Question

A wave $\mathrm{y}=\mathrm{a}\sin (\mathrm{\omega t}-\mathrm{kx})$ on a string meets with another wave producing a node at x = 0. The equation of the unknown wave is

• A. $\mathrm{y}=\mathrm{a}\sin (\mathrm{\omega t}+\mathrm{kx})$
• B. $\mathrm{y}=\mathrm{a}\cos (\mathrm{kx}-\mathrm{\omega t})$
• C. $\mathrm{y}=-\mathrm{a}\cos (\mathrm{kx}-\mathrm{\omega t})$
• D. $\mathrm{y}=-\mathrm{a}\sin (\mathrm{\omega t}+\mathrm{kx})$

Answer 1

The correct option is D

$\mathrm{y}=-\mathrm{a}\sin (\mathrm{\omega t}+\mathrm{kx})$

Step 1: Given parameter

$\mathrm{y}=\mathrm{a}\sin (\mathrm{\omega t}-\mathrm{kx})...\left(1\right)$

Step 2: Find the unknown wave

As the above wave meets with an unknown wave producing a node at $\mathrm{x}=0$, i.e after the two waves superimpose, the amplitude of the resultant wave is zero.

In order for the unknown wave to superimpose with the incident wave, its direction must be opposite to the incident wave and there must be a phase difference of $\mathrm{\pi }$.

Consider the equation of unknown wave as follows:

$\mathrm{y}=\mathrm{a}\sin (\mathrm{\omega t}+\mathrm{kx}+\mathrm{\pi })...\left(2\right)$

$\mathrm{y}=-\mathrm{a}\sin (\mathrm{\omega t}+\mathrm{kx})\left[\because \sin \left(\mathrm{\theta }+\mathrm{\pi }\right)=-\mathrm{sin\theta }\right]$

Hence, the correct option is (D), $\mathrm{y}=-\mathrm{a}\sin (\mathrm{wt}+\mathrm{kx})$