Name: Longitudinal Standing Waves
Uploaded: 2019-07-19
Duration: 13 min 37 s
Description: We have already studied standing waves, but do you know standing waves can be formed in longitudinal waves also, watch this video to know more.

Question

Explain the formation of stationary waves by the analytical method. Show that nodes and antinodes are equally spaced in stationary waves.

Accepted Answer

Consider two simple harmonic progressive waves having same amplitude ( $a$ ), frequency ( $n$ ) and period ( $T$ ) travelling along same medium in opposite direction. The wave along positive direction of $X$ -axis is given by :

$Y_{1} = a s i n 2 π (n t - \frac{x}{λ})$ .....(i)

$Y_{2} = a s i n 2 π (n t + \frac{x}{λ})$ .....(ii)

Using the principle of super position the resultant displacement will be:

$Y = Y_{1} + Y_{2}$

$Y = a s i n 2 π (n t - \frac{x}{λ}) + a s i n 2 π (n t + \frac{x}{λ}) ∴ Y = 2 a s i n \frac{2 π}{2} (n t - \frac{x}{λ} + n t + \frac{x}{λ}) . c o s \frac{2 π}{2} (n t - \frac{x}{λ} - n t - \frac{x}{λ})$

$Y = 2 a s i n 2 πnt . c os 2 π (\frac{- x}{λ}) Y = [2 a \cos \frac{2 πx}{λ}] . \sin 2 πnt Y = A . \sin (2 πnt)$

The points of medium which vibrates with minimum or zero amplitude are called nodes.

The points of medium which vibrates with maximum amplitude are called antinodes.

For nodes put $A = 0$

$2 a \cos \frac{2 πx}{λ} = 0 \cos \frac{2 πx}{λ} = 0 \frac{2 πx}{λ} = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2} ∴ x = \frac{λ}{4}, \frac{3 λ}{4}, \frac{5 λ}{4}$

For antinodes:

$A = \pm 2 a$

$2 a \cos \frac{2 πx}{λ} = \pm 2 a \cos \frac{2 πx}{λ} = \pm 1 \frac{2 πx}{λ} = 0, π, 2 π$

$∴ x = 0, \frac{λ}{2}, λ$

Thus the distance between two successive nodes and antinodes and nodes equal to $\frac{λ}{2}$ . Therefore, antinodes and nodes are equally spaced.

Explain the formation of stationary waves by the analytical method. Show that nodes and antinodes are equally spaced in stationary waves.

Important Questions on Stationary Waves

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