Ocean Acoustic Theory

• Acoustic Wave Equation

• Integral Transforms

• Helmholtz Equation

• Source in Unbounded and Bounded Media

• Propagation in Layered Media

– Reflection and Transmission

• The Ideal Waveguide

– Image Method

– Wavenumber Integral

– Normal Modes

• Pekeris Waveguide

1 COMPUTATIONAL OCEAN ACOUSTICS Lecture 5 13.853

COMPUTATIONAL OCEAN ACOUSTICS

Horizontally stratified environment

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Superposition Principle

Source contribution:

Homogeneous Solution:

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13.853

Mag

nit

ud

e

50 F = 150 Hz

40

30

20

Evanescent Radiating Waves

Waves 10

0 0.2 0.4 0.6 0.8 1.0

Horizontal wavenumber (m-1)

Magnitude of the depth-dependent Green’s function for point source in an

infinite medium. Solid curve: z – z = Ȝ/10; dashed curve: z – zs = 2 Ȝ.s

4 COMPUTATIONAL OCEAN ACOUSTICS Lecture 5

Example: Hard Bottom – c2 > c1

Medium 1 Radiating Evanescent

0 k2 k1

Medium 2 Radiating Evanescent

| R | < 1 | R | = 1 | R | < 1

[See Fig 2.10 in Jensen, Kuperman, Porter and Schmidt. Computational Ocean Acoustics.

New York: Springer-Verlag, 2000.]

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COMPUTATIONAL OCEAN ACOUSTICS

Idealized ocean waveguide model with pressure-release surface and bottom

Ideal Fluid Waveguide Image Method

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COMPUTATIONAL OCEAN ACOUSTICS

7 Lecture 513.853

1234

5

6

7

C

Re(kr)

r)

r

+ Ȧ/c- Ȧ/c

Im(k

Complex k -plane

Singularities of the depth-dependent Green’s function for an ideal waveguide.

8 COMPUTATIONAL OCEAN ACOUSTICS Lecture 5 13.853

Im(kr)

7 Mode Mode Mode

1 2 3 -0 -0 -06 Complex kr-plane

5

4 3 2 1 Re(kr) 50 50 50

C

100 100 100 -1 0 1 -1 0 1 -1 0 1

- Ȧ/c + Ȧ/c Depth dependence of the first 3 normal modes in

Singularities of the depth-dependent Green’s function for an ideal waveguide. ideal waveguide at 20Hz.

Dep

th (

m)

9 COMPUTATIONAL OCEAN ACOUSTICS Lecture 5 13.853

[see Jensen, Fig 2.21] [see Jensen, Fig 2.22]

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[see Jensen, Figs 2.23 & 2.24]

Lecture 5

COMPUTATIONAL OCEAN ACOUSTICS

Pekeris waveguide with pressure-release surface and penetrable fluid bottom

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COMPUTATIONAL OCEAN ACOUSTICS

Mode Mode Mode1 2 3

-0 -0-0

50 5050

Dep

th (

m)

100 100100

150 150150 -1 0 1 -1 0 1 -1 0 1

Depth dependence of acoustic pressure for the 3 normal modes in the Pekeris waveguide at 35Hz.

15 Lecture 5 13.853

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[see Jensen, Fig 2.28]

Lecture 5

COMPUTATIONAL OCEAN ACOUSTICS

[see Jensen, Fig 2.29]

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[see Jensen, Fig 2.30]

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