PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

A Computational Investigation into the Influence of the Shear
Properties of the Seabed on Sound Propagation in Shallow Water.
Ray KIRBY 1; Wenbo DUAN 2;
1

University of Technology, Sydney, Australia

2

Brunel University London, United Kingdom

ABSTRACTThe propagation of sound waves in the ocean is influenced by the acoustic characteristics of the seabed,
especially in shallow water. If the seabed is considered to be an elastic or viscoelastic structure, then soundpropagation will be influenced by the shear and longitudinal properties of this structure. Accordingly, thispaper examines the sensitivity of sound propagation to the shear properties of the ocean floor for shallow
waters and when the sound source lies in the ocean. This is motivated by predictions reported in the structural
mechanics literature, which demonstrate that wave propagation in viscoelastic structures can be highly
sensitive to the shear properties of the material. This investigation uses the semi analytic finite element
method to undertake a set of numerical experiments. This numerical approach provides a computationallyefficient way of solving the fully coupled acoustic problem, with Bergmann’s equation used for the fluid,
and Navier’s elastodynamic wave equation for the seabed. Both elastic and viscoelastic structures are
examined and predictions are reported for a range of shear properties, and the relative sensitivity of soundpropagation on these shear properties is then investigated.
Keywords: Sound propagation, Shallow ocean, Shear waves

1. INTRODUCTION

The propagation of sound in the ocean depends on the acoustic properties of the seabed, as well as
those of the ocean. This forms a coupled system so that sound propagation from a point source locatedin the sea depends on the viscoelastic properties of the seabed. Accordingly, to compute accurately
sound propagation in an ocean waveguide it is necessary to capture the physics of the propagating
viscoelastic wave. This demands the use of the elastodynamic wave equation and the inclusion of bothcompressional and shear waves. This delivers a complex coupled system and developing accurate
theoretical models is challenging, and this is especially true for shallow oceans where the coupling
effects are likely to be strong and complex interactions between the ocean and a layered seabed mustbe accounted for.
The development of computational models for ocean acoustics has largely focused on advancing
the pioneering work of Pekeris (1). This involves separating variables in order to recast the problem
into two separate wave equations in the range (𝑟𝑟 ) and depth (𝑧𝑧 ) directions. To solve the depth
dependent wave equation, a modal expansion is often applied so that the pressure and displacement
are written as a sum over downwards and upwards propagating planar waves (2, 3). This enables
reflections from the interface between the ocean and the seabed to be captured, as well as thosereflections from layers inside the seabed. The range dependence is then captured using Hankel
functions, or large argument approximations of Hankel functions, so that sound propagation from asound source located in the ocean can be computed. This approach has led to the development of alarge number of computational codes (4), the majority of which adopt varying approaches to solve thedepth dependent eigenproblem. Popular examples include the analytic approach of Westwood et al.
(5) and the numerical solutions to the depth dependent wave equation by Porter and Reiss (6, 7). These12

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ray.kirby@uts.edu.au
wenbo.duan@brunel.ac.uk

methods are widely used to compute the sound pressure field inside the ocean, and they are suitablefor including the viscoelastic behavior of the seabed. However, it is challenging to capture the entire
physics of the problem using these methods, especially for shallow waters. This is because of thestrong coupling present in shallow waters, where it is common for large numbers of eigenmodes to berequired in order to obtain accurate predictions. Ensuring all relevant modes have been captured, as
well as accommodating the complex scattering inside the seabed is challenging. Furthermore,
enforcing the correct limiting boundary conditions in the seabed is also important for shallow water
predictions and this is also difficult to capture accurately when using a Pekeris style approach.
This article uses a different approach that is based on the semi analytic finite element (SAFE)
method (8-11). This approach discretizes the depth and then solves an eigenproblem for the range
direction, so that the modes obtained are forward and backwards propagating modes. This approachenables the inclusion of depth dependent acoustic properties in the ocean, as well as the correct
coupling conditions between each layer in the waveguide. The application of a perfectly matched layer(PML) also enables the correct limiting boundary conditions to be applied in the depth direction. Thismethod is then used here to explore sound propagation in relatively shallow oceans and the influenceof shear waves is examined through the modification of the acoustic properties of the seabed. Thisstudy is undertaken in order to gain a better understanding of the importance of shear waves on thecalculation of sound propagation in shallow oceans.

2. THEORY

The geometry of the ocean waveguide us shown in Fig. 1.
𝑧𝑧𝑧𝑧 = 𝑎𝑎𝑚𝑚𝑧𝑧 = 0𝑧𝑧 = −ℎAirΩ1OceanΩ2Seabed

Ω3𝑧𝑧 = −𝑏𝑏𝑚𝑚

𝑥𝑥Figure 1 – Geometry of ocean waveguide.
The waveguide consists of three regions, Ω1 which contains air above the ocean, Ω2 is the ocean
in which depth dependent properties can be included, and Ω3 is a viscoelastic seabed. The wave
equation for sound propagation in a fluid (Ω1 or Ω2) is given as (12)′11 𝜕𝜕 2 𝑝𝑝1,2′
𝜌𝜌1,2 (𝑧𝑧)∇ ∙ �∇𝑝𝑝1,2
�− 2=0𝜌𝜌1,2 (𝑧𝑧)𝑐𝑐1,2 (𝑧𝑧) 𝜕𝜕𝑡𝑡 2(1)

where, 𝑝𝑝′ is acoustic pressure, 𝑡𝑡 is time, 𝑐𝑐 is the speed of sound and 𝜌𝜌 is the fluid density (theseare assumed to be depth dependent only in Ω2). The seabed is considered to be an elastic solid that
supports both shear and compressional waves, and here the elastodynamic wave equation gives (13)𝜕𝜕 2 𝐮𝐮′3(2)=0𝜕𝜕𝑡𝑡 2′′ ]𝑢𝑢3𝑧𝑧
is thewhere 𝛔𝛔′3 is the Cauchy stress tensor, 𝜌𝜌3 is the material density, and 𝐮𝐮′3 = [𝑢𝑢3𝑥𝑥
displacement in region Ω3. These equation are solved by expanding the sound pressure field in each
fluid, and the displacement in the seabed, over a set of eigenmodes propagating in the range direction,
so that∇ ∙ 𝛔𝛔′3 − 𝜌𝜌31903

𝑛𝑛 (𝑧𝑧)𝑒𝑒 𝑖𝑖𝜔𝜔𝜔𝜔−𝑖𝑖𝑘𝑘0 𝛾𝛾𝑛𝑛 𝑥𝑥′ (𝑥𝑥,𝑧𝑧, 𝑡𝑡) = � A𝑛𝑛 𝑝𝑝1,2𝑝𝑝1,2,𝑛𝑛=1∞(3)

𝐮𝐮′3 (𝑥𝑥, 𝑧𝑧, 𝑡𝑡) = � A𝑛𝑛 𝐮𝐮𝑛𝑛3 (𝑧𝑧)𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖−𝑖𝑖𝑘𝑘0 𝛾𝛾𝑛𝑛𝑥𝑥 ,𝑛𝑛=1∞(4)𝑛𝑛
𝑛𝑛𝑛𝑛 (𝑧𝑧),(𝑧𝑧) 𝑢𝑢3𝑧𝑧(𝑧𝑧)] are the eigenfunctions, and 𝛾𝛾𝑛𝑛 the eigenvalues. Inand 𝐮𝐮𝑛𝑛3 (𝑧𝑧) = [𝑢𝑢3𝑥𝑥where, 𝑝𝑝1,2addition, A𝑛𝑛 are the modal amplitudes, 𝑘𝑘0 is a reference wavenumber chosen so that 𝛾𝛾 is nondimensional, where 𝑘𝑘0 = 𝜔𝜔⁄𝑐𝑐0 , 𝜔𝜔 is the radian frequency, 𝑐𝑐0 is an (arbitrary) reference sound speed,
and 𝑖𝑖 = √−1.
To couple each wave equation together, it is necessary to apply the relevant boundary conditions.
These are: continuity of pressure and particle velocity between each fluid region; continuity ofpressure and normal stress, and continuity of normal displacement, between the fluid and the seabed;in addition, non-reflecting boundary conditions for 𝑧𝑧 → ±∞ are implemented using a PML. To apply
these boundary conditions, the ansatz for each variable is substituted back into each wave equationand these are then weighted and integrated over the depth 𝑧𝑧 . A SAFE formulation (8-11) is thenapplied that discretizes the depth dimension only, and in this article three noded quadratic line
elements are used. The addition of the boundary conditions into the SAFE formulation then allows aneigenproblem to be solved for 𝛾𝛾 and the respective eigenfunctions.
Following solution of the eigenproblem, the sound propagation from a point source is calculated.
This is realized using the method of Astley and Cummings (14), with the addition of the biorthogonality relation of Scandrett and Frenzen (15) to accommodate the viscoelastic seabed. This
gives the modal amplitudes of the point source as:A𝑗𝑗 =2
𝜌𝜌1
𝑝𝑝 (𝑧𝑧 ),𝑘𝑘0 𝛾𝛾𝑗𝑗 Λ𝑗𝑗,𝑗𝑗 𝜌𝜌𝑠𝑠 𝜌𝜌2 (−ℎ)𝑐𝑐02 2,𝑗𝑗 𝑠𝑠(4)where, Λ𝑗𝑗,𝑗𝑗 is an orthogonality relation (15), and 𝜌𝜌𝑠𝑠 and 𝑧𝑧𝑠𝑠 are the density and depth of the point
source, respectively. Note that these modal amplitudes have been normalized against a reference point
source, whose amplitude is equal to unity at a distance of 1 m. This then enables the soundtransmission loss (TL) to be computed for the waveguide using a normalized TL common in theliterature, so that

TL = −20log10 |𝑝𝑝(𝑥𝑥, 𝑧𝑧)|.

(5)This expression for TL is then used in the calculations that follow in the next section. Note thatthese calculations are for a spherical point source and include cylindrical spreading.

3. RESULTS AND DISCUSSION

In this section a number of numerical experiments are carried out using the SAFE model describedin the previous section. The SAFE model is a general approach that is well suited to analyzing shallow
oceans, and so the focus in this section is on computing sound propagation in oceans of a depth of
100 m or less. Accordingly, the first set of numerical experiments uses an ocean of depth 100 m, witha source placed at a depth of 25 m, and a receiver at a depth of 75 m. The air above the ocean has aheight of 20 m, and a PML of thickness 10 m. The seabed has an overall depth of 310 m, with a PMLof depth 200 m. The finite element mesh uses 440 elements to solve the eigenproblem, and 400 modes
to obtain the TL; this takes about 6 seconds to solve on a standard laptop.
The numerical tests use either sand or limestone in the seabed and their acoustic properties arelisted in Table 1. In addition, the density of air and water are listed, where for simplicity the density
and speed of sound in the ocean are assumed to be constant. In Fig. 2 the TL is calculated for alimestone seabed with zero damping; the shear velocity in the seabed is then varied in order to explore
the influence of the shear properties of the seabed on the sound pressure field. It can be seen in Fig.
2 that significant changes in behavior are evident when the shear properties in the limestone are altered.
For example, differences of up to 25 dB are seen for relatively modest changes in the shear velocity.
It is however noticeable that for relatively short distances away from the sound source the differences

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in TL are less pronounced, so that the effects of the shear properties on sound propagation in thiswaveguide are more evident for larger ranges.
Table 1 – Properties of waveguideCompressional
Properties

Shear Properties(kg/m3)

(m/s)

0.1

𝑐𝑐𝑇𝑇

(dB/𝜆𝜆)

1650

𝛼𝛼𝐿𝐿

(m/s)

1400

𝑐𝑐𝐿𝐿

(dB/𝜆𝜆)

Sand

𝜌𝜌

400

0.2

Limestone

2300

2500

0.05

1315

0.1

Water

1000

1500

Air

1.2343

Transmission Loss (dB re 1 m)

-30𝛼𝛼 𝑇𝑇

F=50 Hz𝑧𝑧s=−25 m𝑧𝑧r=−75 m-40-50-60-70-80-900500

1000Range (m)Figure 2 – TL for limestone seabed with zero damping.
; 𝑐𝑐𝑇𝑇 = 1000 m/s; .

𝑐𝑐𝑇𝑇 = 1600 m/s;

1500

2000; 𝑐𝑐𝑇𝑇 = 1315 m/s;

; 𝑐𝑐𝑇𝑇 = 1200 m/s.

In Fig. 3, damping is added into back into the limestone, with reference values taken from Table 1(𝛼𝛼 𝑇𝑇 = 0.1 dB/λ). It is seen in Fig. 3 that the addition of damping has little effect on the TL, and thatit is only in the longer ranges that a difference in behavior is noticeable. This lack of influence for thedamping in the limestone is likely to be partly caused by the low excitation frequency, however theseresults do indicate that at least for this waveguide it is not so important to identify accurately the shear
damping coefficient in the seabed.
In Fig. 4 predictions are shown for a more realistic waveguide in which a layer of sand 10 m deepis placed above a much thicker layer of limestone 300 m deep (including the PML). Figure 4 theninvestigates the influence of the shear properties of the sand (with damping) whilst keeping theproperties of the limestone fixed (with damping). It is evident in Fig. 4 that significant differences in
the TL are still evident, even when varying the properties in a relatively thin layer of sand. Thisbehavior is likely to be caused by scattering at the interface between the ocean and the seabed, whichtransfers energy from compressional to shear waves in the region close to the interface. This
demonstrates that it is important to have a reasonably accurate understanding of the acoustic propertiesof the upper layer of the seabed, even for relatively short ranges.

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Transmission Loss (dB re 1 m)

-30
F=50 Hz𝑧𝑧s=−25 m𝑧𝑧r=−75 m-40-50-60-700500

1000
Range (m)Figure 3 – TL for limestone seabed with damping.
; 𝛼𝛼 𝑇𝑇 = 0 dB/λ; .

𝛼𝛼 𝑇𝑇 = 0.2 dB/λ;

1500

2000

; 𝛼𝛼 𝑇𝑇 = 0.1 dB/λ;; 𝛼𝛼 𝑇𝑇 = 0.8 dB/λ.

Transmission Loss (dB re 1 m)

-30
F=50 Hz𝑧𝑧s=−25 m𝑧𝑧r=−75 m-40-50-60-70-80

0500

1000
Range (m)

1500

2000

Figure 4 – TL for seabed with 10 m layer of sand with varying properties, above 300 m layer of limestone
with fixed properties.

; 𝑐𝑐𝑇𝑇 = 400 m/s;

𝑐𝑐𝑇𝑇 = 200 m/s;
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; 𝑐𝑐𝑇𝑇 = 600 m/s;

; 𝑐𝑐𝑇𝑇 = 300 m/s.

Transmission Loss (dB re 1 m)

-30
F=50 Hz𝑧𝑧s=−25 m𝑧𝑧r=−75 m

-40
-50
-60
-70
-80

0500

1000
Range (m)

1500

2000

Figure 5 – TL for seabed with 10 m layer of sand with fixed properties, above 300 m layer of limestone
with varying properties.
; 𝑐𝑐𝑇𝑇 = 1000 m/s; .

; 𝑐𝑐𝑇𝑇 = 1315 m/s;

𝑐𝑐𝑇𝑇 = 1600 m/s;

; 𝑐𝑐𝑇𝑇 = 1200 m/s.

In Fig. 5 the properties of the sand are fixed and those of the limestone layer underneath the sandare changed (with both layers including damping). Figure 5 demonstrates that the properties of asecond layer are still important when attempting to quantify sound propagation; however for thisproblem the influence is not as strong as that seen when varying the properties in the upper layer.

Transmission Loss (dB re 1 m)

-30

-40

-50
F=50 Hz𝑧𝑧s=−30 m𝑧𝑧r=−30 m-60-70

0200

400

600

800

1000Range (m)Figure 6 – TL for seabed with 10 m layer of sand with varying properties, above 300 m layer of limestone
with fixed properties.

; 𝑐𝑐𝑇𝑇 = 400 m/s;

𝑐𝑐𝑇𝑇 = 200 m/s;

; 𝑐𝑐𝑇𝑇 = 600 m/s;

; 𝑐𝑐𝑇𝑇 = 300 m/s.

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Transmission Loss (dB re 1 m)

-10F=250 Hz𝑧𝑧s=−30 m𝑧𝑧r=−30 m

-20
-30
-40
-50
-60
-70
-80

0200

400

600

800

1000

Range (m)Figure 7 – TL for seabed with 10 m layer of sand with varying properties, above 300 m layer of limestone
with fixed properties.

; 𝑐𝑐𝑇𝑇 = 400 m/s;

𝑐𝑐𝑇𝑇 = 200 m/s;

; 𝑐𝑐𝑇𝑇 = 600 m/s;

; 𝑐𝑐𝑇𝑇 = 300 m/s.

In Fig. 6 the depth of the ocean is reduced to 40 m and the sound source and receiver are both
placed at a depth of 30 m. To accommodate a shallower ocean the range is reduced from 2 km to 1 kmin Fig. 6. It can be seen in Fig. 6 that varying the properties of the upper layer of sand again leads tolarge differences in the TL. This difference continues to be significant and this further illustrates thatachieving accurate predictions of TL in ocean acoustics is likely to be challenging because of thesensitivity of sound propagation to the shear properties of the seabed. However, if the frequency ofexcitation is increased to 250 Hz, then in Fig. 7 it is seen that the difference between predictions
reduces. For a relatively short range, sound propagation now becomes less sensitive to the shear
properties of the seabed. This is likely to be caused by the change in modal structure within thewaveguide when the frequency is increased, so that those modes that are less sensitive to the shear
properties are preferentially excited when compared to behavior at 50 Hz. This effect is likely to beproblem dependent, however this further illustrates the complexity of the shallow waveguide problem.

4. CONCLUSIONSThis article uses a SAFE approach to compute the sound pressure distribution in an ocean
waveguide. The SAFE method enables the sound propagation in the ocean to be coupled to viscoelastic
wave propagation in the seabed, and through the use of PMLs it is shown to be possible to properly
account for the limiting transverse boundary conditions. This makes the SAFE method well suited tomodelling sound propagation in shallow oceans, and here it is shown that TL values can be very
sensitive to the shear properties of the seabed. This present challenges for modeling sound propagationin shallow oceans, as measuring accurately the acoustic shear properties of viscoelastic materials overa wide frequency range can be difficult.

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