Chapter 17

Superposition & Standing

Waves

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Superposition & Standing Waves

• Superposition of Waves

• Standing Waves

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Wave Interference

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Constructive Interference

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Destructive Interference

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Acoustic (Sound)

Wave Interference

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Sound Wave

Sound waves can be considered from a pressure variation or an air displacement point of view.

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Constructive Interference

Common source to maintain phase relationship in both speakers.

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Destructive Interference

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Detailed Interference Geometry

Constructive Destructive

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Interference in a Ripple Tank

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Interference & Diffraction

http://www.pas.rochester.edu/~ksmcf/p100/java/Optics/Diffraction.html

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http://www.austincc.edu/mmcgraw/simulations/wave-interference.jar

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Beats

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Beats

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Beat Frequency Example

By placing a small piece of putty on

one of the tuning forks the increased

mass causes its frequency to

decrease slightly.

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The beat frequency is 21 fff −=∆

Two waves of

different

frequency

Superposition of

the above

waves

Beat Frequency

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Standing Wave on a String

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Tunable Standing Wave Generator

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Fourier Analysis

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Fourier Analysis

Every waveform can be broken down into its frequency

components.

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Fourier Analysis - Square Wave

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Frequency Component Amplitude

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Wave Components in Frequency Space

Fourier Analysis

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Musical Instruments

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Pressure Variations in a Pipe

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Open Pipe Resonator

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Closed Pipe Resonator

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fundamental frequency fo

1st harmonic

3rd harmonic f1 = 3fo

5th harmonic f2 = 5fo

fundamental frequency fo

1st harmonic

2nd harmonic f1 = 2fo

3th harmonic f2 = 3fo

Open and Closed Pipes Resonance States

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Pipe Resonator Calculations

Natural frequency dependent on length of pipe

For closed pipe - no "even harmonics”

Fundamental frequency is a half-loop or ¼ L.

Since every harmonic represents the addition of a complete loop, which

contains two half-loops, we can never add just one more half-loop.

Thus, we cannot generate even harmonics in closed pipes.

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Pipe Resonator Calculations

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Open Pipe Resonator

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Closed Pipe Resonator

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Musical Instruments

Frequency Components

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Fundamental Wave and the 4th Harmonic

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Violin Trumpet Clarinet

Musical Instrument Waveforms

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Frequency Component Structure

Violin

Clarinet

Organ Pipes

Piano

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Typical Musical Overtone Structures

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Musical Sound Waveforms

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Musical Sound Frequency Spectrum

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Fourier Analysis

http://www.austincc.edu/mmcgraw/simulations/fourier.jar

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Standing Wave Patterns

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Ringing Bell - Resonant Modes

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Guitar - Resonant Modes

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