Write the equation of a stationary wave and obtain the equations of nodes and antinodes by defining them.

(N/A) The equation of a stationary wave is given by:
$y(x, t) = 2a \sin(kx) \cos(\omega t)$ ... $(1)$
Here,the term $2a \sin(kx)$ represents the amplitude of the oscillation at a position $x$. Since the amplitude depends on $x$,different points on the string oscillate with different amplitudes.
Nodes: These are the points where the amplitude of oscillation is zero.
In equation $(1)$,for nodes,$2a \sin(kx) = 0$.
$\sin(kx) = 0 \implies kx = n\pi$,where $n = 0, 1, 2, 3, \dots$
Since $k = \frac{2\pi}{\lambda}$,we have $\frac{2\pi}{\lambda} x = n\pi$.
Therefore,$x = \frac{n\lambda}{2}$ for $n = 0, 1, 2, 3, \dots$
The distance between two consecutive nodes is $\frac{\lambda}{2}$.
Antinodes: These are the points where the amplitude of oscillation is maximum.
In equation $(1)$,for antinodes,$|\sin(kx)| = 1$.
$kx = (n + \frac{1}{2})\pi$,where $n = 0, 1, 2, 3, \dots$
$\frac{2\pi}{\lambda} x = (n + \frac{1}{2})\pi$.
Therefore,$x = (n + \frac{1}{2})\frac{\lambda}{2}$ for $n = 0, 1, 2, 3, \dots$