Misc

             Pi
              /
         1    |
J (z) = -- * |  cos (n * theta - z * sin (theta))   d theta
 n       Pi   |
              /
              0

        1
J'(z) = - (J   (z) - J   (z))
 n      2   n-1       n+1

         z
J (z) =  -- * ( J   (z) + J   (z))
 n       2n      n-1       n+1

A bessel function describes the motion of a vibrating circular membrane (like that of a drum).

Decibel measures sound intensity.

It either measures absolute sound intensity (=dB SPL = dB Sound Pressure Level) or relative sound intensity.

0 decibels SPL = 10^-12 Watts per m².

If a sound of x decibels increases to x+10 decibels (=1 bel), then the power intensity is mulitplied by 10.

n decibels is hence 10^{n/10 -12} Watts per m².

log₁₀(2) = 0.30103... Therefore, 3 decibels is roughly doubling the power intensity.

When adding two sampled sounds together, you may actually wipe out some frequencies, those that had opposite phases and equal amplitudes. The average amplitude of the resulting sound is (for independent originals) sqrt(a^2+b^2) where a and b are the amplitudes of the original signals.

A sine (having an amplitude and a phase) can be represented by a complex number:
amplitude*cos(phase) + i*amplitude*sin(phase).
Two sines (of equal frequency) can be added by adding these complex numbers.