Page 15 - \Winter 2015

e Equations
Equations
to demonstrate the need to adopt a more rigorous terminol- ogy if we wish our science to be taken seriously as it enters its second century of existence.
By convention, the sonar equations are written in a logarith-
mic form by converting ratios of acoustic intensities to dif-
ferences between the corresponding levels in decibels. The
“intensity” is usually not the true intensity of the sound in
question, but its equivalent plane wave intensity (EPWI),
defined as the magnitude of the time-averaged intensity of
a propagating plane wave with the same root-mean-square
(RMS) sound pressure as that sound (Urick, 1983). De-
noting the time-mean-square sound pressure (MSP) by p2,
and characteristic impedance (Morfey, 2000) by Z, inten- N
sity ratios are therefore formed by dividing the noise EPWI,
and the pertinent question, given the need to define I0 un-
ambiguously, becomes “what are p0 and Z0?” During the
period 1951-1960 one could have answered this question
with some confidence. The then current US acoustical ter-
minology standard ASA Z24.1-1951 (ASA, 1951), published
by the American Standards Association (ASA) – now the
American National Standards Institute (ANSI), permit-
ted both p = 20 μPa and p = 105 μPa. Further, Z24.1-1951 00
specified a standard reference sound speed of c0 = 1500 m/s,
with the reference density (ρ0 ≈ 1023.38 kg/m3) inferred
from specified conditions of temperature and pressure. The
corresponding reference impedance is Z0 = p0 c0 ≈ 1.53507
MPa s/m (see Figure 1), from which the reference intensity
can be calculated as either I ≈ 260.57 aW/m2 (using p = 20 00
μPa) or 6.5144x109 aW/m2 (p = 105 μPa), where 1 aW (one
attowatt) = 1018 W.
Figure 1. Extract from withdrawn American Standard Acoustical Terminology Z24.1-1951, entry 9.040 Standard Sea Water Condi- tions (ASA, 1951). The value inferred for the impedance of seawater under these standard conditions was 1.53507 MPa s/m. The CGS unit of impedance is 1 dyn s/cm3 = 10 Pa s/m. © ASA. This extract, reproduced with permission of ANSI and the Acoustical Society of America, is not part of an approved American National Standard, nor may it be referred to as such. All rights reserved.
(2)
N (x)=p2(x)/Z(x), where p is the noise sound pressure, EPWIN N
0
by a reference intensity I0. In equation form, the noise level according to these conventions, using x to denote the receiv- er position, is
(1)
(1) 𝐿𝐿N 𝒙𝒙 =10log!" 𝐼𝐼
!
𝑁𝑁EPWI 𝒙𝒙
dB.
A similar equation can be written for the signal level, and the difference between these two levels is a logarithmic measure of signal-to-noise ratio.
While the value of I0 in Equation (1) is not needed for the signal-to-noise ratio (because it cancels), when either of sig-
ing and interpretation of that level requires a shared under-
standing of the value of I0. By a convention that dates to the
nal or noise level is reported separately, the correct report-
Second World War (Horton, 1959), the reference intensity
𝑁𝑁EPWI 𝒙𝒙 𝑝𝑝! 𝒙𝒙 𝑍𝑍!
in underwater acoustics is understood to be the magnitude
=!,
𝐼𝐼𝑝𝑝𝑍𝑍𝒙𝒙
of the time-avera!ged inten𝑁𝑁sity! of a𝒙𝒙 propagating plane wave
reference pressure (p0) (Urick 1967, 1983). For example,
with p0 = 1 μPa, the level would be reported in units of “dB re
1 μPa” (or, equivalently, “dB // 1 μPa”), a shorthand used to
mean that the level expressed in decibels is that of the EPWI,
relative to the magnitude of the time-averaged intensity of
a plane wave whose RMS sound pressure is 1 μPa (Urick
1967). The reference intensity according to this convention
is I = p2/Z , where Z is the characteristic impedance of sea- 0000
water. If the local impedance is Z(x) it follows that
𝐿𝐿 𝒙𝒙 =10log EPWI
dB.
N !"
in seawater whose RMS sound pressure is equal to an agreed
𝐼𝐼!
𝑁𝑁EPWI 𝒙𝒙 𝑝𝑝! 𝒙𝒙 𝑍𝑍! (2) =!,
In 1960, ASA Z24.1-1951 was superseded by ANSI S1.1- 1960((1A)NSI, 1960), which made no mention of a standard reference impedance for use in water, and introduced in its place the standard reference intensity of 1 pW/m2, where 1 pW (one picowatt) = 1012 W. In 1969 the modern reference value of sound pressure p0 = 1 μPa was adopted by ANSI S1.8-1969 (ANSI, 1969) for sound in liquids. Today these standard values for sound pressure and sound intensity in liquids are recognized by both the International Electrotech- nical Commission (IEC) (IEC, 1994) and the International Organization for Standardization (ISO) (ISO, 2013). The situation is summarized in Table 1.
𝐼𝐼! 𝑝𝑝!𝑍𝑍𝒙𝒙
(2)
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