What is the equation for decay of natural sound over time?

[mikejm]

Let's say you have an impulse of noise, eg. via a guitar string, or a resonant bandpass filter, and it decays naturally. If "1" is the maximum initial amplitude of sound, and "0" is no sound at all, what is the equation for decay of sound over time (x)?

Is it:
y = 1/c^x

Or y = 1/(x+1)^c

I think it is y= 1/c^x but not sure.

What does nature do?

Thanks.

<moved from General Discussion>

 

[Mark44]

Let's say you have an impulse of noise, eg. via a guitar string, or a resonant bandpass filter, and it decays naturally. If "1" is the maximum initial amplitude of sound, and "0" is no sound at all, what is the equation for decay of sound over time (x)?

Is it:
y = 1/c^x

Or y = 1/(x+1)^c

Most likely the first, not the second. The second equation is the translation 1 unit to the left of the graph of the first equation.
Also, c in your equations would usually be take to be the natural number e $\approx 2.71828$, and named after Leonhard Euler, a Swiss mathematician.

Of course, the exponent base is arbitrary, as any exponential function can be rewritten using any other reasonable base.

I think it is y= 1/c^x but not sure.

What does nature do?

Thanks.

 

[tech99]

The decay time constant depends on the Quality Factor, or Q, of the resonator. See https://en.wikipedia.org/wiki/Q_factor. My interpretation is that the wave decreases exponentially with a certain time constant, T. If the initial amplitude is 1 and the amplitude at time x is a,
a= e^-x/T
The time constant T is equal to 2Q/radial freq.

 

[jbriggs444]

a= e^-x/T

That would be more correct as $e^{-x/T}$. As written above, the standard interpretation would be as $\frac{e^{-x}}{T}$

 

[Klystron]

The equations in this article describe attributes such as energy dissipation associated with acoustic attenuation given data about the waveform and materials: https://en.wikipedia.org/wiki/Acoustic_attenuation