Question

The maximum transverse velocity and maximum transverse acceleration of a harmonic wave in one-dimensional string are $1ms^{-1}$ and $1ms^{-2}$ respectively. The phase velocity of the wave is $1ms^{-1}$ . The waveform is
A. $\sin(x-t)$
B.$\sin(2x-t)$
C. $\sin (x-2t)$
D. $\sin\left(\dfrac{x}{2}-t\right)$
E.$\sin \left(x-\dfrac{t}{2}\right)$

Answer 1

Hint: To find the wave equation, we have first found the value for the various parameters which are involved in the wave equation , such as the amplitude , propagation constant angular velocity and the phase difference of the wave.
Formula used:
$y=A \sin(kx-\omega t+\phi)$

Complete step-by-step solution:
A wave is a propagation of disturbance, which results in the transfer of energy from one particle of the medium to the other. These undergo harmonic oscillation and are also called the sine waves. There are broadly two types of waves, namely the travelling wave and the standing wave. The standing waves are caused due to superimposition of two or more travelling waves.
We know that the wave equation is given as $y=A \sin(kx-\omega t+\phi)$ where, $y$ is the displacement of the wave due to amplitude $A$ , $k$ is the propagation constant at a distance $x$ from the origin , $\omega$ is the angular velocity at time $t$ and $\phi$ is the phase difference of the wave.
To begin with, let us consider that the phase difference is $\phi=0$ with respect to the origin.
Then, we also know that $v_{max}=\omega A$ and $a_{max}=\omega^2 A$ where, $v$ and $a$ is the velocity and acceleration of the wave.
Then, we have $\dfrac{v_{max}}{a_{max}}=\dfrac{\omega A}{\omega^2 A}$
Since, $v_{max}=1ms^{-1}$ and $a_{max}=1ms^{-2}$
$\implies \omega=1 rads^{-1}$
Then, the velocity $v=\dfrac{\omega}{k}$
$\implies k=\dfrac{\omega}{v}=1 rad\; m^{-1}$
Similarly, amplitude $A=\dfrac{v}{\omega}=1m$
Now substituting the values in the wave equation, we have
$y=1\sin(1x-1t+0)$
Thus, the correct answer is option A. $\sin(x-t)$

Note: The wave equation can either be described as a sine wave, as shown above or can also be described as a cosine wave. In this question, we have used almost all the formulas related to a wave equation, hence to solve this sum, one must know the relation between the various parameters of the wave equation.