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Chapter 8 (Hall). Sound Spectra. Introduction. Question : When you hear the music “Danny Boy”, what lets you distinguish between a trumpet and a flute? Answer : Each periodic waveform has its corresponding spectrum , which determines the timbre , or tone quality of the sound.

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Chapter 8 hall

Chapter 8 (Hall)

Sound Spectra

PHY 1071


Introduction
Introduction

  • Question: When you hear the music “Danny Boy”, what lets you distinguish between a trumpet and a flute?

  • Answer: Each periodic waveform has its corresponding spectrum, which determines the timbre, or tone quality of the sound.

PHY 1071


Waveforms and spectra of a flute and a trumpet
Waveforms and spectra of a flute and a trumpet

Flute C Note

Trumpet C Note

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Outline
Outline

  • The harmonic series

  • Prototype steady tones

  • Periodic waves and Fourier spectra

    • Fourier spectrum

    • Fourier components

    • Fourier synthesis

    • Fourier analysis

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The harmonic series
The harmonic series

  • An example of a harmonic series: f1 = 110 Hz, f2 = 220 Hz, f3 = 330 Hz, … f10 = 1100 Hz,…so on.

  • Harmonic series: A Harmonic series contains a group of frequenciesthat are based on a single frequency, f1, which is called the fundamental frequency. The frequencies of the other members are simple multiples of the fundamental.

    • fn = nf1, n = 1, 2, 3,…

    • f1: the fundamental frequency; f2: the 2nd harmonic; f3: the 3rd harmonic, … and so on.

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Prototype of periodic steady tones
Prototype of periodic steady tones

  • (a) Sine wave (b) Square wave (c-d) Pulse wave (e) Triangular wave (f-h) Saw-tooth wave

  • What is the simplest of all wave forms?

  • Answer: Sine waves. They are the “building blocks” for other more complex wave forms.

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Two things to show
Two things to show

  • (1) Take simple periodic sine waves and put them together to form a more complex wave.

  • (2) Take a complex periodic wave and break it down into simple sine wave components.

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Combination of sine waves

T

f1=110 Hz

f = f1= 110 Hz

f2=220 Hz

Combination of sine waves

  • Any set of sine waves whose frequencies belong to a harmonic series will combine to make a periodic complex wave, whose repetition frequency is that of the series fundamental.

+

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Combination of sine waves cont
Combination of sine waves (cont.)

  • In general, for a set of sine waves whose frequencies do not belong to a harmonic series, the combined wave will be non-periodic.

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Breaking a periodic complex wave
Breaking a periodic complex wave

  • Any periodic waveform of period T may be built from a set of sine waves whose frequencies form a harmonic series with fundamental f1 = 1/T. Each sine wave must have just the right amplitude and relative phase, and those can be determined from the shape of the complex waveform.

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Recipe for building a square wave

After 200 selected sine waves added together

Recipe for building a square wave

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Fourier spectrum
Fourier spectrum

  • Fourier spectrum: The recipe of sine wave amplitudes involved in a complex wave.

  • Fourier components: Each sine wave ingredient is called a Fourier component.

  • Fourier synthesis: Putting sine waves together to make complex waves.

  • Fourier analysis: Taking complex waves apart into their sine wave components.

Fourier spectrum of a square wave

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Homework
Homework

  • Ch. 8 (Hall), P. 146, Exercises: #1, 2.

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