Fourier Analysis
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Joseph Fourier (1768-1830) and Fourier Analysis
In 1822, Jean Baptiste Joseph Fourier published a theory that "arbitrarily complicated periodic signals" are composites of sine waves, and combine to create complex signals. These composite waves can be deconstructed into sine waves of different amplitudes, frequencies, and phases. This notion was originally attacked by scientists who insisted that his theory was mathematically impossible.
f(t)=f(t+T)!
f(t)=∑An sin(2πnFt+0n)
Harmonic v. Inharmonic and Overtones v. Harmonics
Harmonic frequencies are simple integer multiples of the fundamental frequency. For instance, if the fundamental frequency (first harmonic) is 440 Hz, the second harmonic (also know as the first overtone) is 880 Hz. However, the vast majority of vibrating objects don't vibrate like musical instruments. While musical instruments generally generate harmonic frequencies, more complex objects are inharmonic, and this is shown in the second figure below. Percussive instruments are also frequently inharmonic.
Principle of Linear Superposition
I found this description online at http://arts.ucsc.edu/EMS/Music/tech_background/TE-04/teces_04.html:
If both have the same frequency and phase, the result is a sine wave of amplitude equal to the sum of the two amplitudes.
If both have the same frequency and amplitude but are 180 degrees out of phase, the result is zero. Any other combinations of amplitude produce a result of amplitude equal to the difference in the two original amplitudes.
If both are the same frequency and amplitude but are out of phase a value other 180 degrees, you get a sine wave of amplitude less than the sum of the two and of intermediate phase.
If the two sine waves are not the same frequency, the result is complex. In fact, the waveform will not be the same for each cycle unless the frequency of one sine wave is an exact multiple of the frequency of the other.
Interference
Constructive and destructive interference are described in detail above, but in general terms:
Constructive interference occurs when two waves have the same frequency and phase, and the resultant wave's amplitude is equal to the sum of the two waves.
Destructive interference occurs when two waves have the same frequency, but are out of phase, the resulting wave's amplitude is less than the sum of the two, and is 0, or completely destructive, if the waves are 180° out of phase.
Radian Frequency
w = 2 π f
w = radian frequency f = frequency
The radian measurement system is used for angles in higher mathematics, and define angles in terms of π. Since the circumference of a unit circle is 2π radians, "an angle of one revolution around a circle is equal to an angle whose measure is 2π".
The sides of the angle are labeled A for Abcissa, O for Ordinate, and H for Hypotenuse. "Trigonometry posits that the proportions among the sides of angles inscribed within a circle correspond to sine, cosine, and tangent relations" as follows:
sin(θ) = O / H cos(θ) = A / H tan(θ) = O / A
Here's a helpful description of radian frequency from usenet:
Think of a rotating wheel. If the wheel is spinning at 14 rotations per second, the "plain" frequency is 14 rotations/second, or 14 Hz. Since one rotation is the same as 2*pi radians, you can also say the wheel is rotating 2*pi*14 radians/second.
Linearity / Nonlinearity of Air
Air behaves linearly with sounds we can hear at relatively low dB levels (< 150 dB). However, air is nonlinear with ultrasonic sound at extremely high decibel levels. By linear, we mean that it responds proportionally to the amount of energy applied to it. If we put two signals into the system, we expect the output to be their sum. By non-linear, we mean a system that has built-in thresholds that, if exceeded, cause the system to respond in a new way, as if a switch had been thrown.
Emitting two ultrasonic waves at high SPL causes audible (< 20KHz) sounds, because the two sounds create sum and difference tones, as in frequency modulation, and the difference tone can be in the audible spectrum. Since the bandwidth of ultrasonic waves is so small in comparison to the size of the speaker that generates it, this can be used to focus audio signals in a way that would usually be impossible.
This technology is used in Audio Spotlight: http://www.holosonics.com/technology.html
Another interesting article on the subject: http://www.spie.org/web/oer/october/oct00/indfocus.html
