Name: Waves and Their Phases
Uploaded: 2019-07-19
Duration: 6 min 51 s
Description: In this video, we will learn how two or more waves can be studied in terms of their phases. Equations of two waves and graphs of two waves. Learn more!

Question

Prove analytically that the distances between two consecutive nodes and antinodes in a stationary wave are equal and each is equal to half the wavelength of the wave.

Accepted Answer

Consider two sines waves of equal amplitude and frequency moving in opposite direction along the same string. The equation of two waves are given.

$y_{1} = A \sin (ω t - k x)$

$y_{2} = A \sin (ω t + k x)$

When these two waves superimpose, the resultant wave called standing wave is given by,

$y = y_{1} + y_{2}$

$\Rightarrow y = A \sin (ω t - k x) + A \sin (ω t + k x)$

$\Rightarrow y = 2 A \sin ω t \cos k x$

$\Rightarrow y = 2 A \cos k x \sin ω t$

This equation implies that each particle of the string vibrates in simple harmonic motion with amplitude $2 A \cos k x$ .

There are some point which have zero amplitude, and they are always stationary. These points are called nodes.

Position of node will be points where,

$\cos k x = 0$

$\Rightarrow k x = n π$ where $n$ is integer.

$\Rightarrow x = \frac{n λ}{2}$ where $λ$ is wavelength of wave and $k = \frac{2 π}{λ}$

Thus position of nodes will be $0, \frac{λ}{2}, λ, . . . . .$

Distance between each consecutive nodes is $\frac{λ}{2}$ .

Anti nodes are the points with maximum amplitude. They are given by,

$\cos k x = 1$

$\Rightarrow k x = (n + \frac{1}{2}) π$

$\Rightarrow x = (n + \frac{1}{2}) \frac{λ}{2}$

thus position of anti nodes are $\frac{λ}{4}, \frac{3 λ}{4}, \frac{5 λ}{4}, . . . . . . . .$

Distance between any two consecutive anti node will be $\frac{λ}{2}$

Prove analytically that the distances between two consecutive nodes and antinodes in a stationary wave are equal and each is equal to half the wavelength of the wave.

Important Questions on Waves

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