Sound Waves

Chapter 1. Physical Sound

1.1. Sound Waves

What are sound waves?

Sound waves are air molecules vibrating under the influence of an initial shock: a string being plucked; a drum head being hit. Any movie with loud explosions in outer space should have their technical staff flogged in public: there are no air molecules in space; no air means no sound.

The movement of the string displaces the air molecules around it; these air molecules form alternating zones of higher and lower pressure. When we say that sound is propagating, we mean that those zones are propagating; the air molecules themselves do not move. If a sensitive membrane (e.g. a microphone) is put in front of the sound source, it will feel those alternating zones of higher and lower pressure as alternating moments of higher and lower pressure.

How do we represent them?

The best way to represent this type of behavior is with an oscillating function, going from some lower boundary measuring troughs in pressure to some higher boundary measuring peaks in pressure, with at some point in the middle values which represent the average pressure felt by the measuring device (see below).

Figure1 The formation of sound waves

Figure 1 Sound Waves: A string S vibrates, creating mobile high (H) and low (L) air pressure areas; this can be represented by a sine wave like function of pressure (P) versus time (t); the device T feels the varying pressure; why T? Read on.

In a simple one-dimensional case, the sound pressure wave P as a function of its location x and the time t takes the form of a sine function :

Equation1 Sound wave represented by a sine functionEquation 1 Sine Wave

where A is the wave’s amplitude, describing how large or small it can be, ν (Greek letter “nu”) is the frequency, λ (Greek letter “lambda”) is the wave’s wavelength (see Equation 2 below); 2*Π*x/λ is sometimes represented by the symbol Φ (Greek letter “phi”) and is called the wave’s phase. P represents the pressure recorded at a certain time t by the microphone which is located at the x position.

What is a wavelength?

The wavelength  represents the distance between two points of the function which have the same characteristics, for example two peaks or two troughs. The frequency ν can be obtained with the following formula which is valid in air (linear medium) but not in water (dispersive medium):

Equation2 Sound waves are described by frequency and wavelength

Equation 2 Wavelength and Frequency

where  is the speed of sound; sound travels at roughly 113 feet per second / 343 meters per second at 20° Celsius / 68° Fahrenheit. A linear medium (like air) is a medium in which the waves with different frequencies all travel at the same speed; in a dispersive medium (like water), the waves with different frequencies travel at different speeds; the sound gets progressively deformed as it moves.

What are frequencies?

Frequencies are measured in Hertz (Hz) and wavelengths in meters (m). Examples of typical wavelengths and corresponding frequencies can be found in the next table.

Wavelength Frequency
1 m 343 Hz
1 ft. 1.12 kHz
17 m = 56 ft. 20 Hz
17 mm = 0.67 in. 20 kHz
0.78 m = 31 in. A (440 Hz)
1.31 m = 4 ft. 4 in. C4 (261 Hz)

Table 1 Typical Wavelengths and Frequencies

The lowest frequency a human ear can hear is called the human hearing threshold (more on that in Section 4.4); its value is 20 Hz and is represented by a wave of length 17 m, while the highest frequency a human ear can hear is 20 kHz (17 mm length). Standard tuning with the A note at 440 Hz is 0.78 m long while the middle C on a piano is 1.31 meters long.

Again, note that air molecules do not travel with sound; they literally oscillate in place in the direction of the sound, which is why sound waves are longitudinal waves and why there is wind coming with them. What does move in the direction of the sound are pressure changes caused by those vibrating air molecules.

 

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