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SUMMARYStationary Waves

Superposition - When two waves meet, the total

displacement at a point is equal to the sum of the individual displacements at that point

The principle of

superposition

reinforcement

cancellation2

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Stationary Waves - a wave that remains in a constant

position. It is formed by the superposition of two waves with the same amplitude and frequency, travelling at the same speed, but in opposite directions.

Stationary wave

Notes:1. Stationary wave also known as standing wave.

2. Stationary waves are waves where there is a NO net transfer of energy from one point to another because the energy cannot flow past the nodes, which are permanently at rest.

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Nodes and Antinodes

- are points within a stationary wave that have the minimum amplitude.

Nodes

- are points within a stationary wave that have the maximum amplitude.

Antinodes

Standing waves have nodes and antinodes.

1. The distance between two successive NODES or between two successive ANTINODES is half a wavelength.

Notes:

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Type of Formation String Open Ends Air Column

Closed Air Column

Fundamental mode of vibration

First Overtone Also called the 2nd Harmonics

Also called the 2nd Harmonics

Also called the 3nd Harmonics

Second Overtone Also called the 3nd Harmonics

Also called the 3nd Harmonics

Also called the 5nd Harmonics

𝑓 1=𝑣 /2𝑙𝑓 1=𝑣 /2𝑙

Note: sometimes the fundamental frequency is written as instead of

𝑓 1=𝑣 /4 𝑙

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Type of Formation String Open Ends Air Column

Closed Air Column

General Formula

Frequency of the nth harmonics, where ,…fundamental frequency

Frequency of the nth harmonics, where ,…fundamental frequency

Frequency of the nth harmonics, where ,…fundamental frequency

Note: sometimes the fundamental frequency is written as instead of

L

nvvfn 2

L

nvvf n 4

L

nvvfn 2

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Differences between Stationary Waves and Progressive Waves

Stationary Progressive

Wave Profile The wave profile does not advance.

The wave profile advances with the speed of the wave.

Amplitude Varies from a maximum of 2A at the antinodes to 0 at the nodes. It depends on the position along the wave.

Same for all particles in the wave motion if no energy is lost.

Frequency All particles vibrate with SHM with the frequency of the stationary wave

All particles vibrate with SHM with the frequency of the progressive wave.

Wavelength Twice the distance between a pair of adjacent nodes or antinodes.

Distance between any two consecutive points on the wave with the same phase.

Phase Particles between two successive nodes oscillate in phase with the same frequency.

Neighbouring points are not in phase, but have the same frequency.

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Resonance Tube Experiment- By adjusting the length of a tube with

open end on one side, different mode of vibration is observed on each tube.

- The antonode is at a distance (end correction) beyond the end of the tube.

For the shorter tube, For the longer tube,

Subtracting the first equation from the second equation gives:

Simplifying gives:

Hence,

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Alternative Method- The loudspeaker produces sound waves, which are reflected from the vertical

board. Superposition of the reflected and incident waves produces stationary waves.

- A small microphone detects the stationary sound wave and its output is displayed on the c.r.o. (cathode ray oscilloscope).

- By moving the microphone along the line between the speaker and the board, it is easy to detect nodes and antinodes. Hence, the wavelength can be determined.

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Alternative Method

𝑙

Distance between successive antinodes = But

Hence,

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Kundt’s Dust Tube

- Kundt’s tube can be used to determine the speed of sound. It resembles a closed air column.

- When a stationary wave is established, the dust (fine powder) vibrates towards the left and right.

- At the nodes, where the movement of the air is zero, the dust accumulates.

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Useful Links:

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

https://phet.colorado.edu/en/simulation/wave-on-a-string

http://en.wikipedia.org/wiki/Standing_wave