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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

- 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

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- Colladon and Sturm (1827)
- Lake Geneva
- 1437 m/s at 8 °C
- Sea water speed is greater

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- 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 ?

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- 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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- 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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- 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

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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

- Tiny sphere expanding
- Higher density – condensation
- Impulse/pulse moves outward
- Longitudinal wave – displacements along direction of wave propagation

ORE 654 L2

- 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

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- Qualitative description of wave propagation
- Points on a become wavefronts b
- Wavelet strength depends on direction – Stokes obliquity factor

ORE 654 L2

- 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

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equal travel time R/C

ORE 654 L2

- 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)

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)

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- Incident, reflected, and diffracted wave fronts
- Diffracted portion fills in shadows
- All three = scattered sound – redirected after interaction with a body

ORE 654 L2

(a) pressure at some time

(b) range dependent pressure at some instant of time

(c) time dependent pressure at a point in space

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- Spatial dependence at large range, pressure ~ 1/R
- Time and space
- Repeat every 2π or 360°
- Period T=1/f

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- 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

- Constructive and destructive interference from multiple sources
- Add algebraically for linear acoustics, not so for non-linear
- Approximations are useful tools

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- At a large distance from source
- If restrict ε ≤ λ/8 (45°)
- Then W ≤ (λR)1/2

ORE 654 L2

- 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

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- Near field – differential distances to source elements produce interference
- Far field beyond interference effects
- Critical range

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- 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

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- 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

- 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

- 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

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- 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

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- 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)

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- 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

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- 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

- 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

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- 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

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- 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

- Derive reflection and transmission coefficients
- Applicable for spherical waves at large range – i.e., waves are locally plane

ORE 654 L2

- Use physical boundary conditions at the interface between two fluids
- BC-1: equality of pressure
- BC-2: equality of normal velocity

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- Velocity BC
- Angles by Snell’s Law

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- Pressure BC
- All time dependencies at the interface the same
- Reflection and transmission coefficients

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- Pressure BC
- Take pi as reference, divide through by it

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- Velocity BC

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- 2 equations, Solving for R and T
- Connected by Snell’s Law

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- 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

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- From air to water
- Pressure doubling interface
- Zero particle velocity
- From air, surface is “hard”

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- 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

- Angle of incidence > critical
- Snell’s Law becomes
- Exp decay into medium 2, skin depth z

ORE 654 L2

- 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

- Can have perfect reflection with phase shift
- Useful: virtual, displaced pressure release surface (R12 = -1)
- Virtual reflector

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- 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

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- 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

- 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

- 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

- 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

- 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

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- Laplacian in 3 coordinate systems

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- 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

- Try exponential forms

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- Substituting in first equation
- Plane wave in +x, +y, +z

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- 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

- Again, separate variables – radial and temporal functions

ORE 654 L2

- 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

ORE 654 L2

- 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

- 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

- 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