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

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 - x λ ) .....(i) Y 2 = a s i n 2 π ( n t + 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 - x λ ) + a s i n 2 π ( n t + x λ ) ∴ Y = 2 a s i n 2 π 2 ( n t - x λ + n t + x λ ) . c o s 2 π 2 ( n t - x λ - n t - x λ ) Y = 2 a s i n 2 πnt . c os 2 π ( - x λ ) Y = [ 2 a cos 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 2 πx λ = 0 cos 2 πx λ = 0 2 πx λ = π 2 , 3 π 2 , 5 π 2 ∴ x = λ 4 , 3 λ 4 , 5 λ 4 For antinodes: A = ± 2 a 2 a cos 2 πx λ = ± 2 a cos 2 πx λ = ± 1 2 πx λ = 0 , π , 2 π ∴ x = 0 , λ 2 , λ Thus the distance between two successive nodes and antinodes and nodes equal to λ 2 . Therefore, antinodes and nodes are equally spaced.

Show that even as well as odd harmonics are present as overtones in the case of an air column vibrating in a pipe open at both the ends.

In fundamental mode, the vibration of air column open at both ends, there are two anti-nodes and one node as shown in the figure. Let λ be the wavelength of the wave, v be the velocity and L be the length. Now, as we can see L = λ 2 or λ = 2 L and n = v λ = v 2 L , this is the frequency of fundamental mode in a pipe open at both the ends. Now for 1 s t harmonic let λ 1 be the wavelength, from the figure, L = λ 1 and n 1 = v λ 1 = v L ∴ n 1 = 2 n Now for 2 n d harmonic let λ 2 be the wavelength, from the figure, L = 3 λ 2 2 and n 2 = v λ 2 = 3 v 2 L ∴ n 2 = 3 n Therefore, n , 2 n , 3 n ..... are the harmonics which are both even as well as odd.

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 8 - Stationary Waves>Maharashtra Board-2020>Q 7

HARD

12th Maharashtra Board

IMPORTANT

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Show that all harmonics are present in case of an air column vibrating in a pipe open at both ends.

Important Questions on Stationary Waves

HARD

12th Maharashtra Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 8 - Stationary Waves>Maharashtra Board-2018>Q 1

In stationary wave, the distance between a node and its adjacent anti-node is____.

HARD

12th Maharashtra Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 8 - Stationary Waves>Maharashtra Board-2019>Q 5

A stretched sonometer wire is in unison with a tuning fork. When the length of the wire is increased by 5%, the number of beats heard per second is 10. Find the frequency of the tuning fork.

MEDIUM

12th Maharashtra Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 8 - Stationary Waves>Maharashtra Board-2020>Q 6

A standing wave is produced on a string fixed at one end and free at other. The length of string must be an _______.

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12th Maharashtra Board

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 8 - Stationary Waves>Maharashtra Board-2019>Q 9

A sonometer wire 1 metre along weighing