The definition of
transmission loss (TL) is, " The accumulated decrease in acoustic intensity as
an acoustic pressure wave propagates outwards from a source." As the acoustic
wave propagates outwards from the source the intensity of the the signal is
reduced with increasing range due to:
1) Spreading
2)
Attenuation
1)
Spreading
a) Spherical (Freefield
spreading)
Consider sound propagating outwards from a source S in a directive beam
as illustrated by the diagram below:
(Diagram to be included later)
The intensity at range R is given by the power P per unit area. It can be seen from the diagram above that the circular area which the power is distributed over at a range R is given by pr^{2}. You will also notice that the radius of the circular area increases in proportion with the range R. Thus, the intensity is given by;
I = P / (pr^{2})
and since r is proportional
to R,
Equation 1 (The inverse square law)
Equation 1 is the inverse square law
which tells us that the acoustic intensity is reduced in proportion to the
square of the range due to spreading alone.
The formal definition of spreading on the decibel scale is:
TL = 10 log(Intensity at a distance of 1m / Intensity at a distance R)
TL = 10 log({P/2p1^{2}} / {P/2pR^{2}}
TL = 10 log(R^{2})
TL = 20 log(R) Equation 2 (Spherical spreading law)
b) Cylindrical
spreading
The spherical spreading law will apply when sound energy spreads
outwards with no refraction or reflection from boundaries (e.g. the sea floor or
surface). However, in a sound channel or shallow water where there are
reflections from the sea floor / surface spreading may be considerably reduced
by refraction and reflection. Under these conditions a cylindrical spreading law
of the following form is more appropriate:
TL = 10 log(R) Equation 3 (Cylindrical spreading law)
However, since sound energy is not perfectly contained by reflection (reflection coefficients less than 1) and refraction the true spreading is often somewhere between the predictions given by equations 2 and 3. Thus, a practical spreading equation which represents an intermediate spreading condition between spherical and cylindrical spreading is:
TL = 15 log(R) Equation 4 (Practical spreading law)
2) Attenuation (or
absorption)
Transmission loss due to attenuation is represented
in the sonar equations in terms of an attenuation coefficient 'a' with the units of dB/m.
There are two primary causes of attenuation:
1) Viscous friction
2) Ionic relaxation
phenomena
Attenuation due to viscous friction refers to the conversion of sound energy to heat due to internal friction at a molecular scale within the fluid. Viscous friction is the dominant mode of attenuation at frequencies above 1MHz. The attenuation coefficient is strongly frequency dependent with attenuation increasing rapidly with frequency. An approximate expression for the attenuation coefficient (a_{1}) for freshwater due to viscous friction only is given by:
a_{1} = (2.1x10^{10}(T38)^{2}+1.3x10^{7}) f^{2} dB/m (equation 5)
Here T is the temperature (centigrade) and f is the frequency (kHz).
At frequencies below about 500kHz the presence of certain dissolved salts in sea water increase the attenuation coefficient to a level which is higher than that predicted by equation 5. The dominant form of absorption below 100kHz is due to the ionic relaxation of magnesium sulphate (MgSO_{4}). The ionic relaxation process involves the dissassociationreassociation of MgSO_{4} ions in sea water due to the pressure fluctuation resulting from the propagation of the sound wave. Although MgSO_{4} makes up only 4.7% by weight it is by far the greatest absorption constituent of sea water (even compared to NaCl). A similar empirical absorption coefficient which accounts for the effect of MgSO_{4} relaxation (only) is given by;
a_{2} = bf_{o}(1+(f_{o}/f)^{2})^{1} dB/m (equation 6, MgSO_{4} relaxation)
where, b = 2Sx10^{5 }and^{ }f_{o} = 50(T+1).
At lower frequencies still (<700Hz) boric acid relaxation becomes significant. The effect of boric acid relaxation is given by:
a_{3} = cf_{1}(1+(f_{1}/f)^{2})^{1} dB/m (equation 7, Boric Acid relaxation)
where, c = 1.2x10^{4} and f_{1} = 10^{(T4)/100}.
The overall absorption due to viscous friction and MgSO_{4 }/ boric acid relaxation is given by the sum of equations 5 to 7:
a = a_{1 }+ a_{2 }+ a_{3 } (equation 8, Total Absorption)
A graphical representation of equations 58 is given below.
Also included on the graph for comparison is a simpler but less accurate equation for the absorption coefficient equation given by:
a = (0.17x10^{3}f^{2}) / (T+18) (equation 9, Simple Solution)
It can be seen from the
diagram below that equation 9 provides a conservative estimate for absorption at
frequencies above 10kHz (10^{4}Hz).
The attenuation coefficient given by the previous equations will decrease with depth (pressure). Some empirical corrections for depth are given by Urick (1975).
The combined affect of spreading and absorption are given by:
TL = 20 log(R) + aR Equation 10 (Spherical spreading + absorption)
This equation is for spherical spreading and can be simply modified for the other types of spreading mentioned earlier.
However, measured
transmission losses are often at variance with those calculated using equation
10 due to the combined effects of other complicating factors such
as:

Multiple path propagation.
 Refraction effects.
 Diffraction and
scattering of sound by particulates, bubbles and plankton within the water
column.
Whilst it is possible to derive theoretical expressions to account for these processes it is customary (and simpler) to combine them in a single term called the transmission anomaly (A). Where A is measured in dB. Thus, the oneway transmission loss becomes:
TL = 20 log(R) + aR + A Equation 11
Note that when A is used it is usual to apply a spherical spreading model. This is because the transmission anomaly is meant to account for the effects of refraction.
The graph below is a plot of TL (predicted by equation 11) versus range. Three different frequencies are shown. In this example a is given by equation 8 and T=14^{o}C, S=35ppt and A=5dB.
You can
experiment with the efffects of these different spreading and absorbtion models
for yourself by clicking here
to opperate the transmission loss calculator written in JavaScript code. Try
experimenting with different spreading and absorbtion models.
The transmission loss is an important term in the sonar equations as it is the only parameter which is a function of the range (R). When designing a sonar system the maximum range can be determined by rearranging the sonar equations in terms of TL, (assuming values can be assigned to all other parameters). For example the active noise background sonar equations give:
SL + DI_{T} + TS  2TL  (NLDI) = DT
Therefore:
TL = 0.5(DT + SL + DI_{T} + TS  [NLDI]) Equation 12
Having evaluated TL the range can be determined by solving equation 11 iteratively. The iterative method is necessary since it is impossible to rearrange equation 11 in terms of R. The iterative method is quite simple. It simply involves systematically choosing values of R (a and A are known) and evaluating TL from equation 11. The result is compared with that given by equation 12 and new value of R is selected based on this comparison. The process is iterated until the difference between the results of equations 11 and 12 is negligible. The resulting value for R is the maximum range of the sonar. (NB An alternative and more efficient approach is to use the NewtonRapson method).