Question

A sinusoidal wave traveling in the positive direction of x on a stretched membrane string has amplitude 2.0 cm, Wavelength 1m, and wave velocity 5.0m/s. At x=0 and t=0. It is given that displacement y=0 and ${\displaystyle \frac{\partial y}{\partial x}}<0$. Express the wave function correctly in the form $\mathrm{y}=\mathrm{f}(\mathrm{x},\mathrm{t})$:-
A. $\mathrm{y}=(0.04\mathrm{m})\left[\sin (\pi {\mathrm{m}}^{-1})x-(10\pi {\mathrm{s}}^{-1})\mathrm{t}\right]$
B. $\mathrm{y}=(0.02\mathrm{m})\mathrm{c}\mathrm{o}\mathrm{s}2\pi (x-5t)$
C. $\mathrm{y}=(0.02\mathrm{m})\left[\sin (2\pi {\mathrm{m}}^{-1})x-(10\pi {\mathrm{s}}^{-1})\mathrm{t}\right]$
D. $\mathrm{y}=(0.02\mathrm{m})\mathrm{c}\mathrm{o}\mathrm{s}\pi (x-5t+{\displaystyle \frac{1}{4}})$

Answer 1

Hint: A wave traveling in the positive direction along x from one point to another point of a medium is known as a progressive wave or a traveling wave. When a progressive wave travels in a medium, then the medium's particles vibrate in the same way. Still, the vibration phase changes from one particle to particle at any instant.
If $\varphi$ be the initial phase angle, then the progressive wave traveling along the positive x-axis direction is represented as:
$\mathrm{y}=\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}(\mathrm{k}\mathrm{x}-\omega \mathrm{t}+\varphi )$

Complete step by step answer:
Now according to the question:
The amplitude of the wave is given as $\mathrm{A}=2.0\mathrm{c}\mathrm{m}=0.02\mathrm{m}$
Wavelength is given as $\lambda =1\mathrm{m}$
Now the wave number $k={\displaystyle \frac{2\pi }{\lambda }}=2\pi {\mathrm{m}}^{-1}$
And the angular frequency $\omega =vk=5m/s\times 2\pi \mathrm{m}{\mathrm{s}}^{-1}=10\pi \mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{-1}$
Therefore, we put the values in the above equation with coordinates x and t:
$\Rightarrow \mathrm{y}(\mathrm{x},\mathrm{t})=(0.02)\mathrm{s}\mathrm{i}\mathrm{n}\left[2\pi (x-5.0t)+\varphi \right]$
We have given that for x=0 and t=0,
$\mathrm{y}=0\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{\delta y}{\delta x}}<0$
$\Rightarrow -0.02\mathrm{s}\mathrm{i}\mathrm{n}\varphi =0(\mathrm{a}\mathrm{s}\mathrm{y}=0)$
$\therefore -0.2\pi \cos \varphi <0$
From these conditions, we include that,
$\varphi =2\mathrm{n}\pi \text{ where n = 0,2,4,6}......$
Therefore, $\mathrm{y}=(0.02\mathrm{m})\left[\sin (2\pi {\mathrm{m}}^{-1})x-(10\pi {\mathrm{s}}^{-1})\mathrm{t}\right]$

Hence, the correct option is (C).

Note:
In SI, the unit of propagation constant or angular wave number (k) radian /meter. The Dimensional formula for the angular wave number is $\left[{\mathrm{M}}^0{\mathrm{L}}^{-1}{\mathrm{T}}^0\right]$ . Also, progressive wave traveling along the positive x-axis with a speed $v$ can be represented as $\mathrm{y}=\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{2\pi }{\lambda }}(vt-x)$