Simple sound propagation distance estimation

0. Brief description. Very basic theory

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

P_TX = P_T · U_a    (1)

Where P_T - antenna's transmitting sensitivity, Pa/V, U_a - 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 α_e. 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:

α_g = 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 · α_e · 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:

P_TX · 10^α/10 = P_N · SNR    (4)

Or, substituting (3) into (4):

10^{(-k · log(r)/(10 · log(10)) - 0.0001 · α_e · r)} = P_N · SNR / P_TX    (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	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]	*α_e*		dB/km

**Table 2. Transmitter antenna and ambient noise parameters**
Parameter	Notation	Range	Units	Description
Transmission sensitivity	*P_T*	0.5 .. 10000	Pa/V	On specified frequency f
Voltage on the antenna	*U_a*	1 .. 1000	V	RMS
Noise pressure	*P_N*	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	*P_TX*		Pa
Desired SNR	*SNR_t*		times

**Table 3. Results**
Parameter	Notation	Range	Units
Maximal distance	*r_max*		m
Number of iterations taken	*N_i*	1 .. 1000
Residual	ε		m

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

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