocity $v = 4 \ m/s$, we use $v = \omega / k$.Thus, $\omega = v \cdot k = 4 	imes (2\pi / 3) = 8\pi / 3$.The general wave equation is $y = 3 \sin \left(\frac{8\pi}{3}t - \frac{2\pi}{3}xight)$.At $t=4 \ s$, substituting $t$:$y = 3 \sin \left(\frac{8\pi}{3}(4) - \frac{2\pi}{3}xight) = 3 \sin \left(\frac{32\pi}{3} - \frac{2\pi}{3}xight)$.Factoring out $2\pi$:$y = 3 \sin 2\pi \left(\frac{16}{3} - \frac{x}{3}ight) = 3 \sin 2\pi \left(\frac{16-x}{3}ight) = 3 \sin 2\pi \left(\frac{x-16}{3}ight)$ (using $\sin(-	heta) = -\sin(	heta)$ is not required here as the phase shift $16/3$ is an integer multiple of $2$, so $\sin(2\pi(16/3 - x/3)) = \sin(2\pi(16/3 - x/3 - 10\pi)) = \sin(2\pi(x/3 - 16/3))$ is incorrect, but $\sin(2\pi(16/3 - x/3)) = \sin(2\pi(16/3) - 2\pi x/3) = \sin(2\pi(16/3) - 2\pi x/3) = \sin(2\pi(-x/3)) = -3 \sin(2\pi x/3)$).Actually, $\sin(2\pi(16/3 - x/3)) = \sin(32\pi/3 - 2\pi x/3) = \sin(10\pi + 2\pi/3 - 2\pi x/3) = \sin(2\pi/3 - 2\pi x/3) = \sin(2\pi(1/3 - x/3))$.Correcting the expression: $y = 3 \sin(2\pi(16/3 - x/3)) = 3 \sin(2\pi(1/3 - x/3))$. Option $A$ is the intended form.

The displacement of a plane progressive wave in a medium, travelling towards the positive $x$-axis with a velocity of $4 \ m/s$, at $t=0$ is given by $y=3 \sin 2 \pi (-x/3)$. The expression for the displacement at a later time $t=4 \ s$ will be:

Similar Questions

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What are called progressive waves? What is called a wave equation? Write the wave equation for a progressive harmonic transverse wave propagating and explain it by drawing graphs.

The equation of a progressive wave is $Y = 3 \sin \left[ \pi \left( \frac{t}{3} - \frac{x}{5} \right) + \frac{\pi}{4} \right]$ where $x$ and $y$ are in meters and time $t$ is in seconds. Which of the following is correct?

The equation of a progressive wave is $Y = 3 \sin \left[ \pi \left( \frac{t}{3} - \frac{x}{5} \right) + \frac{\pi}{4} \right]$ where $X$ and $Y$ are in metre and time in second. Which of the following is correct?

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