Modeling Underwater Acoustic Interface Wavesweb.williams.edu/Mathematics/sjmiller/public_html/hudson/tobak... · OUTLINE Waves Rayleigh Waves Parabolic Equation Concluding Remarks

OUTLINE Waves Rayleigh Waves Parabolic Equation Concluding Remarks

Modeling Underwater Acoustic InterfaceWaves

Laura TobakMarist College

Hudson River Undergraduate Mathematics Conference


INTRODUCTION

I Sound = pressure, pressure causes displacement

I Elastic Potential Theory

I Elastic Parabolic equation (PE)

Goal: Compare Elastic Potential Theory displacement to PEdisplacement


APPLICATIONS

I Nuclear Test Ban Treaty monitoring

I Tsunami warning systems

I Explains anomalies in submarine transmissions


APPLICATIONS





APPLICATIONS





COMPRESSIONAL AND SHEAR WAVESCompressional: (P) Particles travel parallel to the waveShear: (SV) Particles travel perpendicular to the wave

I Both types propagate in elastic media

I Only compressional waves propagate in fluid mediaI Interface waves require simultaneous incidence of P and

SV waves on an interface



I Both types propagate in elastic mediaI Only compressional waves propagate in fluid media

I Interface waves require simultaneous incidence of P andSV waves on an interface



I Both types propagate in elastic mediaI Only compressional waves propagate in fluid mediaI Interface waves require simultaneous incidence of P and

SV waves on an interface


SNELL’S LAW

Reflected WaveIncident Wave i

Transmitted Wave

Transmitted Angle

sin(i)c1

= sin(τ)c2

I τ is the transmitted angle,both c1 and c2 arepropagation speeds

I c1 < c2 → c1c2

sin(i) = sin(τ)


SNELL’S LAW

Reflected WaveIncident Wave i

sin(i)c1

= sin(τ)c2


I c1 < c2 → c1c2

sin(i) = sin(τ)

I i < ic → no transmittedwave


SNELL’S LAW

Critical Wave

Head Wave

Expelled Energy from Head Wavei_c

sin(i)c1

= sin(τ)c2


I c1 < c2 → c1c2

sin(i) = sin(τ)

I i = ic (Head wave)


RAYLEIGH WAVES

Stress components acting on an infinitesimalrectangular parallelapiped

I Stress in elastic media

I σxy = σyx

I σxz = σzx

I σzy = σyz

I Six independentcomponents remain


RAYLEIGH WAVES

I Potential equations for compressional and shear waves in elasticmedia

Φ = Ae[−ωηαz]e[iω(x−t

c )], (1)

Ψ = Be[−ωηβz]e[iω(x−t

c )] (2)

where ηα =√

1α2 − 1

c2 , ηβ =√

1β2 − 1

c2 , and c is wave speed

I Elastic displacement equation

~U = (Φx −Ψz)x + (Φz −Ψx)y + (Φz + Ψx)z (3)

I Apply the free surface boundary conditions z = 0, σzz = 0:

A[(λ+ 2µ)η2α + λ

(1c

)2

] + B(2µηβ

c) = 0, (4)

I and σxz = 0:

A(2ηα

c) + B(

(1c

)2

− η2β) = 0 (5)


RAYLEIGH WAVES



c )], (1)


c )] (2)

where ηα =√

1α2 − 1

c2 , ηβ =√

1β2 − 1





A[(λ+ 2µ)η2α + λ

(1c

)2

] + B(2µηβ

c) = 0, (4)

I and σxz = 0:

A(2ηα

c) + B(

(1c

)2

− η2β) = 0 (5)


RAYLEIGH WAVES



c )], (1)


c )] (2)

where ηα =√

1α2 − 1

c2 , ηβ =√

1β2 − 1





A[(λ+ 2µ)η2α + λ

(1c

)2

] + B(2µηβ

c) = 0, (4)

I and σxz = 0:

A(2ηα

c) + B(

(1c

)2

− η2β) = 0 (5)


RAYLEIGH WAVES



c )], (1)


c )] (2)

where ηα =√

1α2 − 1

c2 , ηβ =√

1β2 − 1





A[(λ+ 2µ)η2α + λ

(1c

)2

] + B(2µηβ

c) = 0, (4)

I and σxz = 0:

A(2ηα

c) + B(

(1c

)2

− η2β) = 0 (5)


RAYLEIGH WAVES

Rewrite equations (3) and (4) as the matrix equation:(λ+ 2µ)η2α + λ

(1c

)2(

2µηβc

)2ηα

c

(1c

)2 − η2β

[AB

]=

[00

](6)

Nontrivial solutions exist where determinant equal to zero:

[(λ+ 2µ) η2

α + λ

(1c

)2]((

1c

)2

− η2β

)− 4µ

(1c

)2

ηαηβ = 0

(7)


RAYLEIGH WAVES[(λ+ 2µ) η2

α + λ( 1

c

)2] (( 1

c

)2 − η2β

)− 4µ

( 1c

)2ηαηβ = 0

Find c that allows a solution for A and BFuture Work: Use roots in eq. 3 to find Rayleigh Wave displacementsfor different media


PARABOLIC EQUATION DERIVATION

I Helmholtz equation(L∂2

∂x2 + M)(

uxw

)= 0. (8)

I L and M are matrices containing depth derivatives

I Multiply by L−1 and factor(∂

∂x+ i(L−1M)1/2

)(∂

∂x− i(L−1M)1/2

)(uxw

)= 0. (9)

I Assume outgoing energy dominates incoming energy

∂

∂x

(uxw

)= i(L−1M)1/2

(uxw

). (10)

This is the (ux,w) parabolic equation for propagation iselastic and fluid media.




∂x2 + M)(

uxw

)= 0. (8)

I L and M are matrices containing depth derivativesI Multiply by L−1 and factor(

∂

∂x+ i(L−1M)1/2

)(∂

∂x− i(L−1M)1/2

)(uxw

)= 0. (9)


∂

∂x

(uxw

)= i(L−1M)1/2

(uxw

). (10)





∂x2 + M)(

uxw

)= 0. (8)

I L and M are matrices containing depth derivativesI Multiply by L−1 and factor(

∂

∂x+ i(L−1M)1/2

)(∂

∂x− i(L−1M)1/2

)(uxw

)= 0. (9)


∂

∂x

(uxw

)= i(L−1M)1/2

(uxw

). (10)



PARABOLIC EQUATION MODEL

I Marching method

I Range dependent media

I Solves for pressure at every point in a range depth grid

Transmission Loss: Measure of signal weakening as itpropagates outward from the source

TL = −20 log10

∣∣∣ prp0

∣∣∣where p0 is pressure near the source and p is pressure at thereceiver


PARABOLIC EQUATION RESULTS

Range (km)

Dep

th (

m)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

50

00

40

00

30

00

20

00

10

00

0

Loss (dB re 1 m)60 70 80 90 100 110 120 130 140

Compressional Source

Range (km)D

epth

(m

)0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

50

00

40

00

30

00

20

00

10

00

0

Loss (dB re 1 m)60 70 80 90 100 110 120 130 140

Shear Source


PARABOLIC EQUATION RESULTSPressure output form PE at both interfaces

FutureWork:Compare these curves to theoretical displacement curves


BIBLIOGRAPHY

I ”Geol 333.” Seismology Fundamentals. N.p., n.d. Web. 06 Mar. 2013.I Jensen, Finn Bruun. Computational Ocean Acoustics. Woodbury, NY: American

Institute of Physics, 1993. Print.I Jones, F. ”Ray Paths in Layered Media.” UBC Earth and Ocean Sciences, 12 Nov.

2007. Web. 11 Mar. 2013.I Kolsky, Herbert. Stress Waves in Solids,. New York: Dover Publications, 1963.

Print.I Lay, Thorne, and Terry C. Wallace. Modern Global Seismology. San Diego:

Academic, 1995. Print.I Samad, Hariz. Breathtaking Image Of Tsunami In Switzerland -. ITYSR, 18 Dec.

2012. Web. 06 Mar. 2013.I Shearer, Peter M. Introduction to Seismology. Cambridge: Cambridge UP, 1999.

Print.I Tolstoy, Ivan, and Clarence S. Clay. Ocean Acoustics; Theory and Experiment in

Underwater Sound. New York: McGraw-Hill, 1966. Print.

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Modeling Underwater Acoustic Interface Wavesweb.williams.edu/Mathematics/sjmiller/public_html/hudson/tobak... · OUTLINE Waves Rayleigh Waves Parabolic Equation Concluding Remarks