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Chapter 8 (Hall)

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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)

Sound Spectra

PHY 1071

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

Flute C Note

Trumpet C Note

PHY 1071

Outline
• The harmonic series
• Prototype steady tones
• Periodic waves and Fourier spectra
• Fourier spectrum
• Fourier components
• Fourier synthesis
• Fourier analysis

PHY 1071

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.

PHY 1071

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.

PHY 1071

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.

PHY 1071

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.

+

PHY 1071

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.

PHY 1071

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.

PHY 1071

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

PHY 1071

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

PHY 1071