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An Introduction to the Sonar Equations with Applications (Revised)

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An Introduction to the Sonar Equations with Applications (Revised) Calhoun: The NPS Institutional Archive Reports and Technical Reports All Technical Reports Collection 1979-02-01 An introduction to the sonar equations with applications (revised) Coppens, Alan B. Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/15240 DSTGRADUATE SCHOOL Monterey, Galifornia --- --- X? IYTRODUCTION TO THE SONAX EQVAT'COXS XITIi iUDPLLCATIONS (REVISED) b Y X.S. Coppens, H.X. 3ahl, and J.V. Sanders NPS Publication Approved for public release; distribution unlimited Prepared for: :\Tavl Postgraduate Schoc.1 Monterey, California 93940 v [d 7,$ $- B *;* i NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral Tyler F. Dedman Jack R. Borsting Superintendent Provost The materials presented herein were produced with the support of the Department of Physics and Chemistry and the Office of Continuing Education, Naval Postgraduate School. Reproduction of all or part of this report is authorized. This report was prepared by: h4.4~ Harvey A. V~ahl ~ssistantProfessor of Physics 4WZ > - ,,'"/ James V. Sanders / Associate Professor of Physics i / Reviewed by: <---- - --7 d& d& 2~ /& K.E. Woehler, Chairman William M. Tolles Department of Physics Dean of Research and Chemistry Unclassified CURITY CLASSIFICATION OF THIS PAGE (When Date Entered) READ INSTRUCTIONS REPORTDOCUMENTATIONPAGE BEFORE COMPLETING FORM REPORT NUMBER 12. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER I I TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED AN INTRODUCTION TO THE SONAR EQUATIONS WITH APPLICATIONS (REVISED) NPS Publication 6. PERFORMING ORG. REPORT NUMBER AU THOR(8) 8. CONTRACT OR GRANT NUMBER(*) Alan B. Coppens, Harvey A. Dahl, and James V. Sanders PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK Naval Postgraduate School AREA 6 WORK UNIT NUMBERS Code 61Sd/Cz/Dh Monterey, California 93940 I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Naval Postgraduate School, Code 61 Febraury 1979 Monterey, California 93940 13. NUMBER OF PAGES 144 4. MONITORING AGENCY NAME & AOORESS(if different from Controlling Office) 15. SECURITY CLASS. (of thia repor() Unclassified ' is.. DECLASSIFICATION/ DOWNGRADING SCHEDULE 6. DISTRIBUTION STATEMENT (of thia Report) Approved for public release; distribution unlimited 7. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, If dlfferent horn Report) 18. SUPPLEMENTARY NOTES 19. KEY WOROS (Continue on reverse eide If neceossry and idsntify by block number) Sonar Transmission Loss Underwater Acoustics Attenuation Sonar Equations Detection Underweter Sound Propagation 10. ABSTRACT (Continue on revase aide ff noceee- and fdentiIy by block number) This report provides an introduction to the SONAR equations for those tnterested in underwater sound as applied to ASW but lacking either the mathematical background or the time for a more rigorous presentation. This is a substantially revised extension of material developed in a previous report (NPS-61Sd-76-071). While this material should be supplemented by lectures or a study guide, we have attempted to design the material so that it is reasonably self-explanatory, communicating many of the essentials Unclassified .~clJUlTY CLASSIFICATION OF THlS PAGEfIYhcn Data Entared) concepts without requiring extensive verbal amplification. The unusual format has been deliberately chosen to facilitate these goals, and our experiences in presenting these materials have seemed to justify this choice. It is assumed that the reader has some familiarity with trigonometric functions and either has or will. develop with the aid of the appendix the facility of handling scientific notation and logarithmic operations. Unclassified SECURITY CLASSIFICATION OF THlS PAGE(Whwn Data Entered) TABLE OF CONTENTS I. Fundamentals of Sound Propagation in the Ocean Hydrostatic Pressure Acoustic Pressure Frequency, Wavelength, and Traveling Waves Hydrophone Sensitivity Displays Intensity Cavitation Sound Pressure Level and Intensity Level Transmission Loss Spherical and Cylindrical Waves TLL and TL g Absorption of Sound in Sea Water Temperature, Salinity, Depth, and the Speed of Sound Rays and Snell's Law Rays in Isogradient Layers Acoustical Reciprocity 11. Transmission Loss Models Propagation Paths for Shallow Source & Receiver Mixed Layer Convergence Zone Bottom Bounce Reliable Acoustical Paths The SOFAR Channel Summary of Transmission Loss Models Surface Interference A Use of Surface Interference Normal Modes Cutoff Frequencies and Losses Coherent & Incoherent Transmission Loss Rays, Normal Modes, and Reality 111. Combining Signals Spectrum Level Band Level Nomograph for Combining Levels Tonals and Band Levels Filters IV. The SONAR Equations Passive SONAR Equation Active SONAR Equation Masking Level ML Detected Noise Level DNL Figure of Merit FOM Source Level SL for Passive SONAR Source Level SL for Active SONAR Directivity Index DI Array Gain for a Receiver AG Target Strength (Active) TS Detected Noise Level DNL Noise Level NL Ambient Noise Spectrum Level NSL(ambient) self-~oise---- - Ssectrum Level NSL ( self) everb be ration Level (Active) RL Total Combined DNL Bandwidth Considerations Doppler Shift for Passive SONAR Doppler Shift for Active SONAR Detection Threshold DT Detection Index d ROC Curves - - P (FA) Representative d and P (FA) for Specified P (D) Detection Thresholds for Passive SONAR Detection Thresholds for Active SONAR Putting It AILTogether PURPOSE One example of a SONAR equation, for the passive detec- tion of a target which is radiating sound into the ocean, is SL - TL NSL + 10 Xogw - DI + DT where SL = Source Level of the target being detected passively TL = Transmission Loss as the signal propagates to the de- tector NSL = Noise Spectrum Level of the ambient noise in the ocean W = Frequency Bandwidth of the detecting system DI = Directivity Index of the detecting system DT = Detection Threshold of the detecting system for a 50 percent probability of detection and a specified pro- bability of a false-alarm The equation states that if the inequality is satisfied, then there is a better than fifty-fifty chance that the de- tecting system will detect the presence of the target, and at the same time no more than some specified chance of say- ing a target is there when it really is not. The purpose of this course is to provide the physical and conceptual background necessary to understand the meaning of each of the above terms and others which are combined to form the SONAR equations. At the end of this course, the student should be able to apply the appropriate SONAR equa- tion to a given problem and estimate such things as maximum detection range, optimum receiver depth, necessary bandwidth, etc. HYDROSTATIC PRELSSURE Suppose a very small hollow glass sphere has a vacuum inside. If it is placed in still water (or any other fluid) there are compressive forces acting at all points of the spherical surface which tend to crush it: L/ FLU t D 6 0 * 0 0 0 C .I - \I- a ,. ~ The magnitude of the compressive force acting on any little element of area of the sphere is essentially constant. This force magnitude divided by the area over which it acts is the hydrostatic pressure, (3,. To get a feel for the magnitudes of hydrostatic pres- sures found in ordinary circumstances, consider the follow- ing : (1) Near sea level the hydrostatic pressure of the air is 14.7 lb/in2 a 1 atmosphere of pressure. (2) Just beneath the surface of the ocean the pressure is 2 also 2 14.7 lb/in . (3) At a depth of 33 ft under the surface of the ocean the 2 hydrostatic pressure is @ = 29.4 lb/in = 2 atmo- spheres. This is the sum of the weight of a column of sea water with 1 in2 cross section and height of 33 ft and the force that the atmos2here exerts on the top of I the water column. - (4) In like manner, at a depth of 66 ft the total hydrosta- tic pressure is 44.1 lb/in2 = 3 atmospheres. ACOUSTIC PRESSURF: Sound is composed of very small pressure fluctuations p(t) disturbing the total average hydrostatic pressure These radiate away from the source into the fluid. A Source radiates Hydrophone receives the sound by sensing p (t) Time +\COUSTIC PRESSURE = p(f1 ia Zhe yuanfiZy aenaed by masZ hydttophanea, aince they ake sennitive Za changes in fhe XutaX pke~nute. MONOFREQUENCY SIGNALS For ease of discussion, we will at first restrict our- selves to consideration of tonals (monofrequency signals). Later on, we will see that more complicated signals can be broken down into a collection of monofrequency signals. The time history of the acoustic pressure at a given point in space for a monofrequency signal can be represented as a sine wave, P = The effective amplitude of the acoustic pressure. This is not the same as the peak amplitude A, but is more convenient. It turns out that P = A/+/?? = 0 7O7A for monofrequency signals. T = The period of the monofrequency signal. The period is the time interval required for the acoustical signal to go through one complete cycle, returning to the confi- guration it had at the beginning of the interval. The frequency f of the wave is the number of cycles per second, and is given by f .= 1/T. Notice that @ and P hepneaent ZotaLLy di66etren-t quanti2iea; @ ia the inatantaneoua hydnoatutie phehhuhe and P ih the e66ecXive amptitude 06 the acouatic pxeaaune. A SINEWAVE IN TIME A SINEWAVE IN SPACE X = the wavelength of the signal. It is the distance over which the signal goes through one complete cycle. SPEED OF SOUND rh1 A wave advancing through space is like a train traveling along the tracks: The length of each boxcar is the wavelength A, and the time it takes each boxcar to pass the observer is the period T. The train advances a-distance X in a time T, so its speed c must be A/T. c = X/T Since the number of cars passing the observer in unit time (I sec) is the frequency f = 1/T, A TRAVELING WAVE At a given point in space, x has fixed value so that the pressure at that point depends only on time.
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