Second Order Differential Equation and Aural Trajectories

by Sniffy

This differential equation

y” + 4y’ + 4y

has repeated roots and one solution might be

y = 2xexp(-2x) + 2exp(-2x)

Just to verify that y'(0) = -2, (see graph)

y'(0) = -4xexp(-2x) + 2(exp(-2x) -4exp(-2x)
= 0 + 2 – 4 = -2

If function y = Y(x), then a family of curves can be generated as kY(x), also see graph. The reason for doing this, is to do a frequency multiplication, in this case, the lower frequency is:

Y(110) = 110 * (2(x)exp(-2(x))) + (2exp(-2*(x)))

Finally we want F = Y(1100) * Y(110)

This is not a frequency; however, we could try this:

sin(F) = sin( Y(1100) * Y(110))

and the second derivative becomes a constant value spectrogram which equals zero at x = -0.5 or y'(-0.5) = 0.

Starting at x = -0.5 the sound file sweeps down creating a penguin sound.

Penguin

; PRAAT FORMULA
; divide numbers by 2pi for radian frequency

sin(1100((2(x)exp(-2(x))) + (2exp(-2(x)))))
… * (sin(110((2(x)exp(-2(x))) + (2exp(-2(x))))))