alt.sci.physics.acoustics

"Keith Carlson" wrote in message
news:ILTSi.148699$Xa3.32544@attbi_s22...
> Well, I know what the definition is, but I'm trying to understand where it
> comes from to help me remember it. I tend to get SPL vs sound power level
> mixed up (as well as when it's 3db or 6db that represents a doubling).
>
> The definition I know is:
>
> SPL = 20 log (p1/p0), or 10 log (p1/p0)^2 (where "^2" means to the
> power of 2)
>
> I understand that the factor of 10 is because it's expressed in decibels
> as opposed to bels. My specific question, though, is why is the ratio
> squared? If p0 is the reference pressure level, and p1 is the measured
> pressure level, why isn't SPL just the log of that ratio, similar to the
> way sound power level is the log of power level ratio?

You may think that sound intensity (I) and intensity level (LI) was defined
first and the purpose was to get values and a scale that correspond with
received loudness (this is why lg(x/x0) is used) changes when the level
value varies (ca. 10 dB increase doubles the loudness).
Sound level meters using microphone as the detector measures sound
pressure, not intensity. The target was that level values measured by
using sound pressure would correspond with intensity level values
(or that Lp = LI).

p = sound pressure [Pa], v = particle velocity [m/s]
Lp = sound pressure level [dB]

Specific impedance of media z0 = p/v [(J/m^3)/(m/s)], this is sound
energy density when particle velocity is 1 m/s, or how much sound energy
there is in the meadia, when particle velocity is 1 m/s.
z0 = p/v = rho c, where (rho = density of media [kg/m^3], c = sound
speed [m/s], for air z0 = ca. 1,21*331 = 400 [(J/m^3)/(m/s)].
v = p/z0 = p/(rho c)

I = pv cos phi
phi = phase angle between p and v.
We assume that phi = 0, when cos phi = 1.

I = pv*1= p*(p/z0) = p^2/(rho c)

We define LI = 10 log (I/I0) [dB], where I0 = 10^-12 [W/m^2].
(about more than 40...50 years ago I0 = 10^-13 was uses as the
reference value in american standards].

LI = 10 log (I/I0) = 10 lg [(p^2/z0)/(p0^2/z0)]
This equivalence expects that
(p0^2/z0) = I0, this means p0 = sqrt(I0*z0)
for air p0 = sqrt(10^-12*400) = ca. 2*10^-5 [Pa].
(but not generally. For examle, in water z0 = ca. 1000*1500).

Finally we define Lp = LI
Lp = 10 lg [(p^2/z0)/(p0^2/z0)] = 10 lg(p^2/p0^2) = 20 lg(p/p0).
- - -

Quite another issue is that, for example, by using definition
Lp = 5 lg(p/p0) we get a scale that can be used as an indicator
(rough metrics) of loudness. Of course numeric values differ
from those get when 20lg(p/p0) is used, but in both the scales
x dB change (from y dB to y + x dB) measures equally well the
change in loudness.

regards
Kari Pesonen