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The propagation of sound in the sea, covering topics such as the surface duct, the deep sound channel, caustics and convergence zones, shallow water ducts, reflection and scattering by the sea surface and bottom, temporal and spatial coherence, and multipaths. It includes theoretical models, experimental verification, and field observations. The document was written for the Defense Advanced Research Projects Agency in 1979 by R.J. Urick, an Adjunct Professor of Mechanical Engineering at The Catholic University of America in Washington, D.C.
Typology: Lecture notes
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Fig. 2. Number of papers and letters to the editor per year in JASA 1977 on the subject "Propagation of Sound in Water. Attenuation. Fluctuation" (Class 13-2 - 30.20 - 43.30B), as obtained from the Cumulative Indices to 1968 and individual Journal volumes beyond. The curve illustrates an exponential increase of published papers beyond 1950.
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Methods of Research A wide variety of techniques are available, and have been used, in propagation studies. Most widely used at the present time is computer modelling, wherein a mathematical model of the ocean is set up and exercised by a com- puter program. The results of such studies, however, while relatively inexpensive to carry out, are often of hardly more than academic interest unless closely tied to real-world data, because of the difficulty of realistically modelling the real ocean. Of a similar nature to computer models are analog models, wherein the ocean is physically modelled in a laboratory tank, usually scaling the frequency according to the size of the tank. An example is the work of Wood (15) on shallow water propagation using a tank 20 feet long, 5 feet wide and 8 inches deep. In the laboratory, too, the velocity of sound in sea water has been measured to an exquisite degree of precision by a variety of techniques (16), while the attenuation coefficient has been determined in the laboratory by mea- suring the rate of decay of sound in a resonant glass sphere containing sea water (17). Concerning sea-going studies and measurements, a variety of different measurement platforms have been used. These include (1) Two ships, one a source ship and the other a receiving ship, changing the range between them and making a transmission run so as to yield level vs. range (18)*. In echo-ranging studies, one ship is the target submarine from which echoes are obtained and recorded. This is the oldest, and now classic, technique, but it is now almost obsolete because of the expense and slowness of using two ships and the availability of other sources, receivers and platforms. (2) One ship and an aircraft, where the latter drops explosive sound signals while flying toward or away from the former (19). (3) One aircraft alone, using sonobuoys for reception and recording on board the aircraft (20). A surface ship, launching a sonobuoy and steaming away from it, may replace the aircraft, though at a sacrifice of the range to which a run can be made and speed of conducting the field exercise. (4) One bottommed hydrophone or hydrophone array, cable-connected to shore, receiving signals transmitted from a ship (21) or the explosive shots dropped by an aircraft (22). (5) Two bottomed transducers, one a source, the other a receiver, in studies of the fluctuation of sound trans- mission between two fixed points in the sea (24).
Rectification of Transmission Run Data When a transmission run has been made, it is desirable to fit the data to some simple rational model that gives the results some physical meaning. A commonly used model is
10 log lr = 10 log lQ - (N log r - Ar)
where 10 log lr is the measured signal level at range r, 10 log lQ is the source level of the source of sound, and N and A are coefficients to be found from the data. The N log r term is the spreading loss; the Ar term, called, during World War II, the transmission anomaly, includes every other source of loss (assumed proportional to range). For physical reasonability, N has to be assumed to be either 10, 20 or 30, corresponding to cylindrical, spherical, or hyperspherical spreading (spherical spreading plus time stretching). In shallow water, there is theoretical justification to take N = 15 under some circumstances (25). No other values of N, while they may fit the data better statisti- cally, are physically meaningful. Transmission run data are readily fitted to this model, after selecting a value for N by plotting TL - N log r against range and determining A from the slope of a straight line through the plotted points. For transmission in sound channels, an appropriate model is
10 log lr= 10loglo-(10logro + 10logr + Ar)
where r 0 is a constant and N is taken as 10. A rectified plot of measured data fitted to this model yields 10 log r 0 from the intercept of the line at r = 1 yd and A from the slope as before. Fig. 4 is an example of field data "rectified" according to this expression. Here 10 log lr + 10 log r is plotted against r;the slope of the fitted line, in any octave frequency band, is the attenuation coefficient in that band. The intercept of the line at 0 miles (actually 1 yd.) is the quantity 10 log l 0 - 10 log r 0 , from which r 0 can be found if the source level 10 log lQ is known.
*The references in this section are to examples reported in literature, where some experimental details can be found.
1-
Navy Sei. Serv. 20, 185, 1965.
♦Throughout this book, the abbreviation JASA is used to denote the Journal of the Acoustical Society of America.
For small changes in density and pressure, and for adiabatic (i.e., rapid) changes, meaning that there is no conduc- tion of heat, this relationship becomes
p = Ks 3)
where p is the incremental pressure, or the difference between the instantaneous pressure and the ambient pressure. K is a proportionality factor relating p to s and is called the bulk modulus of the fluid. (d) Equations of Force. The three components of force per unit volume of a non-viscous fluid are related to the pressure acting on a unit volume according to
/x
3p 3x
JY 3y
These equations say merely that the force in a given direction is the negative of the rate of change of pressure in that direction.
Derivation of the Wave Equation The wave equation is a partial differential equation that combines the above four sets of equations. The synthesis is accomplished as follows: (a) Eliminate fx, fy, fz from 4) and 2). (b) Differentiate 3) with respect to x, y and z and substitute in a). (c) Differentiate again and add. (d) Substitute 1), and obtain a partial differential equation in terms of the condensation s. (e) Use 3) again to eliminate s and obtain an equation in terms of p.
The result is
2
The quantity -£- can be shown to have the dimensions of the square of a velocity and may be replaced by c^2. Our "o final result is a relationship between the temporal and spatial changes of pressure in a sound wave, called the wave equation:
Solutions Thereof For a particular problem, the wave equation we have just obtained must be solved for the boundary and initial conditions that apply in that problem. Boundary conditions are the known pressures and particle velocities existing at the boundaries of the medium. In underwater sound, one ubiquitious boundary condition is that the pressure p is zero along the plane (or wavy) sea surface. Initial conditions are those of the source of sound, whose location is usually taken to be at the origin; p(o,o,o,t) ordinarily is a known function that describes the manner is which the source varies with time. The initial (source) conditions play no part in ray acoustics, in which the sound field is described by rays. Both boundary and initial conditions are needed for a wave theory solution; such a solution is therefore complete, at least to the extent that the model of the real world that it assumes is accurate. The wave theory solution is difficult or impossible to obtain whenever the real surface and sea bottom have to be realistically approximated.
Wave Theory In one dimension the wave equation is
92 p _ 2 92 p 3t^2 C^ 9x^2
This was known to the mathematicians of the 18th century to be satisfied by an (arbitrary) function of (t-x/c) or (t+x/c). This may be seen by differentiating f(t±x/c) twice and noting that the wave equation is satisfied regardless of what the function f may be. Moreover, it can be shown that p=f(t±x/c) is necessary as well as sufficient; that is, any function not of this form cannot satisfy the wave equation. If the pressure p=f (t-x/c) at an arbitrary point xi and an arbitrary time ti happens to be same as that at points x 2 at time t 2 , then we must have tx -X!/c = t 2 - x 2 /c, from which it follows that c = (xj-XjJAtj-ta). The quantity c may therefore be interpreted as the velocity of propagation along the line from x 2 to Xj. Similarly, from p=f(t+x/c), c may be seen to be the propagation velocity from xl to x 2. Therefore, functions with arguments (t-x/c) represent waves travelling in the +x direction; those having arguments (t+x/c) represent waves going in the -x direction. If the 3-dimensional wave equation is transformed to spherical coordinates, it will be found that functions of the form
p =-/(t±-!-)^1 r r c
will be solutions. These represent spherical waves radiating to or from the origin. Commonly selected functions are the cosine, sine and exponential functions which are added to satisfy the source and boundary conditions. Returning to one dimension, we try solutions of the form
p = iMx) eiw(t-^
where \pM is any function of the space coordinate x. One substituting, we find i//(x) has to satisfy the relationship
.,. c^2 d^2 1// (^) n ordx^2
which has the solution
t//(x) = A sin kx + B cos kx
where k = c/co and w = 2wf. Let us now apply this to waves travelling vertically between a plane pressure-release sea surface where p=0 and a plane rigid bottom, where p is a maximum. Take the x coordinate to be in the vertical, and let the surface be at x= and the bottom at x=H. The upper boundary condition requires that p(0)=0, from which it follows that B=0. Our solution is now
p = Asinkx • (^) eiw(t-^
To satisfy the lower boundary condition, we return to the basic equations and eliminate f„ from 2) and 4) and obtain
9u 1 9p 9t p 9x
Since the particle velocity u=0 at x=H, 9u/9r must be zero there also or
SU"' Hence we must have
