ORE 654 Applications of Ocean Acoustics Lecture 2 Sound propagation in a simplified sea

1. ORE 654Applications of Ocean AcousticsLecture 2Sound propagation in a simplified sea 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 L2

2. Sound propagation in a simplified sea • Speed of sound • Pulse wave reflection, refraction, and diffraction • Sinusoidal, spherical waves in space and time • Wave interference, effects and approximations • 1-D wave equation • Plane wave reflection and refraction at a plane interface • 3-D wave equation ORE 654 L2

3. Speed of sound - First • Colladon and Sturm (1827) • Lake Geneva • 1437 m/s at 8 °C • Sea water speed is greater ORE 654 L2

4. Speed of sound - Seawater • 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): c = 1449.2 + 4.6T – 0.055T2 + 0.00029T3 + (1.34 – 0.010T)(S – 35) + 0.016z • Others: Mackenzie, Wilson, Del Grosso, and Chen-Millero-Li – newest – TEOS-10 • Note: for deep ocean, uncertainty is likely ±0.1 m/s at depth, still ? ORE 654 L2

5. Speed of sound – gravity affects pressure • To convert pressure p (dbar) to depth z (m) use Saunders, 1981 • Accounts for variation of gravity with latitude z = (1 – c1)p − c2p2 c1 = (5.92 + 5.25 sin2φ) × 10−3, φ latitude c2 = 2.21 × 10−6 • Assumes T = 0 °C and salinity 35 PSU • Additional dynamic height correction available if necessary ORE 654 L2

6. Speed of sound, range, travel time • C = R/T R = C/T T = R/C • Perturbations • Increase in range increases travel time • Increase in sound speed decreases travel time ORE 654 L2

7. Speed of sound – measuring • Sound velocimeters • Needed for • Navigation • Sonars • Measure the ocean temperature • Inverted echosounders • Tomography • Not so easy • Time and distance accuracy • 1 part in 104 best AppliedMicroSystems ORE 654 L2

8. Speed of sound - Seawater c = 1449.2 + 4.6T – 0.055T2 + 0.00029T3 + (1.34 – 0.010T)(S – 35) + 0.016z • Differentiate gives δC≅ 4.6 δT + 1.34 δS • So δT = 1°C≈ 5 m/s in sound speed • And 1 PSU ≈ 1 m/s • In practice temperature variations are large and far out weight salinity variations (which are typically small) ORE 654 L2

9. Pulse wave propagation • Tiny sphere expanding • Higher density – condensation • Impulse/pulse moves outward • Longitudinal wave – displacements along direction of wave propagation ORE 654 L2

10. Acoustic intensity • Fluctuating energy per unit time (power) passing through a unit area • Joules per second per meter squared • J s-1 m-2 = W / m-2 • Conservation of energy – through spherical surface 1 and through surface 2 • Sound intensity (~ p2) decreases as 1/R2 • Total pulse energy would be integral over time and sphere ORE 654 L2

11. Huygens’ Principle • Qualitative description of wave propagation • Points on a become wavefronts b • Wavelet strength depends on direction – Stokes obliquity factor ORE 654 L2

12. Reflection • Successive positions of the incident pulse wave at equal time intervals (R=cΔt) over a half space • Successive positions of reflected pulse wave fronts • Reflection appears to come from image of source • Law of reflection: θ1 = θ2 ORE 654 L2

13. Snell’s Law of Refraction equal travel time R/C ORE 654 L2

14. Fermat’s Principle • Energy/particles can and do take all possible paths from one point to another, but paths with the highest probability (in our case) are stationary paths, i.e., small perturbations don’t change them. • In practice, these are paths of minimum travel time – principle of least time. ORE 654 L2

15. y (x2,y2) Snell’s Law and Fermat’s Principle P C2 α2 PA θ2 • Travel time • Differentiate and set to zero to find minimum • P is minimum travel time path (x1,y1) α1 C1 θ1 x (0,0) (x1,0) ORE 654 L2

16. Diffraction • Incident, reflected, and diffracted wave fronts • Diffracted portion fills in shadows • All three = scattered sound – redirected after interaction with a body ORE 654 L2

17. Sinusoidal, spherical waves in space and time (a) pressure at some time (b) range dependent pressure at some instant of time (c) time dependent pressure at a point in space ORE 654 L2

18. Sinusoids • Spatial dependence at large range, pressure ~ 1/R • Time and space • Repeat every 2π or 360° • Period T=1/f ORE 654 L2

19. Sinusoids - 2 • Radially propagating wave having speed c • Pick an arbitrary phase at some (t,R). At later t+Δt, same phase will be R+ΔR • With negative sign – waves traveling in positive direction • With Positive sign, negative direction ORE 654 L2

20. Wave interference, effects and approximations • Constructive and destructive interference from multiple sources • Add algebraically for linear acoustics, not so for non-linear • Approximations are useful tools ORE 654 L2

21. Local plane wave approximation • At a large distance from source • If restrict ε ≤ λ/8 (45°) • Then W ≤ (λR)1/2 ORE 654 L2

22. Fresnel and Fraunhofer approximations • Adding signals due to several sinusoidal point sources • Separate temporal and spatial dependence • Fraunhofer – long range • Fresnel – nearer ranges • Convert differences in range to phase - decide ORE 654 L2

23. Near field and far field approximations • Near field – differential distances to source elements produce interference • Far field beyond interference effects • Critical range ORE 654 L2

24. Interference between distant sources:use of complex exponentials ORE 654 L2

25. Interference between distant sources:use of complex exponentials - 2 • Maximum value is 4P2 and minimum is 0 • Interference maxima at k(R2-R1) = 0, 2π, 4π, … and minima at π, 3π, 5π, … • Cause pressure amplitude swings between 0 and 2π ORE 654 L2

26. Point source interference near the ocean surface: Lloyd’s mirror effect • Sinusoidal point source near ocean surface produced acoustic field with strong interference between direct and reflected sound • Above surface image • Function of frequency and geometry • Pressure doubling in near region, • Beyond last peak pressure decays as 1/R2 (vs 1/R) ORE 654 L2

27. 1-D wave equation • Newton’s Law for Acoustics • Conservation of Mass for Acoustics • Equation of state for acoustics • Combine to get wave equation • Small perturbations in pressure and density around ambient ORE 654 L2

28. Newton’s Law for Acoustics • Point source, large R, plane wave • Lagrangian frame • Net pressure • Multiply by area to get net force • Mass is density x volume • Acceleration is du/dt • F = ma ORE 654 L2

29. Conservation of mass for acoustics • Eularian frame • Net Mass flux into volume is difference (over x) between flux in and out where flux is density x velocity x volume element • This must balance rate of increase in mass increase ORE 654 L2

30. Equation of state for acoustics • Relation between stress and strain • Hooke’s Law for an elastic body: stress ~ strain • For acoustics, stress (force/area) = pressure • Strain (relative change in dimension) = relative change in density ρ/ρA • Proportionality constant is the ambient bulk modulus of elasticity E • Holds for all fluids except for intense sound • Assumes instantaneous P causes instantaneous ρ (time lag – “molecular relaxation” – absorption) ORE 654 L2

31. Wave equation • Partial x of F=ma • Partial t of conservation of mass • Combine • Use equation of state to replace density with pressure • Define sound speed • Final standard form equation ORE 654 L2

32. Impedance • Relate acoustic particle velocity to pressure in a plane wave (general form of wave equation solution) • General solution +/- • Wave traveling in +x has velocity • Substitute into F=ma • Integrate over x • Analogous to Ohm’s Law • Pressure ~ voltage • Velocity ~ current • Specific acoustic impedance ρAc ~ electrical impedance ORE 654 L2

33. Mach number • In fluid mechanics dimensionless numbers are often very useful • Ratio of acoustic particle velocity to speed of sound • Take plane wave and conservation of mass • M – measure of strength and non-linearity ORE 654 L2

34. Acoustic pressure and density • Use impedance and Mach number relations • Liquids, equation of state p=p(ρ) is complicated so inverse used – eqn of state calculated from accurate measurements of sound speed ORE 654 L2

35. Acoustic intensity • Intensity (vector) = Flux = (energy / second = power) perpendicular though an area – J/s m-2 = W/m-2 • Remember Power = force x velocity • Intensity = (force/area) x velocity • Electrical analog • Power = voltage2 / impedance • If sinusoid, use rms = 0.707 amplitude ORE 654 L2

36. Plane wave reflection and refraction at a plane interface • Derive reflection and transmission coefficients • Applicable for spherical waves at large range – i.e., waves are locally plane ORE 654 L2

37. Reflection and transmission coefficients - 1 • Use physical boundary conditions at the interface between two fluids • BC-1: equality of pressure • BC-2: equality of normal velocity ORE 654 L2

38. Reflection and transmission coefficients - 2 • Velocity BC • Angles by Snell’s Law ORE 654 L2

39. Reflection and transmission coefficients - 3 • Pressure BC • All time dependencies at the interface the same • Reflection and transmission coefficients ORE 654 L2

40. Reflection and transmission coefficients - 4 • Pressure BC • Take pi as reference, divide through by it ORE 654 L2

41. Reflection and transmission coefficients - 5 • Velocity BC ORE 654 L2

42. Reflection and transmission coefficients - 6 • 2 equations, Solving for R and T • Connected by Snell’s Law ORE 654 L2

43. Reflection and transmission at surface -1 • Important at surface and bottom • Surface • ρwater = 1000 kg/m3 >> ρair = 1 kg/m3 • cwater = 1500 m/s > cair = 330 m/s • ρwatercwater >> ρaircair (~3600) • Take θ ≈ 0° water to air • R≅-1 and T ≅ 4 × 10-4 • pr = -pi so near zero total pressure at interface but ur = 2ui so particle velocity doubles • Water to air interface is a “pressure release” or “soft” surface for underwater sound • Water to air extreme case of c2<c1; always (c2/c1)sinθ1 < 1 and θ2 < 90° for all θ1 ORE 654 L2

44. Reflection and transmission at surface -2 • From air to water • Pressure doubling interface • Zero particle velocity • From air, surface is “hard” ORE 654 L2

45. Reflection and transmission at bottom - 1 • From ocean to bottom • cbottom > cwater c2 > c1 • Possibility of total internal reflection • θi > θc critical angle • θc = arcsin(c1/c2) • If θi > θc rewrite Snell’s Law ORE 654 L2

46. Reflection and transmission at bottom - 2 • Angle of incidence > critical • Snell’s Law becomes • Exp decay into medium 2, skin depth z ORE 654 L2

47. Plane wave reflection at a sedimentary bottom • Shallow water south of Long Island • Assume sediments are fluid • R12 “bottom loss” • BL = -20 log10R12 • “thin” layers – one composite layer; thickness< other distance scales ORE 654 L2

48. Plane wave reflection beyond critical angle • Can have perfect reflection with phase shift • Useful: virtual, displaced pressure release surface (R12 = -1) • Virtual reflector ORE 654 L2

49. Spherical waves beyond critical angle: head waves - 1 • When incident wave is at critical angle, a head wave is produced • Moves at c2, radiates into source medium c1 • Travels at high(er) speed, arrives first • Appears to be continually shed into slower medium at the critical angle sinθc = c1/c2 • Fermat’s Principle Minimum travel time ORE 654 L2

50. Spherical waves beyond critical angle: head waves - 2 • More detailed analytical development yields amplitude • Also for source under ice – plates 100s m, 1-2 m thick • Model latter – scale lengths and properties – 3.3 mm acrylic at 62 kHz = 1 m thick ice at 200 Hz; critical angle 39° • “thin” ice covered by air NOT =simple water-air pressure release interface ORE 654 L2