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Underwater Acoustics Concepts - SI units and decibels in underwater acoustics (with some airborne parallels)

The fundamental units for physical measurements are defined by the Systeme Internationale (SI). These underpin the commonly used acoustic decibel levels, by providing reference values (e.g. Po), to generate the logarithmic ratios required.

Names of measured quantities and linking equations Decibel formats
(examples in the literature)
SI units + optional prefixes
e.g. μPa (microPascal)
Sound pressure level (SPL)
Acoustic pressure (P)
SPL = 20 log (P/Po)
dB re 1 μPa
dB re 20 μPa
Pa (Pascal)
-  measured at a defined position
-  bandwidth must be specified for
    broadband sound
Note the different references for:
-  underwater (Po =  1 μPa)
-  airborne (Po = 20 μPa)
Sound power level (WL)
Sound power (W) (in band B)
WL = 10 log (W/Wo)
dB re 1 pW W (Watt)
pW (picoWatt, 10-12 W)
Mainly used when describing airborne sound sources, usually over the whole audio bandwidth
Sound power spectrum level
WSL = WL - 10 log B
  W/Hz (Watt/Hertz) Power per unit frequency can be measured for broadband sound
Sound pressure spectrum level dB re μPa2/Hz
dB re μPa/√Hz
dB re 1 μPa in a 1Hz band
(Pa/√Hz is used for convenience, but is not a physical quantity)
Integration of broadband sound pressures gives the mean P2 value, based on sound power
Sound intensity level (IL)
Sound intensity (I)
IL= 10 log (P2/Z)
dB re pW/m2 W/m2 (Watt/square metre) The airborne 0 dB reference level of 1 pW/m2 is deemed to be the lowest audible level (at ~2 kHz)
Acoustic impedance (Z)
Z = ρ· c
  kg/(m2·s) or Rayl
(Pa·m)2/W is a useful equivalent
The product of density ρ (kg/m3) and sound speed c (m/s)
Source level (SL)
Source output (S)
SL = 20 log (S/So)
    = 20 log ( P·r )
= 20 log (P/Po) + 20 log (r/ro)
= 20 log P + 20 log r
(when Po,ro both unity)
dB re 1 μPa at 1m
dB re 1 μPa·m
dB re 1 μPa-m
Pa·m (Pascal·metre)
Source output S should be
-  measured in a defined direction
-  have a specified bandwidth
   (except for single tones)
Source output S determines the power per unit solid angle, WA (watts per steradian).
WA = S2/Z
(see row above)

Analysis of transmission losses

Decibels are especially convenient for the analysis of transmission loss TL. This conveniently combines the exponential reduction of intensity due to absorption in the fluid, with the inverse square reduction of intensity due to spherical spreading. However, the reference distances must be kept consistent with the quantities used elsewhere.

This table only considers the spherical spreading of coherent signals in non reverberant conditions (i.e. no echoes). Other spreading models (e.g. cylindrical losses) can be applied to more complex environments.

Names of measured quantities and linking equations Decibel formats
(examples in the literature)
SI units + optional prefixes
e.g. μPa (microPascal)
Absorption coefficient a
a = 10 log (I1/I2)
Intensity I1 is reduced to I2 by absorption over a distance ro
Absorption losses over range r,
LA = a · r (losses are positive)
dB/m, dB/km, dB/kiloyard,
The reference distance ro is applied here in a different way to other decibel quantities.
It is not "dB re 1m", but dB per m or dB/m
This is the attenuation in dB, over the reference distance ro. Whilst attenuation is unitless, the absorption coefficient a has units of inverse distance (e.g. m-1) The attenuation due to absorption is an exponential decay.
Doubling the distance doubles the decibel loss
Spherical spreading loss, LS (excluding absorption)
LS = 20 log ( r / ro)
    = SL - SPL
dB re 1m, dB//m
A reference distance ro must be defined to avoid ambiguity.
The double slash provides a convenient short form
The reference is a distance ro , usually given in m.
This must be compatible with the ratio of S/P
e.g. Pa·m/Pa
Doubling the distance divides the pressure by two,the intensity by four, and reduces the SPL by 6dB
Transmission Loss, TL
TL = LS + LA
= 20 log r + a r ( when ro=1)
dB re 1m, dB//m
NB: The inclusion of the exponential absorption term means that the normal rules are extended here
The reference distance ro must be defined with extra care, given the complex relation, especially if ro is defined differently for each term The convenient way in which the decibel transmission loss covers both loss mechanisms is a major benefit, but can lead to serious errors

Some more examples of decibel quantities which sonar design engineers find useful, with their SI reference units

Names of measured quantities and linking equations Decibel formats
(examples in the literature)
SI units + optional prefixes
e.g. μPa (microPascal)
Hydrophone receive voltage sensitivity (M) dB re 1 V/μPa V/Pa
Typical -206 dB re V/μPa
or 50 μV/Pa
Transmit voltage response
(TVR or SV) from a source
dB re 1 μPa at 1m per Volt
dB re 1 μPa·m/V
Pa·m/V Typical 140 dB re 1 μPa·m/V
or 10 Pa·m/V (for a piezosphere)
Transmit current response (SI) dB re 1 μPa at 1m per Amp
dB re 1 µPa·m/A
Pa·m/A The current response is significant for electrical matching and for reciprocity calibration
Power per unit solid angle   W/sr (Watts/steradian)
A steradian is a unit solid angle which intercepts an area r2 on a sphere of radius r, centred on the source
This helps to define directivity, and is given by the source output squared S2 (Pa·m)2, divided by the fluid acoustic impedance Z
Directivity index (DI)
Directivity factor (D)
DI = 10 log D
No reference unit is needed because D is a ratio
For a source:
Equation siunits1
DI is also a simplified form of array gain, used to calculate signal/noise ratios of hydrophone arrays