ORE 654 Applications of Ocean Acoustics Lecture 2 Sound 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 /2/20151ORE 654 L2

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 11/2/2015ORE 654 L22

Speed of sound - First Colladon and Sturm (1827) Lake Geneva 1437 m/s at 8 °C Sea water speed is greater 11/2/2015ORE 654 L23

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 = T – 0.055T T 3 + (1.34 – 0.010T)(S – 35) z 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 ? 11/2/2015ORE 654 L24

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 – c 1 )p − c 2 p 2 c 1 = ( sin2φ) × 10 −3, φ latitude c 2 = 2.21 × 10 −6 Assumes T = 0 °C and salinity 35 PSU Additional dynamic height correction available if necessary 11/2/2015ORE 654 L25

Speed of sound, range, travel time C = R/TR = C/TT = R/C Perturbations Increase in range increases travel time Increase in sound speed decreases travel time 11/2/2015ORE 654 L26

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 10 4 best 11/2/2015ORE 654 L27 AppliedMicroSystems

Speed of sound - Seawater c = T – 0.055T T 3 + (1.34 – 0.010T)(S – 35) z Differentiate gives δC ≅ 4.6 δT 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) 11/2/2015ORE 654 L28

Pulse wave propagation Tiny sphere expanding Higher density – condensation Impulse/pulse moves outward Longitudinal wave – displacements along direction of wave propagation 11/2/2015ORE 654 L29

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 (~ p 2 ) decreases as 1/R 2 Total pulse energy would be integral over time and sphere 11/2/2015ORE 654 L210

Huygens’ Principle 11/2/2015ORE 654 L211 Qualitative description of wave propagation Points on a become wavefronts b Wavelet strength depends on direction – Stokes obliquity factor

Reflection 11/2/2015ORE 654 L212 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: θ 1 = θ 2

Snell’s Law of Refraction 11/2/2015ORE 654 L213 equal travel time R/C

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. 11/2/2015ORE 654 L214

Snell’s Law and Fermat’s Principle 11/2/2015ORE 654 L215 Travel time Differentiate and set to zero to find minimum P is minimum travel time path (0,0) θ1θ1 (x 1,0) (x 1,y 1 ) (x 2,y 2 ) θ2θ2 α2α2 α1α1 C2C2 C1C1 x y P PAPA

Diffraction Incident, reflected, and diffracted wave fronts Diffracted portion fills in shadows All three = scattered sound – redirected after interaction with a body 11/2/2015ORE 654 L216

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 11/2/2015ORE 654 L217

Sinusoids Spatial dependence at large range, pressure ~ 1/R Time and space Repeat every 2π or 360° Period T=1/f 11/2/2015ORE 654 L218

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 11/2/2015ORE 654 L219

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 11/2/2015ORE 654 L220

Local plane wave approximation At a large distance from source If restrict ε ≤ λ/8 (45°) Then W ≤ (λR) 1/2 11/2/2015ORE 654 L221

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 11/2/2015ORE 654 L222

Near field and far field approximations Near field – differential distances to source elements produce interference Far field beyond interference effects Critical range 11/2/2015ORE 654 L223

Interference between distant sources: use of complex exponentials 11/2/2015ORE 654 L224

Interference between distant sources: use of complex exponentials /2/2015ORE 654 L225 Maximum value is 4P 2 and minimum is 0 Interference maxima at k(R 2 -R 1 ) = 0, 2π, 4π, … and minima at π, 3π, 5π, … Cause pressure amplitude swings between 0 and 2π

Point source interference near the ocean surface: Lloyd’s mirror effect 11/2/2015ORE 654 L226 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/R 2 (vs 1/R)

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 11/2/2015ORE 654 L227

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 11/2/2015ORE 654 L228

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 11/2/2015ORE 654 L229

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/ρ A 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) 11/2/2015ORE 654 L230

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 11/2/2015ORE 654 L231

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 ρ A c ~ electrical impedance 11/2/2015ORE 654 L232

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 11/2/2015ORE 654 L233

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 11/2/2015ORE 654 L234

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 = voltage 2 / impedance If sinusoid, use rms = amplitude 11/2/2015ORE 654 L235

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 11/2/2015ORE 654 L236

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 11/2/2015ORE 654 L237

Reflection and transmission coefficients - 2 Velocity BC Angles by Snell’s Law 11/2/2015ORE 654 L238

Reflection and transmission coefficients - 3 Pressure BC All time dependencies at the interface the same Reflection and transmission coefficients 11/2/2015ORE 654 L239

Reflection and transmission coefficients - 4 Pressure BC Take p i as reference, divide through by it 11/2/2015ORE 654 L240

Reflection and transmission coefficients - 5 Velocity BC 11/2/2015ORE 654 L241

Reflection and transmission coefficients equations, Solving for R and T Connected by Snell’s Law 11/2/2015ORE 654 L242

Reflection and transmission at surface -1 Important at surface and bottom Surface ρ water = 1000 kg/m 3 >> ρ air = 1 kg/m 3 c water = 1500 m/s > c air = 330 m/s ρ water c water >> ρ air c air (~3600) Take θ ≈ 0° water to air R ≅ -1 and T ≅ 4 × p r = -p i so near zero total pressure at interface but u r = 2u i so particle velocity doubles Water to air interface is a “pressure release” or “soft” surface for underwater sound Water to air extreme case of c 2 <c 1 ; always (c 2 /c 1 )sinθ 1 < 1 and θ 2 < 90° for all θ 1 11/2/2015ORE 654 L243

Reflection and transmission at surface -2 From air to water Pressure doubling interface Zero particle velocity From air, surface is “hard” 11/2/2015ORE 654 L244

Reflection and transmission at bottom - 1 From ocean to bottom c bottom > c water c 2 > c 1 Possibility of total internal reflection θ i > θ c critical angle θ c = arcsin(c 1 /c 2 ) If θ i > θ c rewrite Snell’s Law 11/2/2015ORE 654 L245

Reflection and transmission at bottom - 2 Angle of incidence > critical Snell’s Law becomes Exp decay into medium 2, skin depth z 11/2/2015ORE 654 L246

Plane wave reflection at a sedimentary bottom Shallow water south of Long Island Assume sediments are fluid R 12 “bottom loss” BL = -20 log 10 R 12 “thin” layers – one composite layer; thickness< other distance scales 11/2/2015ORE 654 L247

Plane wave reflection beyond critical angle Can have perfect reflection with phase shift Useful: virtual, displaced pressure release surface (R 12 = -1) Virtual reflector 11/2/2015ORE 654 L248

Spherical waves beyond critical angle: head waves - 1 When incident wave is at critical angle, a head wave is produced Moves at c 2, radiates into source medium c 1 Travels at high(er) speed, arrives first Appears to be continually shed into slower medium at the critical angle sinθ c = c 1 /c 2 Fermat’s Principle Minimum travel time 11/2/2015ORE 654 L249

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 11/2/2015ORE 654 L250

Spherical wave reflection from a finite reflector: Fresnel zones - 1 Reflection form a circular plane Circular rings – Fresnel zones Radii such that ½ λ difference magnitude of reflection = f(λ, h, r, R 12 ) 11/2/2015ORE 654 L251

Spherical wave reflection from a finite reflector: Fresnel zones /2/2015ORE 654 L252 Different rings, different distances from source Can be cancellation or increased signal Finite disk, sum over all elements dS Phase of wavefront traveling distance 2R Interested in phase change 2kR; smallest (reference) value is 2kh Relative phase difference ΔΦ Solve for R and then r as function of ΔΦ Interest in large separation (first term in r 2 only) First phase zone (central white circle) 0 – π – positive In next (first dark ring) phase is π - 2π - negative

Spherical wave reflection from a finite reflector: Fresnel zones - 3 Formula for radii of the n rings n = 1, reflected signal 2x pressure as infinite plane for n = 2, reflected ~ 0 For infinite plate (r = ∞), P r equivalent to virtual image h behind disk/reflector, factor 1/2h – pressure inversely proportional to range for spherical divergence 11/2/2015ORE 654 L253

3-D wave equation D plane wave not adequate in many cases Shallow water – cylindrical Fish – cylinders Scattering by spheres – spherical (or expand in terms of plane waves) General equation – divergence of the gradient of p = Laplacian 11/2/2015ORE 654 L254

3-D wave equation - 2 Laplacian in 3 coordinate systems 11/2/2015ORE 654 L255

Continuous waves in rectangular coordinates Use separation of variables Each term function of only one variable, so each of terms must = one constant (factor of c 2 between space and time) 11/2/2015ORE 654 L256

Continuous waves in rectangular coordinates Try exponential forms 11/2/2015ORE 654 L257

Continuous waves in rectangular coordinates Substituting in first equation Plane wave in +x, +y, +z 11/2/2015ORE 654 L258

Omnidirectional continuous waves in spherical coordinates Assume spherical symmetry so no angular dependence – dependence only on R and t (e.g., point source) Solution analogous to 1- D rectangular with p replaced with Rp and x by R ± = - outgoing, + incoming P 0 usually at unit distance R 0 11/2/2015ORE 654 L259

Acoustic pressure for sinusoidal, omnidirectional waves Again, separate variables – radial and temporal functions 11/2/2015ORE 654 L260

Particle velocity continuous waves Start with acoustic force equation – radial component Close to source, small kR, quadrature component which lags pressure by 90° Explosion – motion lags pressure pulse Large kR – like plane wave u ~ p 11/2/2015ORE 654 L261

Far field intensity From earlier, conservation of energy showed intensity proportional to 1/R 2 For kR >> 1, particle velocity ~ p Now using functional dependence (ct-R) At long range i R is simple product of p and u R In the far field of a point source, sound pressure and velocity decrease as 1/R and intensity as 1/R 2 Now shown from first principles, wave equation 11/2/2015ORE 654 L262

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 11/2/2015ORE 654 L263

Next week 8/30 Tuesday – Transmission and attenuation along ray paths – Energy transmission in ocean acoustics – Ray paths and ray tubes – Ray paths in refracting media – Attenuation – SONAR equation – Doppler shifts 11/2/2015ORE 654 L264