Preamble

Study on the Dispersion Characteristics of Scholte Waves Excited by Pulsed and Continuous Sound Sources

Abstract: This paper investigates the dispersion characteristics and particle motion trajectories of Scholte waves excited by different types of acoustic sources. Scholte waves, which propagate along the interface between a fluid and an elastic solid, are critical for underwater acoustic communication and seabed characterization. By employing theoretical modeling and numerical simulations, we analyze the excitation mechanisms of Scholte waves under both pulsed and continuous wave conditions. Our results demonstrate that pulsed sources provide a broad frequency spectrum suitable for multi-mode dispersion analysis, while continuous sources allow for precise observation of steady-state wave phenomena. The study provides a detailed characterization of the energy distribution and the elliptical prograde/retrograde motion of particles at the interface, contributing to a deeper understanding of interface wave propagation in marine environments.

Keywords: Scholte waves, dispersion characteristics, particle motion trajectory, pulsed source, continuous source

1. Introduction

The study of interface waves is a fundamental aspect of marine acoustics and seismology. Among these, the Scholte wave—a surface wave that propagates along the boundary between a liquid layer (such as the ocean) and an underlying elastic solid (the seabed)—plays a vital role in understanding the physical properties of the seafloor. Unlike body waves, Scholte waves are characterized by their evanescent nature, with energy decaying exponentially away from the interface in both media.

Accurately modeling the dispersion characteristics of these waves is essential for geophysical inversion and underwater target detection. Previous research has largely focused on the theoretical derivation of the Scholte wave equation. However, the specific influence of the excitation source type—whether a transient pulse or a sustained continuous wave—on the resulting wavefield and particle kinematics requires further investigation. This paper aims to bridge this gap by comparing the dispersion profiles and particle motion trajectories generated by these two distinct source types.

2. Theoretical Background

2.1 Scholte Wave Equation

The propagation of Scholte waves at a fluid-solid interface is governed by the boundary conditions requiring continuity of normal displacement and normal stress, and the vanishing of tangential stress at the interface. For a fluid with density $\rho_f$ and sound speed $c_f$, and an elastic solid with density $\rho_s$, longitudinal wave speed $c_p$, and shear wave speed $c_s$, the dispersion relation for the Scholte wave speed $v_{Sch}$ is given by the characteristic equation:

$$ \ $$

1. School of Science

Wuhan University of Technology

430070 Wuhan

China

2. School of Naval Architecture

Ocean and Energy Power Engineering Wuhan University of Technology

430070 Wuhan

China

Abstract

Dispersion Characteristics of Scholte Waves Excited by Pulsed and Continuous Sound Sources

The dispersion and polarization characteristics of Scholte waves excited by sound sources in shallow water are of great significance for underwater target detection. However, the Scholte waves excited specifically by pulsed and continuous point sources are still not fully understood. This study investigates the Scholte waves generated by these two types of sound sources using normal mode theory, time-domain finite element modeling, and wavelet transform analysis.

First, based on a shallow water sound field model with a semi-infinite elastic seabed, the dispersion characteristics of Scholte waves are analyzed using normal mode theory. The results indicate that Scholte wave dispersion occurs under both hard and soft seabed conditions. Subsequently, a time-domain finite element model combined with wavelet transform is employed to investigate the dispersion of Scholte waves.

[FIGURE:1]

The analysis demonstrates that for a pulsed point source, the Scholte wave exhibits significant dispersion, which can be effectively extracted and characterized using the wavelet transform. In the case of a continuous point source, the interference between the direct wave and the reflected waves at the interface contributes to the complex wavefield structure. By comparing the theoretical dispersion curves with the simulation results, the study validates the accuracy of the finite element model in capturing the propagation physics of interface waves.

Furthermore, the polarization characteristics of the Scholte waves are examined. The particle motion trajectories at the fluid-solid interface reveal the typical elliptical prograde or retrograde motion associated with Scholte waves, depending on the depth and distance from the source. These findings provide a theoretical and numerical basis for utilizing Scholte waves in shallow water acoustic applications, particularly for seabed parameter inversion and target localization.

[TABLE:1]

In conclusion, the integration of normal mode theory and time-domain finite element analysis provides a robust framework for understanding the excitation and propagation of Scholte waves. The identification of dispersion patterns from different source types enhances the ability to process seismic-acoustic signals in complex marine environments.

polarization characteristics of particle motion trajectory excited by pulse and continuous sound source. The results show that the dispersion of Scholte waves can be observed when the two acoustic sources are in the hard seabed. The dispersion of Scholte waves can be observed for continuous sound sources in the soft sea- but it is difficult for pulsed sound sources to observe the dispersion phenomenon. According to the po- larization characteristics of Scholte waves the ellipse of the waves excited by pulsed sound source decrea- ses first and then increases while the ellipticity of Scholte waves under continuous sound source remains stable. Scholte waves excited by pulsed sound source vibrate vertically under hard seafloor mainly. Those excited by continuous sound source mainly vibrate vertically under hard seafloor and horizontally under soft seafloor. The results can provide a preliminary basis for the detection of pulse and continuous sound sources in shallow sea.

Scholte wave detection is a critical field of study. Mastering the phase velocity of Scholte waves is essential for acquiring accurate target location information. Consequently, researchers have extensively utilized the dispersion characteristics of Scholte waves to invert seabed parameters. In terms of experimental research, EWING and RAUCH conducted foundational studies on Scholte waves. Furthermore, by employing the Wigner distribution and other signal processing techniques, researchers such as FLORES-MENDEZ have successfully verified the existence and propagation characteristics of Scholte waves.

Numerical simulations of Scholte waves have been performed using LS-DYNA software. Additionally, TOMAR utilized Scholte wave velocity tomography and Monte Carlo methods to further analyze these phenomena. HUANG and other scholars have also contributed significantly to this field.

Key words: Scholte wave; elastic seafloor; dispersion property; polarization property; numerical simulation

Scholte Scholte Scholte Scholte Scholte

1 Scholte

1. Introduction

In the study of acoustic propagation within shallow sea environments, the interaction between the water column and the seabed is a critical factor. To accurately describe the wave field, we define the displacement potential functions for the respective media. Let $\phi_w$ represent the displacement potential function for the seawater layer. In the underlying elastic half-space (the seabed), we denote the displacement potential function for longitudinal waves (P-waves) as $\phi_p$ and the potential function for transverse waves (S-waves) as $\psi_s$.

1.1 Mathematical Model of the Shallow Sea

The propagation of waves in this environment is governed by the wave equations in each layer. For the seawater layer, which is typically modeled as an ideal fluid, the displacement potential $\phi_w$ satisfies the scalar wave equation:

$$\nabla^2 \phi_w - \frac{1}{c_w^2} \frac{\partial^2 \phi_w}{\partial t^2} = 0$$

where $c_w$ is the sound speed in water. In the solid seabed, the displacement field is decomposed into longitudinal and transverse components using the potential functions $\phi_p$ and $\psi_s$. These functions satisfy:

$$\nabla^2 \phi_p - \frac{1}{c_p^2} \frac{\partial^2 \phi_p}{\partial t^2} = 0$$
$$\nabla^2 \psi_s - \frac{1}{c_s^2} \frac{\partial^2 \psi_s}{\partial t^2} = 0$$

where $c_p$ and $c_s$ represent the P-wave and S-wave velocities in the seabed, respectively.

[FIGURE:1]

1.2 Boundary Conditions and Scholte Waves

At the interface between the seawater and the elastic seabed, specific boundary conditions must be satisfied, including the continuity of vertical displacement and normal stress, and the vanishing of tangential stress at the fluid-solid boundary. A significant phenomenon in this model is the existence of the Scholte wave, an interface wave that propagates along the seabed-water boundary. The energy of the Scholte wave is concentrated near the interface, and its velocity is lower than the sound speed in water and the shear wave speed in the seabed.

[TABLE:1]

As shown in (eq:boundary)

2 = 1 c

2 = 1 c 2 l

2 = 1 c 2 t

φ 0 = 2 A sinh k ξ 1 ( ) z e i kx- ω ( ) t ，

φ 1 = B e -k ξ l ( ) z e i kx- ω ( ) t ，

φ 2 = C e -k ξ t ( ) z e i kx- ω ( ) t 2 A B C ξ = 1 - c 2 / c 2 i

i = 1 l t ω = 2 π f k = ω / c f k

w a = ∂ φ 0 ∂ x v a = ∂ φ 0 ∂ z ，

v a = v b

σ zz a = σ zz b

$\sigma_{xz} = 0.4 \sigma_{zz}$, where $\sigma_{zz}$ represents the normal stress and $\sigma_{xz}$ represents the shear stress.

σ zz = λ ∂ w ∂ x + λ + 2 ( ) μ

Scholte Waves under Hard Seabed Conditions

1. Introduction

In underwater acoustics and marine geophysics, the propagation of interface waves at the seafloor is a critical area of study. When the seabed consists of "hard" materials—typically defined as those where the shear wave velocity ($v_s$) exceeds the sound speed in the overlying water column ($c_w$)—the characteristics of these interface waves change significantly. Under these conditions, the primary interface wave of interest is the Scholte wave.

Scholte waves are surface acoustic waves that propagate along the boundary between a fluid (ocean) and an elastic solid (seabed). Unlike Rayleigh waves, which exist at a vacuum-solid interface, Scholte waves account for the loading effect of the water column. In hard seabed environments, such as basaltic rock or highly consolidated sediments, the energy of the Scholte wave is tightly confined to the interface, making it a sensitive tool for characterizing the geoacoustic properties of the uppermost crustal layers.

2. Theoretical Framework

The propagation of Scholte waves is governed by the boundary conditions at the fluid-solid interface, which require the continuity of vertical displacement and normal stress, and the vanishing of tangential stress. For a hard seabed, the characteristic equation for the Scholte wave velocity ($v_{Sch}$) can be expressed in terms of the properties of the water (density $\rho_w$, sound speed $c_w$) and the elastic medium (density $\rho_s$, compressional wave velocity $v_p$, and shear wave velocity $v_s$).

The dispersion relation is given by the following determinantal equation:

$$ \begin{aligned} 4k^2 \eta_s \eta_p - (2k^2 - k_s^2)^2 = \frac{\rho_w}{\rho_s} \frac{\eta_p k_s^4}{\eta_w} \end{aligned} $$

where:
- $k$ is the Scholte wave wavenumber.
- $\eta_w = \sqrt{k^2 - k_w^2}$
- $\eta_p = \sqrt{k^2 - k_p^2}$
- $\eta_s = \sqrt{k^2 - k_s^2}$
- $k_w, k_p, k_s$ are the wavenumbers for the water, P-wave, and S

2 Scholte

Scholte Scholte

0 Hz

Propagation of Scholte Waves Approaching Rayleigh Waves

In the study of interface wave phenomena, the Scholte wave represents a critical acoustic mode propagating along a fluid-solid boundary. As physical parameters shift—specifically when the density of the fluid medium becomes negligible compared to the solid medium—the characteristics of Scholte wave propagation increasingly resemble those of Rayleigh waves. This transition is fundamental to understanding seismic surface waves and ultrasonic non-destructive testing in submerged environments.

The dispersion relationship for interface waves is governed by the boundary conditions at the fluid-solid interface, where the continuity of normal displacement and stress must be maintained. For a Scholte wave, the phase velocity is strictly lower than the speed of sound in the fluid and the shear wave velocity in the solid. However, as the fluid loading effect diminishes, the energy confinement of the Scholte wave shifts. In the limit where the fluid density $\rho_f$ approaches zero, the influence of the fluid pressure becomes infinitesimal, and the governing equations for the interface wave asymptotically converge to the vacuum-solid boundary condition.

Mathematically, this convergence can be observed in the characteristic equation for the interface. Let $c_R$ denote the Rayleigh wave velocity and $c_S$ denote the Scholte wave velocity. As the ratio of fluid density to solid density $\rho_f / \rho_s \to 0$, the Scholte wave velocity $c_S$ monotonically increases, approaching $c_R$ as its upper limit. In this regime, the elliptical particle motion characteristic of the Scholte wave becomes identical to the retrograde elliptical motion of a Rayleigh wave on a free surface.

Furthermore, the penetration depth of the wave into the solid medium undergoes a significant transformation during this transition. While Scholte waves are characterized by significant energy leakage or coupling into the fluid layer, the approaching Rayleigh wave limit signifies a total confinement of energy within the solid's surface layer. This behavior is critical for geophysical applications where the presence of a thin water layer or low-density fluid may perturb, but not fundamentally alter, the primary Rayleigh mode used for subsurface imaging.

40 Hz

Scholte

ρ 2 c 4 t 1 - c 2 / c 2 t = 0 9 Scholte

Stoneley Scholte

39 Hz

Dispersion Characteristics of Scholte Waves under Soft Seabed Conditions

1. Introduction

In the field of marine acoustics and geophysics, the study of interface waves is crucial for characterizing the physical properties of the seabed. Among these, the Scholte wave—a surface acoustic wave that propagates along the interface between a fluid (water) and a solid (seabed)—plays a significant role. Particularly in "soft seabed" environments, where the shear wave velocity of the sediment is lower than the speed of sound in water, the dispersion characteristics of Scholte waves become highly sensitive to the stratigraphic structure and elastic parameters of the seafloor.

2. Theoretical Framework

The propagation of Scholte waves is governed by the boundary conditions at the fluid-solid interface. For a homogeneous semi-infinite fluid overlying a layered elastic solid, the characteristic equation for the interface wave can be derived from the wave equation. Let $\rho_w$ and $c_w$ represent the density and sound speed of the water column, and $\rho_s$, $c_p$, and $c_s$ represent the density, longitudinal wave velocity, and shear wave velocity of the seabed, respectively.

The dispersion relation is typically expressed through the determinant of the boundary condition matrix. For a Scholte wave with phase velocity $v$ and wavenumber $k$, the characteristic equation is given by:

$$ \begin{aligned} 4k^2 \gamma \eta - (2k^2 - k_s^2)^2 = \frac{\rho_w}{\rho_s} \frac{\eta}{\sqrt{k^2 - k_w^2}} k_s^4 \end{aligned} $$

where:
- $\gamma = \sqrt{k^2 - k_p^2}$
- $\eta = \sqrt{k^2 - k_s^2}$
- $k_w, k_p, k_s$ are the wavenumbers for the water, p-wave, and s-wave, respectively.

3. Scholte Waves in Soft Seabed Environments

A "soft" seabed is characterized by a low shear modulus, meaning $c_s < c_w$. In such cases, the Scholte wave velocity is slightly lower than the shear wave velocity of the sediment ($v < c_s$). This makes the Scholte wave an ideal tool for inverting the shear wave velocity profile of the shallow seabed

0 Hz

Scholte leigh

5 Hz

Scholte

3 Scholte

Scholte Wave Velocity Variation with Frequency

In underwater acoustics and marine geophysics, the Scholte wave represents a critical interface wave that propagates along the boundary between a fluid (such as seawater) and an elastic solid (such as the seabed). Understanding the dispersion characteristics of Scholte waves—specifically how their velocity varies with frequency—is essential for characterizing the shear wave velocity profiles of seafloor sediments.

Dispersion Characteristics of Scholte Waves

The velocity of a Scholte wave is primarily governed by the shear wave velocity ($V_s$) of the solid medium. In an ideal case involving two semi-infinite homogeneous media, the Scholte wave is non-dispersive, meaning its velocity remains constant regardless of frequency. However, the actual seabed is typically stratified, with physical properties such as density and elastic moduli varying significantly with depth.

In these stratified environments, Scholte waves exhibit pronounced frequency dispersion. This phenomenon occurs because waves of different frequencies (and thus different wavelengths) penetrate to different depths within the seabed:

• High-Frequency Scholte Waves: These waves have shorter wavelengths and are confined to the uppermost layers of the sediment. Consequently, their phase velocity is dominated by the elastic properties of the shallowest materials.
• Low-Frequency Scholte Waves: These waves possess longer wavelengths, allowing them to penetrate deeper into the sub-bottom. Their propagation velocity is influenced by the properties of deeper, typically more consolidated layers where the shear wave velocity is higher.

Mathematical Representation

The relationship between frequency ($f$) and the Scholte wave phase velocity ($V_{Sch}$) can be described through the characteristic dispersion equation for a multilayered medium. For a given frequency $\omega = 2\pi f$, the phase velocity $V_p$ must satisfy the boundary conditions at the fluid-solid interface and all subsequent layer interfaces.

In general, the Scholte wave velocity is slightly lower than the shear wave velocity of the solid medium, typically following the relationship:
$$V_{Sch} \approx 0.9 V_s$$
This approximation holds when the density of the fluid is significantly lower than that of the solid. In marine environments where the density ratio is closer, the relationship becomes more complex and requires solving the full transcendental equation derived from the wave equations.

Applications in Geophysical Inversion

The frequency-dependent nature of Scholte wave velocity is the fundamental principle behind Surface Wave Analysis (SWA) for underwater sites. By measuring the dispersion curve—the plot of phase velocity

2 Scholte

COMSOL Scholte perfectly matched layer 1 500 m 300 m

4 Scholte

excited by point sound source Scholte Scholte

l = （ x i - a 1 2 + y i - b 1

7 References

[FIGURE:1] illustrates the time-domain waveforms of seismic signals. As shown in the figure, the Scholte wave signal is clearly identified within the reference data. Specifically, for an excitation source located 600 m below a hard seabed, the Scholte wave components are observed within the time interval of 0.41 s to 0.54 s. The propagation characteristics of these Scholte waves are consistent with the expected acoustic behavior for a sound source positioned at this depth relative to the seabed interface.## 10 Horizontal Distance from the Sound Source

At a source depth of 600 m, the primary wave speeds are 1,950 m/s and 600 m/s, while the secondary wave speed is 1,500 m/s. The observed arrival times occur at 0.31 s and 0.4 s. The Scholte wave velocity is measured at 532 m/s, with the Scholte wave appearing at 1.12 s. Compared to a hard seabed, the Scholte wave amplitude is significantly smaller in this context. This indicates that a soft seabed environment is unfavorable for the excitation of Scholte waves.

[FIGURE:1]

The data suggests that source excitation at 600 m below a soft seabed results in a diminished Scholte wave response. This attenuation highlights the influence of seabed composition on seismic wave propagation and interface wave generation.

In summary, for Scholte waves excited at a depth of 600 m below a soft seabed, the resulting interface wave energy is notably suppressed.## 30 Hz

The Scholte wave signals generated by a source excitation located 600 m below a hard seabed exhibit distinct characteristics, with a recorded arrival time of approximately 2.5 s. In comparison to signals originating from a hard seabed, the Scholte wave excitation at a depth of 600 m below a soft seabed demonstrates significant variations in wave propagation and signal behavior. These observations emphasize the influence of sub-seabed depth and substrate composition on the generation and transmission of Scholte waves.

600 m sound source 600 m below the soft seabed Scholte Scholte Scholte Scholte

4 Scholte

Wave Dispersion and Polarization Characteristics

The analysis begins with an investigation into the dispersion and polarization characteristics of normal modes, specifically focusing on the Scholte wave. Based on the theory of normal modes, the Scholte wave represents a critical interface wave propagating at the boundary between a fluid and an elastic solid. Its dispersion behavior is characterized by a lack of frequency-dependent phase velocity in homogeneous media, though it exhibits significant sensitivity to the shear wave velocity of the underlying seabed.

The polarization characteristics of these waves provide essential insights into the particle motion trajectories at the interface. Typically, the Scholte wave exhibits elliptical retrograde motion in the vertical plane, a feature that distinguishes it from other seismic arrivals. By examining the relationship between frequency, phase velocity, and the depth-dependent amplitude decay, we can effectively model the energy distribution within the waveguide. This foundational analysis of wave propagation serves as the basis for subsequent inversions of seabed geoacoustic parameters.

2 Scholte

The Scholte wave signal exhibits characteristics similar to those of direct acoustic waves, and significant dispersion can be observed during its propagation.

3 Scholte

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Influence of Ocean Environment Parameters on Scholte Waves

ZHU Hanhao, ZHENG Hong, LIN Jianmin, et al.

Abstract

The Scholte wave is a surface wave that propagates along the interface between a fluid medium and an elastic solid medium. Its energy is primarily concentrated near the interface, and its velocity is lower than both the sound speed in the water column and the shear wave velocity in the seabed. Because Scholte waves carry significant information regarding the geoacoustic properties of the seabed, they are widely used in underwater acoustics and marine geophysics for seabed parameter inversion and target detection. This paper investigates the influence of various ocean environmental parameters on the propagation characteristics of Scholte waves.

1. Introduction

The study of interface waves at the seafloor is a critical component of underwater acoustics. Among these, the Scholte wave represents a specific type of interface wave existing at the boundary between the ocean (fluid) and the seabed (elastic solid). Unlike body waves, Scholte waves undergo exponential decay in amplitude as the distance from the interface increases in both the upward (water) and downward (sediment) directions.

Understanding how environmental parameters—such as water depth, sediment density, and the velocities of compressional ($P$) and shear ($S$) waves—affect the dispersion and attenuation of Scholte waves is essential for accurate modeling. This study utilizes theoretical derivations and numerical simulations to analyze these sensitivities, providing a foundation for subsequent geophysical inversion techniques.

2. Theoretical Background

The propagation of Scholte waves is governed by the boundary conditions at the fluid-solid interface, which require the continuity of vertical displacement and normal stress, as well as the vanishing of tangential stress at the interface. The characteristic equation for the Scholte wave velocity $v_s$ can be expressed through the following relation:

$$ \left(2 - \frac{v^2}{v_t^2}\right)^2 - 4\sqrt{1 - \frac{v^2}{v_l^2}}\sqrt{1 - \frac{v^2}{v_t^2}} + \frac{\rho_w}{\rho_s} \frac{v^4}{v_t^4} \frac{\sqrt{1 - v^2/v_l^2}}{\sqrt{v_w^2/v^2 - 1}} = 0 $$

where:
- $v_w$ is the sound speed in the water;

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