Modelling sound particle motion in shallow water^a) Open Access

Many aquatic organisms have evolved ways of detecting and producing sound (Song , 2015). Marine animals use sound to sense their surroundings, as sound travels relatively far and fast in water (Ainslie, 2010). Studies of the impact of anthropogenic sound on aquatic animals are often based on sound pressure; however, sound particle motion is expected to be more important for most fishes and invertebrates than sound pressure (Nedelec , 2016; Popper and Hawkins, 2018). Therefore, it is of interest to be able to model the particle motion component of the sound field. Studies on the impact of, for example, shipping noise on marine life, could benefit from the availability of maps of sound particle motion components as well as sound pressure (Sertlek, , 2019; Sertlek 2021; Putland , 2022). Therefore, the EU Horizon 2020 research program SATURN (SATURN) the relationship between sound pressure and sound particle velocity acoustic field components, and to develop the capability to produce maps of sound particle motion.

Underwater sound maps (Sertlek , 2019; Sertlek, 2021; Putland , 2022) provide a geographical view of the sound field, at a specified depth or averaged over depth, and in a specified frequency band. Such maps are generally calculated by combining acoustic propagation models with source level models or data for the relevant sound sources. This paper proposes solutions for three model benchmark scenarios for shallow water sound propagation. Shallow water here means that sound propagation is affected by interaction of the sound with sea surface and seabed. The proposed scenarios include a step-by-step increase in complexity, from a fluid to a solid sediment (adding shear properties) and from range-independent to range-dependent (adding horizontal variation in water depth). The scenarios are described in Sec. II A.

The literature (for example, Flamant and Bonnel, 2023) describes a relationship between the potential and kinetic energy in waveguides. The focus in this paper is on sound particle velocity because of its direct connection with acoustic impedance and intensity. As the acoustic intensity is related to the kinetic energy, similar conclusions are to be expected. For practical application, the best practice guide for underwater particle motion measurements (Nedelec , 2021) advises us to report sound particle acceleration, when possible, to ensure compatibility of results between studies. Sound particle acceleration is the time-derivative of sound particle velocity and the Fourier transform of sound particle acceleration at frequency $f$ equals $2πif$ times the Fourier transform of sound particle velocity.

The calculated metrics are described in Sec. II B. Based on the results from the different scenarios, a discussion is provided of the occasions where dedicated sound particle motion modelling provides value over an estimation of sound particle velocity from sound pressure via the characteristic specific acoustic impedance of the medium.

Previous work on sound particle motion modelling considered parabolic equation (PE) models for range-dependent and range-independent environments (Collis , 2007; Smith, 2008; Smith, 2010; Dall'Osto ., 2012; Deal and Smith, 2016). PE models are far-field approximations of the Helmholtz equation, which can compute solutions very quickly, but ignore some physics, such as backscattering, which may be relevant for particle motion modelling. Other modelling approaches are possible, each with its own advantages and disadvantages. In this study, a PE code is compared with a finite element (FE) code, a wavenumber integration (WI) code, and an image source (IS) code. This study aims to provide systematic verification of the applicability, implementation, and configuration of different models for underwater sound particle motion. Details of the models are described in Sec. II C.

Because reference solutions are generally lacking, a comparison of different model solutions to the same problem is an accepted means to confirm the validity of numerical models (Felsen, 1990). This work builds on recent efforts for benchmarking models for the sound pressure field (Collins, 1990; Jensen and Farla, 1990; Collins and Evans, 1992; Sertlek ., 2019; Binnerts , 2019; Küsel and Siderius, 2019).

Three different shallow-water sound propagation scenarios are proposed. The environmental properties are shown in Table I, and Fig. 1 shows sketches of the geometry. Three frequencies are considered for the sound propagation calculations for each scenario: 25 100 and 400 Hz.

The first scenario is for a range-independent Pekeris waveguide (Pekeris, 1948). The same scenario has been used for recent benchmarks for sound pressure field models (Sertlek and Ainslie, 2014; Binnerts , 2019; Küsel and Siderius, 2019). The water has a uniform depth of 50 m and the seabed is modelled as a fluid, only supporting compressional waves, with the properties of a “sandy” sediment (Ainslie, 2010). The sound source is at 5 m depth below the water surface. This is a representative scenario for, for example, a merchant vessel in the North Sea (de Jong , 2021). All four considered models (Sec. II C) can produce solutions for this scenario.

The geometry for the second scenario is the same as that of the first scenario, but in this case, the seabed supports shear waves as well as compressional waves. The shear properties are tabulated in Table I (bottom row) and represent a rock with a low shear speed (Davies, 1965; Ludwig, 1970). Solutions are presented for finite element and WI models (Sec. II C).

The third scenario is for a range-dependent waveguide, for which the parabolic equation and the finite element models (Sec. II C) can produce pressure and particle motion solutions. This scenario is the “Acoustical Society of America (ASA) wedge” benchmark case (Felsen, 1987), in which the water depth decreases from 200 to 0 m over a distance of 4 km. The sound source is placed at 100 m below the water surface (mid depth).

Typically, the scaled impedance in the sound field deviates from one in bounded environments, such as in laboratory tanks and aquaria, and in open waters at distances within about one acoustic wavelength from sources and discontinuities.

Four propagation models are selected that either produced sound particle motion as a standard output or could be modified to calculate sound particle motion.

For the FE calculations, the COMSOL Multiphysics software (COMSOL) is used. For the physics module, the pressure acoustics in the frequency domain was chosen, and for the shear scenario, the solid mechanics and multi-physics were added. The order of the finite elements is quadratic. A free triangular mesh is used, where the elements are at most λ/6 in size, with wavelength λ chosen for each respective local material. In the sediment, the λ value of the shear waves is used (when applicable) resulting in a finer mesh inside the sediment for the shear environments. Furthermore, a perfectly matched layer (PML) with a size of eight elements was used. Since the FE model does not simplify the wave equation, it is assumed to be the most accurate method for generating a solution for both pressure and particle motion.

The parabolic equation model used in this work is RAM. For the RAM implementation, a split-step Padé solution method is used (Collins, 1993) with eight Padé terms, a range step of 0.25λ, and a depth step of 0.01λ. The sediment is modelled as two layers. The upper layer contains the sediment defined by the scenario, with a thickness of 20λ. The lower layer is a PML with a size of 8λ. The particle motion is calculated from the spatial derivatives of the calculated sound pressure field.

The WI model used in this work is the propagation loss module OAST in the OASES code (Schmidt and Jensen, 1985). OAST was set up with separate runs to calculate the horizontal and vertical particle velocity directly. The environment was specified as comprising a vacuum upper half-space, the water layer, and the sediment lower half-space. The minimum and maximum phase velocities were 700 and 10 000 m/s, respectively. Values for the number of sampling points in wavenumber space and the complex WI contour offset were calculated automatically in OASES.

As can be seen, the images of the sound pressure field and the radial sound particle velocity field inside the water column look almost identical to this impedance scaling. Outside the region close to the water surface, the vertical sound particle velocities are generally smaller than the radial velocities, and decay much faster with increasing range. Solving for the sound pressure field for this scenario therefore appears to provide an acceptable proxy for solving the radial sound particle velocity field, and the radial sound particle velocity can be estimated as $U r≈ P / ρ c$ at most locations (see Sec. III D).

Figure 3 shows the depth-averaged sound pressure and sound particle velocity propagation loss $DA N P L , u z$ as a function of range and frequency, from the FE model calculations for the three scenarios. For the range-independent shallow water scenarios (I and II), the loss at 25 Hz is larger than the loss at higher frequencies. This is because the 60 m wavelength sound at 25 Hz fits poorly inside the 50 m deep waveguide. The depth-averaged propagation factor of pressure for scenario II (solid seabed) increases somewhat more steeply with range than for scenario I, due to energy lost into sediment shear waves. For the range-dependent scenario III (ASA wedge), the depth-average propagation loss is very similar for the three frequencies, because the water is deeper, and the source is further from the water surface. Towards larger ranges in scenario III, where the water depth decreases to 0 m, the calculated losses for the three frequencies show some spread. The number of grid points over which the depth-average is calculated decreases with decreasing water depth, leading to an increasing uncertainty.

For scenarios I and II, the PE and WI solutions overlap with the FE solution to within 0.1 dB at distances greater than 200 m from the source (four times the water depth). At large distances, where the u_z component of the sound field gets small, numerical rounding errors in the PE implementation lead to some scatter in the solutions, which is considered insignificant. At shorter distances, the PE and WI solutions start to deviate from the FE solutions. The FE solutions are more than 1 dB above the PE solutions at distances shorter than about 30 m, while the FE solutions at the lowest frequency (25 Hz) are more than 1 dB below the WI solutions at distances shorter than about 150 m. The image source model solutions are close to the FE solutions up to a distance of about 90 m (about two water depths) from the source. At larger distances, the IS solution is close to the FE solution at the highest frequency (400 Hz), while the deviation increases towards lower frequencies, where the far-field assumptions underlying the images source model approach are no longer valid. Nevertheless, the deviation is limited to about 3 dB for these scenarios.

The PE model is the only model against which the FE results for scenario III are compared. For 25 and 100 Hz, the PE solutions are very close to the FE results. For 400 Hz, there is a difference larger than 1 dB over the first 150 m from the source. At large distances, where the components of the sound field as well as the water depth get small, numerical rounding errors lead to scatter in the solutions. The FE results for sound pressure propagation loss match very closely with the results from Collins and Evans (1992), as shown in the supplementary material.

The previous sections give confidence in the modelling of sound pressure and sound particle velocity loss. The different models agree well for the scenarios in which they can be applied, with differences being less than a decibel at most distances from the source. Hence, the model results presented in Figs. 2 and 3 can be used as benchmark solutions for the verification of other model implementations.

The model calculations provide insight into the question of where sound pressure or sound particle motion dominate the sound field. The scaled impedance shows where the sound pressure is dominant $Z s c > 1$⁠, when the sound particle velocity is dominant $Z s c < 1$⁠, or when neither is dominant $Z s c = 1$⁠. When neither is dominant, then the sound particle velocity can be estimated by the free-field solution $U= P / ρ c$⁠. The scaled impedance fields from the FE model calculations for the three scenarios are shown in Fig. 5, and zoomed in to horizontal distances up to 500 m from the source in Fig. 6. Three regions with deviating scaled impedance are determined: the near surface region, the near source region, and the near seabed region. Flamant and Bonnel (2023) also concluded that these are the regions where the potential energy and kinetic energy ratio differs.

The particle velocity is dominant $Z s c < 1$ in the region near the air–water surface (depth 0), where the pressure release boundary condition affects the sound field. As can be seen in Fig. 5, this region exists, irrespective of the bathymetry and range. The depth below the surface for which the scaled impedance reaches 0.99 can be computed from the range average of the scaled impedance over the full computation range, as function of depth in the water. This computation was done for scenarios I and II for the frequencies [12.5, 25, 50, 100, 200, 400, 800] Hz. The scaled impedance is smaller than 0.99 at depths smaller than a mean depth of about a quarter of a wavelength (⁠ $0.24 λ 0$⁠, with standard deviation $0.04 λ 0$⁠) below the water surface. Closer to the water surface, the scaled impedance decreases in proportion with $k 0z$ towards $Z sc=0$ at the water surface.

The pressure is dominant $( Z s c>1)$ in the region near the seabed (depth $H$⁠), where reflections affect the sound field. For a fluid-like sediment, the strength of the reflections depends on the ratio of the impedances of sediment and water. Similar to that at the water surface, the effect region extends to about a quarter of a wavelength above the seabed. For the three selected frequencies (25, 100, and 400 Hz), this is at 15, 4, and 1 m, respectively.

As shown in the previous sections, the sound particle velocity can be estimated from scaling the sound pressure with the characteristic specific impedance of water, particularly if one is not interested in detailed results at single locations and single frequencies. To illustrate this, Fig. 7 presents the error in the estimation of depth-averaged sound particle velocity from depth-averaged sound pressure for the three scenarios and three frequencies. This error is calculated as the difference between the calculated depth-averaged sound particle velocity propagation loss and the sound pressure propagation loss + 63.5 dB, according to Eq. (10). At horizontal ranges beyond about 20 m (less than one water depth) from the source, the error is less than 1 dB for the magnitude of the sound particle velocity. At ranges beyond 200 m, where the radial component of the sound particle velocity dominates the magnitude, the error is less than 1 dB for the radial component as well.

Figure 8 shows the sound power flow into fluid waveguide and seabed as a function of range, from FE calculations for the three scenarios at the three selected frequencies. This confirms that the total power flow does not vary strongly with range, which is a good check of the correct implementation of the models.

At short ranges (up to about 100 m), the power flows mainly in the water. At larger ranges, the power flow in the water waveguide decays with range, due to sound power transmission into the seabed. Shear in the sediment increases this decay. In the deeper range-dependent scenario III, the power flow into the sediment is less prominent.

There is an increasing interest in the particle motion component of underwater sound, because this is what many fish and invertebrate species sense. In addition to guidelines for measurements (Nedelec , 2021), there is a need for guidelines for modelling sound particle motion. Different existing models for underwater sound propagation have the potential to calculate sound particle motion (displacement, velocity, or acceleration) in addition to sound pressure. To support a reliable implementation of such models, this work presents model benchmark solutions for three shallow water scenarios, including a range-independent scenario, a range-dependent scenario with a fluid sediment, and a range-independent scenario with a sediment with shear properties, calculated at three frequencies (25, 100, and 400 Hz). For the proposed axisymmetric geometries, the radial component of the sound particle velocity is generally larger than the vertical component. Four different types of models were tested: a finite element (FE) model, a parabolic equation (PE) model, a WI model, and an image source (IS) model. All models produced very similar results of the depth-averaged sound pressure and sound particle velocity propagation loss for the scenarios in which they can be applied. Close to the source, at ranges smaller than a few water depths, PE and WI solutions deviate from the FE and IS solutions, which are considered more applicable at short range. At larger ranges, the IS solutions at the lowest frequencies (25 and 100 Hz) deviate from the others, because the approximations in that approach limit the applicability to higher frequencies.

Calculation of both sound particle velocity and sound pressure allows for evaluation of sound intensity and power flows. This work demonstrates how such calculations can provide additional information and confidence in the reliability of the model implementation.

The scaled impedance relates the sound pressure and sound particle velocity fields. It was calculated for the three scenarios to determine whether there are regions where particle motion modelling is considered necessary and where a simple free-field solution can yield a good approximation. It is shown that the scaled impedance field in these scenarios is very close to unity at most locations in the water waveguide. Deviations from a value of unity mainly occur within about a quarter wavelength from the air–water interface and the seabed, and within a distance of about one wavelength from the source, extending to about three water depths if the water depth is smaller than the wavelength. Outside these regions, the sound particle velocity field in a shallow water waveguide can be reasonably estimated by dividing the sound pressure by the characteristic specific impedance of the fluid.

See the supplementary material for a comparison of the FE model and PE model with benchmark pressure field results from the literature (Collins and Evans, 1992) and the tabulated depth averaged pressure and particle velocity for the three considered scenarios.

This study was completed as part of the EU Horizon 2020 funded SATURN project (Grant No. 101006443). We thank Jan-Willem Vrolijk for performing the FE calculations.

The authors have no conflicts of interest to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.