1. Dr. Jie ZouPHY 10711 Chapter 8 (Hall) Sound Spectra

2. Dr. Jie ZouPHY 10712 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.

3. Dr. Jie ZouPHY 10713 Waveforms and spectra of a flute and a trumpet Flute C NoteTrumpet C Note

4. Dr. Jie ZouPHY 10714 Outline The harmonic series Prototype steady tones Periodic waves and Fourier spectra Fourier spectrum Fourier components Fourier synthesis Fourier analysis

5. Dr. Jie ZouPHY 10715 The harmonic series An example of a harmonic series: f 1 = 110 Hz, f 2 = 220 Hz, f 3 = 330 Hz, … f 10 = 1100 Hz,…so on. Harmonic series: A Harmonic series contains a group of frequencies that are based on a single frequency, f 1, which is called the fundamental frequency. The frequencies of the other members are simple multiples of the fundamental. f n = n  f 1, n = 1, 2, 3,… f 1 : the fundamental frequency; f 2 : the 2 nd harmonic; f 3 : the 3 rd harmonic, … and so on.

6. Dr. Jie ZouPHY 10716 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.

7. Dr. Jie ZouPHY 10717 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.

8. Dr. Jie ZouPHY 10718 f = f 1 = 110 Hz Combination of sine waves + f 2 =220 Hz f 1 =110 Hz T 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.

9. Dr. Jie ZouPHY 10719 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.

10. Dr. Jie ZouPHY 107110 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 f 1 = 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.

11. Dr. Jie ZouPHY 107111 Recipe for building a square wave … After 200 selected sine waves added together

12. Dr. Jie ZouPHY 107112 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

13. Dr. Jie ZouPHY 107113 Homework Ch. 8 (Hall), P. 146, Exercises: #1, 2.