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# ORE 654 Applications of Ocean Acoustics Lecture 2 Sound propagation in a simplified sea - PowerPoint PPT Presentation

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 2011. Sound propagation in a simplified sea. Speed of sound

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

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## 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

### 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

### Speed of sound - First

• Lake Geneva

• 1437 m/s at 8 °C

• Sea water speed is greater

ORE 654 L2

### 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

### 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

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### 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

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### Speed of sound – measuring

• Sound velocimeters

• Needed for

• Sonars

• Measure the ocean temperature

• Inverted echosounders

• Tomography

• Not so easy

• Time and distance accuracy

• 1 part in 104 best

AppliedMicroSystems

ORE 654 L2

### 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

### Pulse wave propagation

• Tiny sphere expanding

• Higher density – condensation

• Impulse/pulse moves outward

• Longitudinal wave – displacements along direction of wave propagation

ORE 654 L2

### 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

### Huygens’ Principle

• Qualitative description of wave propagation

• Points on a become wavefronts b

• Wavelet strength depends on direction – Stokes obliquity factor

ORE 654 L2

### 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

### Snell’s Law of Refraction

equal travel time R/C

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### 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

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

### 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

### 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

### Sinusoids

• Spatial dependence at large range, pressure ~ 1/R

• Time and space

• Repeat every 2π or 360°

• Period T=1/f

ORE 654 L2

### 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

### 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

### Local plane wave approximation

• At a large distance from source

• If restrict ε ≤ λ/8 (45°)

• Then W ≤ (λR)1/2

ORE 654 L2

### 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

### 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

ORE 654 L2

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Reflection and transmission coefficients - 2

• Velocity BC

• Angles by Snell’s Law

ORE 654 L2

### Reflection and transmission coefficients - 3

• Pressure BC

• All time dependencies at the interface the same

• Reflection and transmission coefficients

ORE 654 L2

### Reflection and transmission coefficients - 4

• Pressure BC

• Take pi as reference, divide through by it

ORE 654 L2

• Velocity BC

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### Reflection and transmission coefficients - 6

• 2 equations, Solving for R and T

• Connected by Snell’s Law

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### 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

### Reflection and transmission at surface -2

• From air to water

• Pressure doubling interface

• Zero particle velocity

• From air, surface is “hard”

ORE 654 L2

### 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

### 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

### 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

### 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

### 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

### 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

### 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, R12)

ORE 654 L2

### Spherical wave reflection from a finite reflector: Fresnel zones - 2

• 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 r2 only)

• First phase zone (central white circle) 0 – π – positive

• In next (first dark ring) phase is π - 2π - negative

ORE 654 L2

### 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 = ∞),

• Pr equivalent to virtual image h behind disk/reflector, factor 1/2h – pressure inversely proportional to range for spherical divergence

ORE 654 L2

### 3-D wave equation - 1

• 1-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

ORE 654 L2

### 3-D wave equation - 2

• Laplacian in 3 coordinate systems

ORE 654 L2

### 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 c2 between space and time)

ORE 654 L2

### Continuous waves in rectangular coordinates

• Try exponential forms

ORE 654 L2

### Continuous waves in rectangular coordinates

• Substituting in first equation

• Plane wave in +x, +y, +z

ORE 654 L2

### 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

• P0 usually at unit distance R0

ORE 654 L2

### Acoustic pressure for sinusoidal, omnidirectional waves

• Again, separate variables – radial and temporal functions

ORE 654 L2

### Particle velocitycontinuous waves

• Close to source, small kR, quadrature component which lags pressure by 90°

• Explosion – motion lags pressure pulse

• Large kR – like plane wave u ~ p

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### Far field intensity

• From earlier, conservation of energy showed intensity proportional to 1/R2

• For kR >> 1, particle velocity ~ p

• Now using functional dependence (ct-R)

• At long range iR is simple product of p and uR

• In the far field of a point source, sound pressure and velocity decrease as 1/R and intensity as 1/R2

• Now shown from first principles, wave equation

ORE 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

ORE 654 L2

### 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

ORE 654 L2