Mathematics and Sound

1. Mathematics and Sound The basic physics behind beautiful music By: Amy, Peter and Marissa

2. Sound Perceptual Aspects Physical Aspects • Amplitude • Frequency • Phase • Loudness • Pitch

3. Frequency • Sound is perceived when fluctuations in air pressure cause structures inside our ears to vibrate. • We refer to the rate at which pressure fluctuates cyclically from higher to lower to higher and so forth as its frequency. • Typically we express frequency in cycles per second or equivalently Hertz.

4. Amplitude • In addition to the frequency of a sound, we can describe its amplitude. • The next figure shows another sound which is lower in amplitude that the previous example because the pressure varies less extremely over time. This figure shows a sound which also differs in frequency from the sound illustrated in the previous figure. • Note that frequency and amplitude vary independently.

5. Phase • One other property called phase is important in describing the physical properties of sound. • To illustrate what is meant by phase, the next figure shows two 200 Hz sinusoids, one drawn with a solid line and the other drawn with a dotted line. However, they represent snippets of functions which do not have beginning and ending points. • The phase differences shown in the present figure do not reflect the notion that one function started at a different time than the other. Rather, the phase differences represent the way the two functions are aligned with respect to each other at all times, including those which lie outside the bounds of the present graph.

6. Sound relations • The physical properties of amplitude, and frequency correspond to the sensory/perceptual qualities of loudness and pitch. • The normal young human auditory system is sensitive to a range of frequencies from about 20 Hz to about 20,000 Hz. • The amplitude range is substantially broader, beginning at a level so low that we can almost "hear" the fluctuations in air pressure due to random motion of air molecules near the ear drum and extending to the threshold of pain at about 10 million times that level..

7. More Relations • A second important difference between the perceptual properties of sound and its physical properties is that they aregenerally non-linear. • For example, if we increase the amplitude of a sound in a series of equal steps, the loudness of the sound will increase in steps which seem successively smaller. • Similarly, increasing the frequency of a sound in equal steps will lead to perceived increases in pitch that seem to grow smaller with each step.

8. Last Relations • We often describe sounds using scales that reflect equal perceptual differences. For frequency, one such scale is the Mel scale • Forloudness, it is most convenient to describe sound over the enormous range of perceptible amplitudes in logarithmic units called Decibels and abbreviated dB. • On the decibel scale, 0.0 dB corresponds to about the normal threshold of hearing and 130 dB to the point at which sound becomes painful.

9. Simple vs. Complex Sound • Despite their differences in amplitude and frequency, the sounds shown and heard above depict simple sounds because the pressure fluctuations associated with these sounds are sinusoidal. • Fortunately, it turns out that such complex sounds can be described mathematically as combinations of simple sounds. Consider for example, the sound illustrated in the next figure which simply alternates between a region of constant high pressure and a region of constant low pressure. • This particular waveform does have a name, it is called a square wave because of its boxy shape. • This square wave is very similar to the 200 Hz sine wave shown in the first figure in that it too repeats a single pattern two hundred time a second • This complex square wave can be described as a summation of a set of simple frequency since wave components. • The components in this sequence are called overtones or harmonics and by definition can only occur at integer multiples of F0.

10. Standing Waves: Strings A standing wave by definition is a wave that is oscillating but remains in one place. So it isn't actually standing, it just appears to be.

11. Resonance and the Standing Wave Resonance is the condition in which an object or system is subjected to an oscillating force having a frequency close to its natural frequency. The Standing Wave is simply a resonating system; the oscillating force is the finger that plucks it.

12. Frequency and Wavelength • The frequency and wavelength of a waveform are inversely proportional. • A mathematical equation relates the two to the speed of sound: We know that the speed of sound is about 434 m/s*

13. Natural Frequency Some system's Natural Frequency, F1, is the frequency at which it will oscillate to create exactly half of a wavelength, as shown in the image below. If the distance from the two nodes is L, one can see that the wavelength of this incomplete wave is exactly 2L, or Fundamental wave, or 1st harmonic, thus n=1

14. Subsequent harmonics The second and third harmonic are shown in blue and yellow. One can see that the second harmonic includes one more half wavelength in the standing wave. Thus: Similarly, the third harmonic has one more half wavelength than the previous harmonic; therefore

15. Ratios This phenomenon of resonance is clearly deeply rooted in the mathematical concept of ratios. Whenever the length of the string is divided into any number of equal lengths (which becomes half of the wavelength) , as dictated by the nodes, the string resonates. This fact, however, is not true for all kinds of standing waves. Marissa will now talk about how Standing Waves exist in pipes closed at one end.

16. . Open Closed Pipe An open-closed pipe instrument includes most wind instruments. There are a few exceptions such as flute, however, most wind instruments such as trumpet and bassoon are considered an open closed pipe. A summary of the first three harmonics for an open-closed vibrating system are shown on the next slides.

17. harmonic. First Fundamental Frequency Notice the open end remains “free” to move. Observe that there are no "even harmonics" among the resonance states of this type of vibrating system. This stems from the fact that the fundamental frequency is a half-loop or ¼ of a wavelength. Since every overtone 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. The closed end remains fixed

18. First Overtone 3rd Harmonic The closed end remains fixed The second fundamental frequency must add a full loop of half of a wavelength to the fundamental frequency, producing uneven harmonics. This is why the harmonics of a wind instrument are only odd numbers. Notice the open end remains “free” to move.

19. Second Overtone 5th Harmonic The closed end remains fixed Notice the open end remains “free” to move. Again, a full loop or half of a wavelength is added to the previous wavelength, causing the harmonic to be uneven.

20. Homework • How do we perceive amplitude? How do we perceive frequency? • If eight cycles of a sound wave occur in 10 milliseconds, what is the frequency of the tone? • If the sound is increasing in pitch, what is happening to the wavelength? • If a sound wave has a frequency of 1000 Hz at 20 degrees Celsius, what is the wavelength? • Given an open-closed pipe that is 1.0 meter in length, what is the wavelength if the sound is oscillating at the second overtone?