<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section{The ocean acoustic environment}

\subsection{Ocean Acoustic Waveguide}

The ocean is an acoustic waveguide limited above by the sea surface
and below by the seafloor.  The speed of sound in the waveguide plays
the same role as the index of refraction does in optics. In the ocean,
density is related to static pressure, salinity, and temperature. The
sound speed in the ocean is an increasing function of temperature,
salinity, and pressure, the latter being a function of depth. 
In general the temperature and salinity are close to constant at
depth, but vary at the surface due to heating and cooling by the
atmosphere and freshwater runoff from rivers. This depth-dependence
is the dominant feature, and for most applications it may be assumed that
the environment is range-independent, i.e. constant in the horizontal.

The Munk profile is an idealized ocean sound-speed profile which
captures this depth-dependence and allows us to illustrate many
features that are typical of deep-water propagation. In its general
form, the profile is given by

\begin{equation}
   c(z) = 1500.0
   \left[
      1.0 + \epsilon
      \left(
         \tilde{z} - 1 + e^{ -\tilde{z} }    \right)
   \right] \,.
\end{equation}

\noindent
The quantity $\epsilon$ is taken to be

$$   \epsilon = 0.00737\,, $$

\noindent
while the scaled depth $\tilde{z}$ is given by

$$   \tilde{z} = { 2\, ( z - z_C ) \over z_C } $$

\noindent
where $z_C$ is the depth of the minimum sound speed, the so-called
SOFAR Channel depth, typically in the range 1000 - 1500 m depth.

The resulting profile for $z_C = 1300 $m/s is plotted in Figure
\ref{fig:5-9}.

\vspace{0.1in}
\begin{figure}[ht]
  \centering 
 \includegraphics[width=0.35\textwidth]{figures/fig5_9.png}
\caption{The Munk sound-speed profile.}
\label{fig:5-9}
\end{figure}

%***************************************************************************

\subsection{Ray tracing}

As in optics, a ray of sound propagates in a depth-dependent medium
according the Snell's law

\begin{equation}
  \frac{\cos \theta(z)}{c(z)} = \mbox{Const.}
  \label{eq:snell}
\end{equation}

\noindent
where $\theta(z)$ is the grazing angle, i.e. the angle with horizontal
for a ray at depth $z$. For a sound speed profile that is linear in
depth, i.e. when temperature and salinity are constant, then this
leads to ray paths that are circle section as described below.  For
the general sound speed variation, the ray tracing is performed by
solving the coupled ordinary differential equations.  In cylindrical
coordinates $(r, z)$ these ray equations are of the form:

\begin{eqnarray*}
      {dr \over ds} &amp; = &amp; c\, \xi ( s ) \,, \qquad
      {d \xi  \over ds } =  -{ 1 \over c^2 }\, { dc \over dr }\:, 
\label{eq:rays2D1}\\
      {dz \over ds} &amp; = &amp; c\, \zeta ( s ) \,, \qquad
      {d \zeta \over ds } =  -{ 1 \over c^2 }\, {  dc \over
        dz }\:,
\label{eq:rays2D2}
\end{eqnarray*} 

\vspace{0.1in}
\noindent
where $[ r(s), z(s) ]$ is the trajectory of the ray in the
range--depth plane, with $s$ being the arc-length, shown schematically
in Figure \ref{fig:3-2}. Note that we assume here that the sound speed
profile is range-independent, i.e. $dc/d r = 0$.  We have introduced
the auxiliary variables $\xi (s)$ and $\zeta (s)$ in order to write
the equations in first-order form.  Recall that the tangent vector to
a curve $[ r(s), z(s) ]$ is given by $[ dr / ds,\, dz /ds ]$.  Thus
from the above equations the tangent vector to the ray is $ c\, [\xi(
  s ),\zeta( s)]$.
 

%****************************************************************************
\vspace{0.1in}
\begin{figure}[ht]
  \centering 
  \includegraphics[width=0.6\textwidth]{figures/raygeometry.png}
\caption{Schematic of 2-D ray geometry.}
\label{fig:3-2}
\end{figure}
%****************************************************************************

This set of ordinary differential equations is solved numerically
using numerical methods such as Euler's method
and Runge-Kutta integration. However, to complete the specification of
the rays we also need initial conditions.  As indicated in
Figure \ref{fig:3-2}, the initial conditions are that the ray starts at
the source position $(r_0, z_0)$ with a specified take-off angle
$\theta_0$. Thus we have 

\begin{eqnarray*} 
  r &amp; = &amp; r_0\,,\qquad \xi = { \cos
  \theta_0 \over c( z_0 ) } \:,\\ z &amp; = &amp; z_0\,,\qquad \zeta = { \sin
  \theta_0 \over c( z_0 ) } \:.
   \label{eq:rayics}
\end{eqnarray*}


The source coordinate is of course a given quantity whereas the take-off angle
$\theta_0$ for the moment is an unknown variable. 

\subsection{Part  3. Acoustic Pressure}

The pressure field amplitude along each ray path is 
\begin{equation}
   p( s ) = { 1 \over 4 \pi }
   \left|
      { c( z(s) ) \cos \theta_0 \over c( z_0 )\, J( s ) }
   \right|^{1/2}
  \label{eq:p_s}
\end{equation}

\noindent
where $J(s)$ is a measure of the relative cross-sectional area of the
'ray tube', varying as the ray propagates, and $\theta_0$ is the
launch angle of the ray at the source.  One can see from the geometry
in Figure \ref{fig:3-4} that the cross-sectional area is just the
hypotenuse:

\begin{equation}
   J  = r \left[  
             \left( { \partial z \over \partial \theta } \right)^2
           + \left( { \partial r \over \partial \theta } \right)^2
          \right]^{1/2}\!. \label{eq:J2D}
\end{equation}

\noindent
The extra factor of $r$ accounts for the fact that we have assumed
cylindrical symmetry so that Figure \ref{fig:3-4} is really just
showing a slice through a ray tube that is rotated around the $z$
axis.

%****************************************************************************
\vspace{0.1in}
\begin{figure}[ht]
  \centering 
  \includegraphics[width=0.7\textwidth]{figures/raytube.png}
\caption{The ray-tube cross section.}
\label{fig:3-4}
\end{figure}
\vspace{0.1in}

%****************************************************************************

\noindent
The cross-sectional area may be calculated by putting 
calipers on the ray trajectories, i.e. using finite-difference approximations. 
This yields the approximation:

\begin{equation}
   J( s ) = { r_i( s ) \over \sin \theta } \,{ r_{i+1}(s) - r_i(s) \over d\theta_0 } \:,
\end{equation}

\noindent
where $r_i(s)$ and $r_{i+1}(s)$ are bracketing rays forming a ray
tube, and $\theta$ is the local grazing angle at arc-length $s$.

As in geometric optics, the rays hitting the surface or seabed will
reflect. Assume the reflection is governed by the rule that reflected
angle is equal to the incident angle on the interface. For your
pressure computation, assume a ray reflected from the seabed undergoes
a loss of 6 dB.


The acoustic pressure is commonly expressed in terms of {\em
  Transmission Loss}, which is a logarithmic measure of the relative
amplitude change along the ray path, with the reference being the
pressure at 1 m distance, i.e.

\begin{equation*}
 TL(s) = -20  \log_{10} \left(\frac{p(s)}{p(1)}\right)
\end{equation*}

\noindent
with $p(1) = 1/4 \pi$ when using the source normalization Eq.(\ref{eq:p_s}).

\subsection{Linear Sound Speed Profile}
\label{sec:lin_svp}
%****************************************************************************

\vspace{0.1in}
\begin{figure}[ht]
  \centering 
  \includegraphics[width=0.85\textwidth]{figures/circleray.png}
\caption{Ray path in a medium with linear sound-speed variation.}
\label{fig:3-18}
\end{figure}
%****************************************************************************

\noindent
It can be shown \cite{COA_11}, that in an ocean with a sound speed
that is linear in depth,

\begin{equation*}
c(z) = c(0) + g z \; ,
\end{equation*}

\noindent
the ray paths will be circular arcs. The circles are centered along
the line where the sound speed vanishes,

\begin{equation*}
z = - c(0)/g
\end{equation*}
 
\noindent
and have an arbitrary radius

\begin{equation*}
R = \frac{1}{ag} \; ,
\end{equation*}

\noindent depending on the choice of $a$, as shown in Figure \ref{fig:3-18}.

The value of a is determined by the initial conditions for the
ray. Thus, for a ray launched from depth $z_0$ at grazing angle
$\theta_0$, Snell's law, Eq. \ref{eq:snell}, requires the sound speed
at the deepest point $z_{\max}$ of the array, the turning point, to be

\begin{equation*}
c(z_{\max}) = c(0) + g z_{\max} = \frac{c(z_0)}{\cos \theta_0} \; ,
\end{equation*}

\noindent yielding

\begin{equation*}
z_{\max} =  \frac{c(z_0)}{g \cos \theta_0} - \frac{c(0)}{g} \; .
\end{equation*}

The last term is the depth of the center of the circular arcs, and the
first term is therefore the radius, yielding

\begin{equation}
R = \frac{c(z_0)}{g \cos \theta_0} \; .
\end{equation}

Thus, the arbitrary parameter $a$ is simply the constant in Snell's
law, Eq.~\ref{eq:snell},

\begin{equation*}
a = \frac{1}{gR} = \frac{\cos \theta_0}{c(z_0)} \; .
\end{equation*}

\noindent
Now one can easily compute the ray paths in range and depth. Thus, the arc length is

\begin{equation*}
s = R(\theta_0 - \theta) \; ,
\end{equation*}

\noindent
yielding, together with simple projection of the circular arc,

\begin{eqnarray*}
r(s) &amp; = &amp; R \sin \theta_0 - R \sin \theta \nonumber \\
     &amp; = &amp; R \left[\sin \theta_0 + \sin ( s/R - \theta_0)\right] \\
z(s) &amp; = &amp; R \cos \theta - c(0)/g \nonumber \\
     &amp; = &amp; R \cos ( s/R - \theta_0) - c(0)/g
\end{eqnarray*}

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