Simple sound propagation distance estimation

0. Brief description. Very basic theory

We want to estimate the distance at which, for a given pressure PTX developed by an underwater acoustic transmitting antenna at a given frequency f, and a given noise pressure PN at the receiving point, the residual pressure arriving at the receiver will have a given signal-to-noise ratio SNR.
The pressure PTX developed by the transmitting antenna is defined as:

PTX = PT · Ua    (1)

Where PT - antenna's transmitting sensitivity, Pa/V, Ua - RMS voltage applied to the antenna.

In this simplified calculation, we assume that there are only two types of losses: the first, associated with the absorption of sound in the medium, described by the absorption coefficient αe. The absorption coefficient in the general case is a function of the parameters of the medium: temperature t, salinity s, acidity pH, depth h and frequency of the signal f. The dependence of the absorption coefficient is described, for example, in the work of Francois-Harrison[1].

The second type of loss is associated with the geometric spreading of the signal energy and is described as follows:

αg = k · 10 · log10(r)    (2)

where r - propagation distance in meters, k - coefficient of the signal's front shape (1 - for cylindrical, 2 - for spherical).

The total path loss can now be described as follows:

α = k · 10 · log10(r) + 0.001 · αe · r    (3)

A factor of 0.001 is used to convert dB/km to dB/m.

Now we can write the equation from which we define r:

PTX · 10α/10 = PN · SNR    (4)

Or, substituting (3) into (4):

10(-k · log(r)/(10 · log(10)) - 0.0001 · αe · r) = PN · SNR / PTX    (5)

Obviously, r, for which equality (5) holds, is the desired distance.
Equation (5) in its general form cannot be solved analytically, and the use of numerical methods is required. In this document, Newton's method is used.

1. Calculation

Table 1. Absorption due to the medium
Parameter Notation Value Range Units
Frequency f 1 .. 1000 kHz
Water temperature t -2 .. 40 °
Depth h 1 .. 104 m
Water salinity s 0 .. 42 PSU
Acidity pH 7.7 .. 8.3
Absorption coefficient[1] αe dB/km

Table 2. Transmitter antenna and ambient noise parameters
Parameter Notation Value Range Units Description
Transmission sensitivity PT 0.5 .. 10000 Pa/V On specified frequency f
Voltage on the antenna Ua 1 .. 1000 V RMS
Noise pressure PN 0.001 .. 103 Pa On the receiving antenna
Desired SNR SNR -5 .. 20 dB On the receiving point
Geometric spreading coefficient K 0.1 .. 10 1 - cilyndrical, 2 - spheric
Transmitter pressure PTX Pa
Desired SNR SNRt times

Table 3. Results
Parameter Notation Value Range Units Description
Maximal distance rmax m
Number of iterations taken Ni 1 .. 1000
Residual ε m

  1. Francois & Garrison, J. Acoust. Soc. Am., Vol. 72, No. 6, December 1982.

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