subject: The decibel [db] [print this page] The decibel [db] The decibel [db]
The decibel, abbreviateddB, is used to denote a ratio that is ten times, or "deci" times, a unit the "Bel," as so named by its celebrated devisor whose name it bears, Alexander Graham Bell, the inventor of the telephone.The decibel,dB, thus became the commonly used unit of measure for expressing a change in power from an original setting to that being measured, that is to say, adB is a relative power measurement.
SincedB expressesa dimension-less ratio between two power-levels, the caveat for the measurement is that it must be taken at the same point as the reference so as to have the same "acoustical impedance."Whereas, acoustical impedance, symbolized c, is the product of the density,, of the medium of propagation, and the speed,c, of sound in it.
ThedB is therefore used to express arelative increase or decrease in acoustic power or pressure, and their corresponding electric power-levels, as a ratio with either a previous level or a specified standard, or a reference.Such reference typically is in the style of a minimum discernible signal, aMDS, at a sound receptorsuch as the human ear, whose minimum audible field, theMAF, for the human ear is referenced as0 dB.
ThisMAF for the human ear has been measured by experimentation to be at an Intensity,I, of1.0 x10-16 W/m2 expressed as aPower per unitArea,This equates to a minute pressure,p, of2.0 x 10-4 dyn/cm2, which most often is expressed asref 20Pa, whereas20 micro-Pascals is a pressure inMKS units.[1]
The smallestchange in sound-power level,P, the normal human ear can detect, or "sense," is about1 dBref Po.More than likely, such hearing sensitivity was considered by Alexander Graham Bell as the minimum change to which all other sound should be referenced.
Thus, by definition, a decibel,1 dB ref Po, is ten times the base-ten logarithm of a power ratio equal to the one-tenth root of ten.For instance, the power per unit area, the "Sound Intensity Level,"SIL, of a quiet whisper is measured to be18 dB, that is, its "volume" was 18 dB above the MAF.If that particular volume-setting is increased in intensity by 25.9%, then by this convention it is said to be "up"1 dBref Po, whereref Po denotes the original setting 18 dB.
This widely acknowledged convention is shown below in notational form:
(The number of) dB ref Po = 19 dB 18 dB = 1 dBref Po; and,
converting1 dB by dividing by 10 yields the exponent of ten as100.1,
which equals 1.259 and denotes a 25.9% increase above the original.
Further, to be precise,+3dB ref Po by logarithmic calculation is the result of a doubling of its original power level, such that:
rounding the exponent to 0.3 the calculation approximates
10log10 [(100.1)3] = 3 dB ref Po, which is a cube of its base value.
Moreover, a +10 dBgain in power level implies a 10-fold increase over the original level,10 :: 1,whereas a -20 dBloss implies a 100-fold decrease,1/100 :: 1, such that:
10log10 [101 100] = 10 x (+1 - 0) = +10 dB ref Po; and,
10log10 [10-2 100] = 10 x (-2 0) = -20 dB ref Po.
The decibel is also used to express either voltage or current ratios, as either an electro-motive force,E, in units of volts,V, or a magneto-motive force,I, in units of amperes,A. Notably, these electrical forces are squared terms in their respective power-expressions, acting as if their motive force was an "electrical-pressure," whereP = E2/R = I2R; and,
P2, in dB ref P1, = 10log10 [(V2 /V1)2],and 10log10 [(A2 /A1)2]; or,
Strictly though, when the decibel is used to express voltage or current ratios in lieu of power ratios, then the voltages or currents in the expressionmust be measured at places having identical electrical impedances, that is,R1 R2.
Further, by extension, the relation between the number of decibels and the corresponding ratios of voltages and currents are sometimes applied where the values in the ratios are not the square roots of the corresponding electrical power ratios, that is, not from the initialE22/R2andE12/R1 expressions.To preclude confusion, a specific statement of the particular application should accompany such usage.Preferably, such extensions of terms should be avoided.
Intensity,I, is defined in units of power,P, applied over an area, A, which is in units of square-length unit, such asm2. Whereas,P is work per increment of time,t, in units of seconds,s, and work is a force,F, applied in a given distance, or length,l, thenP is in units for force-length per time, Fl/t,such as, ftlbf/s, dyncm/s, or Nm/s.
As defined inNewton's Laws,F is the instantaneous rate of change of momentum with respect to time; whereas, momentum is the inertia of a body-mass,m, moving with some velocity,v, and defined in units of mass-length per increment of time,t.
By calculus, the time-derivative of this defining product for momentum, as it undergoes an instantaneous rate of change with respect to an infinitesimal increment of time,dt, yields an expression that defines force,F, with two additive terms.The first term is the multiplication of the mass,m, by the time-derivative of the velocity,v, which yields,m (dv/dt).The second additive term is the multiplication of the velocity,v, by the time-derivative of the mass,m, which yields,v (dm/dt).
Notably, for momentum, if only the velocity term is undergoing an instantaneous rate of change with respect to time, but not its mass, thendm/dt = 0, andthusthe additive term ofv(dm/dt) = 0.Therefore, classically,F = ma, wheredv/dt = a, which isacceleration in units of length per square-time,ft/s2, cm/s2, or m/s2, wherem isthe symbol for mass in units oflbf/ft/s2,g orkg, whereas Force,F, is expressed in units oflbf, dyn, or Na la,a Newton of force.
Definition. Sound is a distinguishing physical wave, a sound-wave,per se. Lord Raleigh in his work, "Theory of Sound," volumes 1 and 2, Dover Publications, New York, 1945, defined a sound-wave as an alteration in pressure, stress, particle displacement, or particle velocity that is propagated in an elastic material, or the superposition of such propagated alterations in that medium.Further, a sound-sensation is produced through the ear by the above alterations.
Van Nostrand'sScientific Encyclopedia defines sound somewhat more physically as a longitudinal elastic wave-motion propagated by alternate compressions and rarefactions of the medium.The analogy stated therein is that sound is like the propagation of a "bump," or a "jerk," from a freight-train's engine to its caboose.
Thus,a sound-wave of acoustical energy only can propagate in a medium, being it a gas, a liquid, or a solid; and, it is either deflected or refracted, or both, at the laminar boundary between media of differing densities.In the denser media, or in a heated gas, the molecules orbit in closer proximity to one another such that the "bumps" propagate faster.Distinctly, the closer the orbiting molecules are to one another then the better the propagation of the sound-- that is, the sound is demonstrably "louder."Conversely, void of any molecules to "bump," such as in a vacuum, sound cannot propagate; in other words,you cannot hear yourself scream in space.
Sound-intensity is defined as Power per unit area, which is the average rate (time) of sound-energy transmitted in a specified direction as it impinged on anarea normal to this direction of propagation.In notational form, sound-intensity,I, of a spherical-wave, or even a plane-wave, in the direction of propagation can be expressed as being directly proportional to the square of its impinging pressure and indirectly proportional to the acoustic impedance in which it is propagating, that is:
I = [(pressure) 2 (medium-density) (sound-speed)].
I is expressed in terms of the square of the impinging sound-pressure,p, with respect to the acoustical impedance of the medium,c. Accordingly, the resultant-product of the density of the medium,, multiplied by the speed of sound,c, in that medium,is(kg/m3) (m/s) inMKS-units, which further reduces to Newton-seconds per cubic-meter,Ns/m3.The square ofp is in(N/m2)2.And, to be a comparable sound-intensity level, the sensedI must be in a ratio with a previously sensed level, or a reference-level,Iref, where both arein units of power peran area common to both,P/Ao,a la, the sensing area of the ear, or the sensing area of an underwater transducer.
In either case,Ao is a unit-area equal to1, sinceA1 A2, such thatA1 / A2 = 1.Therefore, sound-intensity,I, inSI-units,[2]is Watts perunit-square-meter,W/mo2, that is:
I, inW/m2 = [(Nm/s)/m2][(N/m2)(m2/N)(s/s) = [(N/m2)2] [Ns/m3]; which is I,inW/m2 = [(N/m2)2][(kgm/s2)s/m3)] = [(N/m2)2] [(kg/m3)(m/s) = p2/c.
Particularly, the squared sound-pressure,p2, is expressed in units of (N/m2)2, and the acoustic impedance,c, is expressed in units ofNs/m3.Thus,I reduces to(Nm/s)/m2, which relates to power perunit-area,P/Ao, which inW/m2can be converted to CGS-units by multiplying byW/m2 by a conversion factor of107, and conversely by10-7.
Discernibly though, when sound-intensity,I, asP/Ao, is expressed indBref, then it is known as a Sound-Intensity Level,SIL; and, by decibel-definition is expressed as apower ratio for a common area,a la, aunit-area, in that,SILdB = 10log10 [P1 :: Po], wherePo issomePref MDS.
Typically, the measure ofSILdB in any medium is referenced, that is, "zeroed," to some set standard, which is not necessarily theMDS that the acoustical receptor can detect,a la, "sense," in that medium.For veritable comparison of differing sound-intensity levels given in decibels,dB, it is imperative that this reference-level be noted.
Simply though, the ratio ofI2 / I1reducesto p22 / p12, wherec /c = 1. Thus, for a Sound-Pressure Level,SPLdB,in air, the reference,prefair, is the Minimum-Audible-Field, theMAF, for the human earin air, which is:
pref MAF = 2.0 x 10-4 dyn/cm2 = 2.0 x 10-5 N/m2 = 20 x 10-6 Pa
= 20 Pa, where N/m2 is defined inMKS units as a Pascal,Pa.
Remarkably though, for sensing acoustical sound-pressure levelsin water, modern-day electrostriction-ceramic transducers,[3]coupled with advanced digital, number-crunching, acoustic signal processors, are "zeroed" to1 Pa, which is20 times more sensitive as a reference than theMAFair; whereas,
20log10 [(1/20)Pa] = -26 dB ref20 Pa"down" from that for20 Pa.
The characteristic acoustical impedances for differing media are experimentally determined, and the measurement of each is certified as a physical constant for universal reference.As such, the gaseous density of air and the speed of sound in it are delineated below-- as measured in the sonic frequency range at0 degrees Celsius, C, and760 millimeters of mercury,mmHg, with0.03-mole-percent content of CO2.Furthermore, from0o C to about+20o C, the speed of sound in air,cair,demonstrably varies by a factor of[60.7 x Tdegrees C]. For reference some comparable values are shown below:
Density of medium:oair = 1.2931 x 10-3 g/cm3at 760 mmHg; and,
1air 200 C = 1.2078 x 10-3 g/cm3at 760 mmHg.
Speed of Sound:coair = 3.3145 x 104 cm/sec at0oC; and,
c1air 20oC = 3.4359 x 104 cm/sec at20oC; such that,
Acoustic Impedance:ocair = 4.2860 x 101 dyns/cm3, and,
1cair 20oC = 4.1499 x 101 dyns/cm3.
Moreover,SILref air is derived from thepref MAF, which is2.0 x 10-4 dyn/cm2, thus:
I air, in W/cm2 = (2.0 x10-4)2 (4.2860 xl01) = 9.3327 x10-10; convert withx10-7,
(2.244x10-5 Pa) (6.8945 xl03 Pa/lbf/in2) = 3.255 x l0-9 lbf/in2.[4]
This is the math that proves that our binaural hearing system can detect very minute changes in sound-pressure levels,within our audible frequency-range.By convention, that audible frequency-range is known as the [our]sonic band. Its range is from16 Hz to 16 kHz, with a maximum sensitivity at about2 kHz, from which our 2,000 Hz conversational band extends to about4 kHz .Also, by convention, frequenciesbelow 16 Hz aresub-sonic, whereas thoseabove 16 kHz areultra-sonic, and thereby denote sound-frequencies that areinaudiblefor us.Notably,super-sonic isa speed greater than the speed of sound,c, in reference to the medium of propagation.
Sound percussions, "beats and bumps," vary in intensity.As an example, consider an explosion of50 pounds ofTNT, which results in achange ofSPL equal toone atmosphere,14.6972 lbf/in2.TheSPLdBfor this near-instantaneous change of pressure-- measured10 feet from the source, reference0.0002 dyn/cm2, or20 Pa, is as follows:
Some examples of sound-intensityin air,[5]referenced to10-16 W/m2, are:
(1)The threshold of painful sound is130 dB, or about0.009 lbf/in2.
(2)The subway-express passing the station emits102 dB, or about0.0004 lbf/in2.
(3)Normal conversational speech at one meter is70 dB, or about0.000009 lbf/in2.
(4)A quiet whisper heard at five feet is18 dB, or about0.00000002 lbf/in2.
Notably, it is painful to feel (sense) a change in pressure on your ear drum of 9/1000th of pound per square inch.
For a denser media, no pun intended, consider seawater at15 degrees Centigrade, and a salinity of36 ppt, parts per thousand, which equates to aSpecific Gravity, also a unit-less ratio, of1.025; such that,
Density,oseawater = 1.025 g/cm3; and,
Speed of Sound, cseawater = 1.505 x l05 cm/s; such that,
Discernibly, the acoustical impedance of seawater,ocseawater, is about3600 times greater thanocair; in that,(1.5426x l05) (4.2860 x 101) 3600:
10log10 [3600] = 35.5630 +36 dBrefc air "up" from air.
In that the speed of sound,c, varies directly with the density of the medium, the acoustical impedance varies accordingly.Notably, if thesame sound-pressure, pair, is applied in seawater as intensely as it was in air, then the correspondingSILseawater will be more due to the greater acoustical impedance in the denser medium.
Notably, the sound in seawater will be +36 dBref0c air "louder" than it was in air.Thus, sound-intensities in different media vary directly with the characteristic acoustical impedance of the propagating medium,cref medium.And, for example, theSPLair ofnormal conversational speechheard at4 feet, or about120 cm, is0.645 dyn/cm2, therefore:
If that same sound-pressure of0.645 dyn/cm2 in air is applied in seawater, then for aSILseawater, aSIL ref for that denser medium must be referenced to theMAF in air, such that:
Albeit thedB levels are the same, the references are different, that is,ref 2.6 x10-20 W/m2in seawater, differs fromref 10-16W/m2in air, and therefore, one deduces that the human ear is better suited for sensingSounds in the Air than it is forSounds in the Sea. Neither is the comparison below veritable, in that the minimum sound-pressure level sensed by the human ear in air is not comparable to the "zeroed" reference level for a modern ceramic transducer in seawater:
= [(-24 dB) (-120 dB)] = 96 dBref 1 Pa, the difference of the references.
With respect to the sensitivity of the acoustical receptor, consider that an earlier design of a magnetostriction[6]electro-acoustic transducer,a la a hydrophone, could be "zeroed" to1 dyn/cm2@4ftas itspref in seawater.In comparison, today's electrostriction electro-acoustic ceramic transducers can be "zeroed" to1 Pa, which is100000 times more sensitive, in that,1 dyn/cm2 =0.1 Pa = 1 x10-5 Pa, a technological advance of+50 dB ref 1 dyn/cm2.
Summation-- with an example. Our binaural hearing system has a low threshold for sensing acoustic energy levelswithin our sonic frequency-band.Moreover, we can discern relatively small changes in those incoming acoustic levels.
Patently, by advances in modern technology, ceramic electrostriction-transducers coupled with powerful digital-signal processors have much lower detection thresholds than we do just hearing through our ears; and, can discern much smaller increments of level-changes.
Some say, perhaps for marketing hype, that their hearing-assisted amplification devices can sense, "hear," a sparrow's heartbeat across the street.Nonetheless, there are devices that can "hear" normal conversation inside a room from across the streetor, from a helicopter patrolling overhead.
In regard to measurement, it is somewhat more difficult [more $$$] to measure changes in sound-intensity, or sound-power levels, and record theSIL indB for the respectiveI, than it is to measure changes in sound-pressure, and simply note theSPL indB as indicated on the meter-face for the impingingp. Similarly, dB can be measured for reciprocatory transducer voltages, as referenced to the electro-mechanical measuring instrument's "zeroed" setting for aMDS.
Practically,SPL in dB isthe preferred measurement forSounds in the Sea.
To close with an intriguing example of a somewhat foreboding man-made sound in the Sea, consider a coal-oil powered [diesel-electric] submarine-warship running submerged at about200 feetmaking170 RPM [8 knots]and, radiating broadband noise from water-cavitations caused by the thrashing rotation of theship's propulsion screws.
Markedly though, the processed sound-pressure spectrum peaks at about28 dBref 1 dyn/cm2@ 4 ft, and is centered around1-kHz. ThisSPLdB equates topseawaterof 2.55 x 101dyn/cm2. ItsSILdB is comparable to102 dBref 10-16W/m2 in the air at the passenger-platform as the subway-express passes through the station; whereas, theIairfor the subway-express is1.58 x 10-6 W/cm2.
Notably,102 dBref 10-16W/m2 is just -3 dB "down" from105 dBref 10-16W/m2, the sound-intensity level at which the US Navy requires the donning of double-ear protection.
Q:Is that close enoughfor government work, or is it a doubling of the sound-intensity?
A: Well now, you know precisely how muchthat is, to wit:2(1.6 x 10-16W/m2).qed.
Most importantly:Always note the dBreference forApple-to-Apple comparisons.
"O, God, Thy sea is so great,
and my boat is so small."
Table of Sound Intensities.[7]
[Note: For IW/m2 = p2/0cair, where 1cair =41.15Ns/m3; and, p ref = 1 dyn/cm2 @ 4 ft for SPLdB.]
SoundSILdB I inp inSPLdB
Type ref10-16W/m2 W/m2 dyn/cm2ref 1 dyn/cm2 @ 4ft
Saturn Rocket1942.4 x103 1.01 x106 120
Flight Deck Ops1401.0 x10-2 2.04 x103 66
Excruciating Pain1301.0 x10-3 6.45 x102 56
Missile Tube Vent1201.0 x10-4 2.04 x102 46
Rock Concert1153.2 x10-5 1.14 x102 41
Marine Diesel1101.0 x10-5 6.45 x101 36
Radial Saw[8]1053.2 x10-6 3.68 x101 31
Subway Express1021.6 x10-6 2.55 x101 28
Paint Chipper1001.0 x10-6 2.04 x101 26
Lawn Mower953.2 x10-7 1.14 x101 21
Niagara Falls921.6 x10-7 8.08 x100 18
Shouted Speech901.0 x10-7 6.45 x100 16
Forklift[9]853.2 x10-6 3.68 x100 11
Conversation701.0 x10-9 6.45 x10-1 -4
Average Office553.2 x10-11 1.14 x10-1 -19
Average Home401.0 x10-12 2.04 x10-2 -34
Rustling Leaves201.0 x10-14 2.04 x10-3 -54
Quiet Whisper186.3 x10-15 1.62 x10-3 -56
MAF reference level 01.0 x10-16 2.00 x10-4 -74
Sample Calculations:
SILdB = 10log10 [1.6 x 10-6 W/cm2] 10log10 [1 x 10-16] [+2 dB -60 dB] [-160 dB] = 102 dB ref10-16 W/m2.
SPLdB = 20 log10 [2.55 x 101dyn/cm2] -10 log10 [1 dyn/cm2] [+8 dB +20 dB] [0 dB] = 28 dB ref 1 dyn/cm2 @ 4ft.
PW/mo2 = p2 1cair = [(2.55 x 101dyn/cm2)2 x 10-3 x 10-4] [(41.1551Ns/m3)] = 1.580W/m2; where,
1cair = 41.1551Ns/m3is for the extant air-density in the subway-express station at the time of measurement.
