Answer the following in Brief

Question

Derive an expression for equation of stationary wave on a stretched string. Show that the distance between two successive nodes or antinodes is / 2.

Vidyadip · Accepted Answer

When two progressive waves having the same amplitude, wavelength and speed propagate in opposite directions through the same region of a medium, their superposition under certain conditions creates a stationary interference pattern called a stationary wave.
Consider two simple harmonic progressive waves of equal amplitudes
$(a)$ and wavelength $(\lambda)$ propagating on a long uniform string in opposite directions $($remember $2 \pi / \lambda=k$ and $2 \pi n=\omega ).$
The equation of wave travelling along the $x-$axis in the positive direction is
$y _1=a \sin \left\{2 \pi\left(n t-\frac{x}{2}\right)\right\} ....(1)$
The equation of wave travelling along the $x-$axis in the negative direction is
$y _2=a \sin \left\{2 \pi\left(n t+\frac{x}{\lambda}\right)\right\} ....(2)$
When these waves interfere, the resultant displacement of particles of string is given by the principle of superposition of waves as
$y=y_1+y_2$
$y=a \sin \left\{2 \pi\left(n t-\frac{x}{\lambda}\right)\right\}+a \sin \left\{2 \pi\left(n t+\frac{x}{\lambda}\right)\right\}$
Using the trigonometrical identity,
$\sin C+\sin D=2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$,
we get $y =2 a \sin (2 \pi n t) \cos \frac{2 \pi x}{\lambda}$
$y =2 a \cos \frac{2 \pi x}{\lambda} \sin (2 \pi n t)$ or,$ ....(3)$
Using $2 a \cos \frac{2 \pi x}{\lambda}= A$ in equation $3,$
we get $y=A \sin (2 \pi n t)$
As $\omega=2 \pi n$,
we get, $y=A \sin \omega t$.

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