This document provides an overview of sound from a mathematical perspective. It discusses how sound is created through vibration, its propagation as a longitudinal pressure wave, and how frequency and wavelength are related through the speed of sound. The speed of sound varies according to the medium and environmental factors like temperature. Sound intensity decreases with the inverse square of distance from the source and is measured on the decibel scale. Common sources of sound production are also examined, including vocal cords, speakers, and musical instruments like the French horn.

1. A Mathematical Approach
By: Alex Barrington

2. Contents:
 Sound
 Measuring Sound
 Frequency and Wavelength
 Speed of Sound
 Intensity and Decibel Range
 Producing Sound
 Bibliography

3. Sound
 Sound is created by an object vibrating
 Mechanical wave
 Particle-to-Particle interaction
 Requires a medium
○ Through air: 344 meters/second, 1130 feet/sec. or 770
miles per hour at room temperature of 20oC (70oF).
 Longitudinal waves
 Particles of the medium through which the
sound is transported vibrate parallel to the
direction that the sound wave moves
 Pressure Wave

4. Measuring Sound
1 Hertz = 1 vibration/second

5. Frequency and
Wavelengths
 wavelength and frequency equation:
λ = v/f or v = fλ
λ (lambda) = wavelength
v = velocity
f = frequency
λ = v/f
12.5 = v/27.5
v = 12.5*27.5
v = 343.75 m/s

6. Piano key Wavelength
Name of note Frequency (Hz)
number (meters)
-- -- 20 17.2
1 (first key) A0 27.5 12.5
40 C4 (middle C) 261.63 1.31
49 A4 440 0.79
64 C6 (soprano C) 1046.5 0.329 (32.9 cm)
88 (last key) C8 4186.01 0.082 (8.2 cm)
-- -- 20,000 0.0172 (1.72 cm)

7. Speed of Sound
 speed = distance/time
 Elastic Properties vs. Inertial Properties
 Elastic properties: the tendency of a material to maintain its
shape and not deform whenever a force or stress is applied
to it.
 Elasticity:
 vsolids > vliquids > vgases
 Inertial properties: the material's tendency to be sluggish to
changes in its state of motion. For example: Mass Density.

8. Gases
• Humidity & Temperature vs. Air Pressure
• Sound travels slower at higher altitudes
because the temperature and humidity are
lower, not because the air pressure is lower
higher altitudes.
• v = 331.4 + 0.6TC m/s
v = velocity
TC = Celsius temperature.
331.4 = v at 0o C

9. The speed of sound for various gases at 0° C:
Speed
Gas
(meters/second)
Air 331
Carbon Dioxide 259
Oxygen 316
Helium 965
Hydrogen 1290

10. Intensity and Decibel Range
 Intensity: the amount of energy a wave transports
in a given area per unit of time.
 Intensity = Watts/meter2
 The intensity of a sound wave decreases as
distance from the source increases.

11. Distance Intensity
1m 160 units
2m 40 units
3m 17.8 units
4m 10 units
Inverse square relationship:
• The intensity varies inversely with the square of the distance from the
source.

12. Decibel Scale:
# of Times Greater
Source Intensity Intensity Level
Than TOH
Threshold of Hearing
1*10-12 W/m2 0 dB 100
(TOH)
Rustling Leaves 1*10-11 W/m2 10 dB 101
Whisper 1*10-10 W/m2 20 dB 102
Normal Conversation 1*10-6 W/m2 60 dB 106
Busy Street Traffic 1*10-5 W/m2 70 dB 107
Vacuum Cleaner 1*10-4 W/m2 80 dB 108
Large Orchestra 6.3*10-3 W/m2 98 dB 109.8
Walkman at
1*10-2 W/m2 100 dB 1010
Maximum Level
Front Rows of Rock
1*10-1 W/m2 110 dB 1011
Concert
Threshold of Pain 1*101 W/m2 130 dB 1013
Military Jet Takeoff 1*102 W/m2 140 dB 1014
Instant Perforation of
1*104 W/m2 160 dB 1016
Eardrum

13. Producing Sound
 Vocal Cords
 Speakers
 Instruments

14. Vocal Cords
 Phonation: The process of converting the air
pressure from the lungs into audible vibrations
 The pitch produced depends upon the length,
mass and tension of the vocal folds.
 The male speaking voice averages about 125 Hz,
while the female voice averages about 210 Hz.
Children's voices average over 300 Hz.

15. Speakers
• Alternating current
• The frequency and amplitude of an electrical audio signal dictates the rate
and distance that the voice coil moves. This, in turn, determines the
frequency and amplitude of the sound waves produced by the diaphragm.

16. Instruments
French Horn:
 made in the key of F with a length of about 12ft and in
a key of C with about 8ft of length.
 resonant peaks out to about 1500 Hz
 A trigger valve, thumb valve, is used to change
between the F and the B-flat horn.
 Embouchure, air pressure, key, length of tubing, and
hand position all have an effect on the pitch and
intensity of the sound wave being produced.

17. Thank You!

18. Bibliography
 http://electronics.howstuffworks.com/speaker6.htm
 http://www.physicsclassroom.com/class/sound/
 http://www.school-for-champions.com/science/sound.htm
 http://hyperphysics.phy-astr.gsu.edu/hbase/music/voice.html