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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 prefixese.g. μPa (microPascal) Comments Sound pressure level (SPL)Acoustic pressure (P)SPL = 20 log (P/Po) dB re 1 μPa dB//μPadB 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 levelWSL = 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/HzdB re μPa/√Hz dB re 1 μPa in a 1Hz band Pa2/Hz(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 )                 (Po·ro)= 20 log (P/Po) + 20 log (r/ro)= 20 log P + 20 log r (when Po,ro both unity) dB re 1 μPa at 1mdB re 1 μPa·mdB 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 prefixese.g. μPa (microPascal) Comments Absorption coefficient aa = 10 log (I1/I2)Intensity I1 is reduced to I2 by absorption over a distance roAbsorption 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//mA 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/Pe.g. Pa·m/Pa Doubling the distance divides the pressure by two,the intensity by four, and reduces the SPL by 6dB Transmission Loss, TLTL = LS + LA= 20 log r + a r ( when ro=1) dB re 1m, dB//mNB: 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 prefixese.g. μPa (microPascal) Comments Hydrophone receive voltage sensitivity (M) dB re 1 V/μPa V/PaμV/Pa Typical -206 dB re V/μPaor 50 μV/Pa Transmit voltage response(TVR or SV) from a source dB re 1 μPa at 1m per VoltdB re 1 μPa·m/V Pa·m/V Typical 140 dB re 1 μPa·m/Vor 10 Pa·m/V (for a piezosphere) Transmit current response (SI) dB re 1 μPa at 1m per AmpdB 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 dBNo reference unit is needed because D is a ratio For a source: DI is also a simplified form of array gain, used to calculate signal/noise ratios of hydrophone arrays