# relationship of frequency with range in Underwater acoustic channel

I am interested to know what range of frequencies can be used/ or are effective for wireless communication, within the underwater acoustic channel. I do know a table exists from \cite{akyildiz2004challenges, akyildiz2005underwater, kheirabadi2013greedy}, which is reproduced below. I want to add another column in this Table to add the range of frequencies that would be effective for the given range of propagation distances. But cannot get an idea of how to get it done.

What determines the effective range of frequencies for a given frequency and distance in underwater acoustic channels are the frequency and distance dependent path loss, and the power spectral density (PSD) of the noise.

The path loss is given by

$$A(f,d)=d^{\kappa}\left[\alpha(f)\right]^d$$

where $$d$$ is the distance between the transmitter and receiver in Km, $$f$$ is the frequency in KHz, $$\kappa$$ is the spreading factor which is usually assumed 1.5 for practical consideration, and $$\alpha(f)$$ is the absorption coefficient which usually calculated using the Thorp's formula as (measured in dB/Km)

$$10\log_{10}\alpha(f) = 0.11\frac{f^2}{1+f^2}+44\frac{f^2}{4400+f^2}+2.75\times 10^{-4}f^2+0.003$$.

The PSD of the noise is the sum of the PSD of the noise sources in UWA channels, which are: turbulence, thermal, shipping, and wave noise. You can find the equation of each noise in the literature.

Having the path loss and the PSD of the total noise, you can find the 3-dB bandwidth as following: the narrow-band signal-to-noise ratio (SNR) is given by

$$\gamma(f,d)=\frac{P/A(f,d)}{N(f)\Delta f}$$

where $$N(f)$$ is the PSD f the total noise, $$P$$ is the transmit power, and $$\Delta f$$ is the frequency over which the channel is frequency-flat. For a given distance $$d$$, there is an optimum $$f$$ that maximizes the SNR (which basically minimizes the term $$\frac{1}{A(f,d)N(f)}$$). Let that frequency denoted by $$f_d$$. Then you find the range of frequencies for a given distance such that

$$\gamma(f,d) \geq \frac{1}{2}\gamma(f_d,d)$$.

This way you can find the range of frequencies that can can be used for UWA communication for a given distance.