Mathematical Modeling: Music and Acoustics

The frequency of a vibrating string is given by the equation f = (1/2L) * sqrt(T/m), where L is the length of the string, T is the tension in the string, and m is the mass per unit length of the string.

The wavelength of a sound wave is given by the equation λ = v/f, where v is the velocity of the sound wave and f is the frequency of the sound wave.

The intensity of a sound wave is given by the equation I = P/A, where P is the power of the sound wave and A is the area over which the sound wave is spread.

The reverberation time of a room is given by the equation RT = 0.161 * V/A, where V is the volume of the room and A is the total absorption of the room.

The frequency of a musical note is given by the equation f = 2^(n/12) * 440, where n is the number of half steps above or below the note A4 (440 Hz).

The loudness of a musical note is given by the equation L = 20 * log(I/I_0), where I is the intensity of the note and I_0 is the reference intensity (10^-12 W/m^2).

The envelope of a musical note is given by the equation E = A(t) * D(t) * S(t) * R(t), where A(t) is the attack function, D(t) is the decay function, S(t) is the sustain function, and R(t) is the release function.