ORE 654 Applications of Ocean Acoustics Lecture 3a Transmission and attenuation along ray paths

1. ORE 654Applications of Ocean AcousticsLecture 3aTransmission and attenuation along ray paths Bruce Howe Ocean and Resources Engineering School of Ocean and Earth Science and Technology University of Hawai’i at Manoa Fall Semester 2011 ORE 654 L3

2. Prologue • Newton’s “corpuscular energy” moving along “rays” helped describe propagation, reflection and refraction. • In WWII – Ewing and Worzel discovered the SOFAR channel – sound fixing and ranging • Perth-Bermuda 1960 shots Ewing and Worzel, 1948 ORE 654 L3

3. Transmission and attenuation along ray paths • Energy transmission in ocean acoustics • Ray paths and ray tubes • Ray paths in refractive medium • Attenuation • The “SONAR Equation”; source level, sound pressure level, transmission loss ORE 654 L3

4. Energy transmission in ocean acousticsImpulse sources • Explosions • Sparkers and air guns • Implosions (light bulbs) • Often milliseconds • Finite, discrete delta function • Continuous – Direc delta function ORE 654 L3

5. Light bulb impulse source • “100 W” light bulb • At 18.3 m depth Heard et al., 1997 ORE 654 L3

6. Nuclear testing Umbrella 1 kiloton TNT = 4 × 109 J Yield: 8 kilotonsLocation: EniwetokDate: 8June 1958 Depth: 48 m Underwater crater: 92 m across 6 m deep ORE 654 L1

7. Pressure, particle velocity, and intensity in a pulse • Far field, spreading spherically from point source • p0 at R0 • Waveform shape of outward traveling p same as p0; u follows same form • Just time delay and amplitude change ORE 654 L3

8. Pressure in a pulse - 2 • Typical pulse shape – satisfies two conditions • Step/delta with exponential decay, finite for t ≥ R/c • τs – time to decay to 1/e • tg – duration of integral or “gate time”; 98% correct for tg > 4τs ORE 654 L3

9. Particle velocity and intensity – transient signal • Radial particle velocity and intensity in far field • Message energy (Joules) pass through element ΔS at range R in gate open time tg, function of (t-R/c) • Over full sphere ΔS = 4πR2, total energy Em (J) • Energy flux at range R transmitted during time interval pressure squared (tips) tg (J/m2) ORE 654 L3

10. Power radiated by continuous wave signal • Continuous wave sinusoid, peak value P0 • Source power over all angles is average energy over period T • Same procedure for other signals ORE 654 L3

11. Ray paths and ray tubes • Starting point -Spherical wave solution • p0(t) is temporal function • Time for message to arrive is R/c • R0/R is spatial function • p(t,R) – output of receiver at R ORE 654 L3

12. Reflections along ray paths • Reflected rays appear to come from image source • Pressure signal for bottom reflecting ray • Pressure signal for the bottom and surface reflected ray ORE 654 L3

13. Multiple ray paths • Signal pressures add vectorally and can constructively and destructively interfere at receiver ORE 654 L3

14. Conservation of energy in ray tubes • Equality of energy • E1 = E2 ORE 654 L3

15. Sound pressures in ray tubes • Sound pressures at 1 and 2 follow 1/R spherical divergence • In homogenous media properties are constant, e.g., density and sound speed • Change t-R/c to τ • Assume tg large enough to include 1 and 2 • Assume p1 proportional to p2, so integrand must = 0 • P changes as sqrt of area ratio, or cross path ray tube diameter/scale ORE 654 L3

16. Ray paths in a refracting medium • Sounds speed / index of refraction usually function of space and time – can be complicated • Use Snell’s Law to trace path of a small portion of a wave front – determines direction • Ray is perpendicular to the local wave front • Gives direction and time • Valid for high frequencies: • Changes in sound speed over scales large compared to wavelength • Water depth and range to receiver >> wavelength • Very good approximation when diffraction absent (wave effect) • Fails where rays cross (caustics), or shadow zones – use wave theory to patch • Specular reflection at interfaces (mirror, flat) • Intensity losses along rays through geometric divergence (spherical divergence modified by refraction), through absorption along paths, and reflections on interfaces ORE 654 L3

17. Speed of sound - Seawater • From previous lecture • Sound speed (c or C m/s) is a complicated function of temperature T °C, salinity S PSU, and pressure/depth z m • Simple formula by Medwin (1975): • Latest: http://www.teos-10.org/ ORE 654 L2

18. Ocean stratification • In general, sound speed = c(x,y,z,t) • Sound speed and other ocean properties such as temperature and salinity (and pressure) have the largest variation in the vertical. • Horizontal variations are typically smaller, over larger distances • Assume c = c(z) ORE 654 L3

19. Variation in sound speed – horizontal, vertical • Sargasso Sea at 750 m – main thermocline - +/- 5 m/s (1 °C) • Perth to Bermuda – Blue 1,470 m/s, red 1,550 m/s ORE 654 L3

20. Snell’s Law again • Different layers • Define Ray parameter a – constant for one ray no matter where along the ray you are • Start ray at z0 and c0. At depth z, angle is θz and cz ORE 654 L2

21. Formulating Ray integrals dr ds dz θ • To determine ray paths (r,z) and travel time, must integrate, using Snell’s Law • Dependent on z because c(z) only • Differential distances dz, ds, dr and time dt • Between “initial” and “final” positions ORE 654 L2

22. Ray tracing approximations • Constant sound speed layers: • c(z) = cn for zn ≤ z ≤ zn+1 • Constant sound speed gradient layers, i.e., sound speed varies linearly in each layer: • c(z) = c(z1)+b(z-z1) for z1 ≤ z ≤ z2 • b = d(c(z))/dz ORE 654 L3

23. Rays through constant sound speed layers • Real c(z) is continuous but simplest solution - break into constant c layers • Replace integrals with sums Use Snell for next θ ORE 654 L2

24. Rays through slowly changing sound speed layers • Assume constant gradient in each layer ORE 654 L2

25. Rays through slowly changing sound speed layers • Start with c(z) integrals • discretize • change of variables to simplify ORE 654 L2

26. Integral tables ORE 654 L2

27. Rays through slowly changing sound speed layers • Perform integrals • Sum over intervals to get total time and range ORE 654 L2

28. Rays through slowly changing sound speed layers • With linear sound speed, ray paths are arcs of circles ORE 654 L2

29. Rays through slowly changing sound speed layers • Curvilinear path length • s = Rθ ORE 654 L2

30. Ray example - Arctic • Nearly isothermal • Linear sound speed gradient = b ≈ 0.016 / s ORE 654 L3

31. RayNorth Atlantic • From Ewing and Worzel,1948. • Minimum at 1300 m ORE 654 L3

32. RaysNorth Atlantic • From Howe et al., 1987. • Minimum at 1300 m • 18°C mode water in upper layer • eigenrays ORE 654 L3

33. RaysNorth Pacific • From Dushaw et al., 1994. • Minimum at 1000 m • Surface layer develops as summer progresses • Transition from surface reflecting to barely refracting ORE 654 L3

34. Philippine Sea – 2009-2011 • ONR deep water acoustic propagation/tomography experiment ORE 654 L3

35. Philippine Sea – profiles ORE 654 L3

36. Philippine Sea - rays ORE 654 L3

37. Philippine Sea – timefronts ORE 654 L3

38. Attenuation • Seawater is a dissipative propagation medium • Through viscosity or chemical reactions to heat • Local amplitude decrease proportional to the amplitude itself • Acoustic pressure decreases exponentially with distance • In terms of 1/e – neper/m • Then attenuation coefficient – in terms of dB/km (relative power loss per km) ORE 654 L3

39. Absorption losses in sea water: viscosities and molecular relaxation • Coefficients of viscosity • μ dynamic or absolute coefficient of shear viscosity – the viscosity =ratio shear stress to rate of strain • Each component of stress is due to a shearing force parallel to area A, caused by velocity gradient (shear modulus) • Bulk viscosity appears only in compressible media – i.e., acoustics – in compressible N-S, term proportional to rate of change of density – related to extensional modulus ORE 654 L3

40. Absorption losses in sea water: Molecular relaxation • Bulk viscosity – finite time for real fluid to respond to p change, or to relax back to a base state • Ionic dissociation – activated, deactivated by condensation, rarefaction • Magnesium sulfate and boric acid, even though minor parts of “salinity” • Affects speed of propagation slightly (dispersive) – usually ignore this aspect • Relaxation frequency • Damped oscillator ORE 654 L3

41. Molecular relaxation - 2 • Attenuation per wavelength • For fresh water • Goes to zero very high and very low frequency • Very high f – molecules can’t respond • Very low – follow • Near fr – activated molecules transfer energy from condensation to rarefaction – heat ORE 654 L3

42. Molecular relaxation - 2 • The fresh water curves • Relaxation frequency off scale to right ORE 654 L3

43. Molecular relaxation in sea water • During WWII, unexplained large attenuation at sonar frequencies – 20 kHz • In 1950s – careful lab experiments by Leonard, Wilson and Bies • Drive water filled sphere vs frequency - measure amplitudes of modes of oscillation and therefore damping, with various salts • Determined magnesium sulfate, a relatively minor constituent by weight, was responsible ORE 654 L3

44. Molecular relaxation in sea water • Similar effects found by Mellen and Browning, 1970, for boric acid, moderated by pH • Relaxation frequency ~ 1 kHz ORE 654 L3

45. Molecular relaxation in sea water • Final, including boric acid, magnesium sulfate and pure water ORE 654 L3

46. Molecular relaxation in sea water • Boric acid • pH function of location • A1 in Pacific half of Atlantic ORE 654 L3

47. Molecular relaxation in sea water • Magnesium sulfate ORE 654 L3

48. Molecular relaxation in sea water • Pure water ORE 654 L3

49. TL • Absorption dominant at long ranges, balance at R = Rt ORE 654 L3

50. Current interest in pH • Change in sound absorption at 440 Hz due to CO2/pH Hester et al., 2008 ORE 654 L3