What are stationary waves? Obtain their equation.

(N/A) Stationary waves: When two waves of the same amplitude,frequency,and wavelength travel in opposite directions along the same line and superpose,the resultant wave formed is called a stationary wave.
These resultant waves do not propagate in any direction; hence,they do not transport energy.
To obtain the equation,consider two waves of the same amplitude $a$ and angular frequency $\omega$ traveling in opposite directions:
$1$. Wave traveling in the positive $x$-direction: $y_{1}(x, t) = a \sin(kx - \omega t)$
$2$. Wave traveling in the negative $x$-direction: $y_{2}(x, t) = a \sin(kx + \omega t)$
By the principle of superposition,the resultant displacement $y$ is:
$y = y_{1} + y_{2}$
$y = a \sin(kx - \omega t) + a \sin(kx + \omega t)$
Using the trigonometric identity $\sin(A) + \sin(B) = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$:
$y = 2a \sin(kx) \cos(\omega t)$
This is the equation of a stationary wave,where $2a \sin(kx)$ represents the amplitude of the particle at position $x$.