Non-dispersive Wave Research Articles

Effect of bathymetry data on tsunami wave ray tracing in the western Banten sea

The lack of tsunami wave warning during the Anak Krakatau collapse in 2018 resulted in devastating damage to coastal areas. The tsunami wave arrived in the coastal areas approximately half an hour after the collapse. As a tsunami wave can travel abruptly, an early warning system should warn faster. However, a warning system based on a sophisticated hydrodynamic model in real-time would take time to conduct the numerical tsunami wave simulation. Hence, this study proposes a fast and reliable estimation of tsunami wave propagation through a classic ray tracing analysis. We use two ray-tracing methods to investigate the tsunami wave propagation from Anak Krakatau to the western Banten Sea. The first method follows Snell’s law, considering dispersive waves, and the second assumes non-dispersive waves based on the ray tracing equations, considering the Earth’s sphericity. Both methods are quantitatively evaluated by comparing the travel time measured at Anyer and Marina Jambu. This study finds that non-dispersive wave tracing performs a shorter computational time and slightly better prediction of tsunami wave propagation than dispersive-based wave tracing, with a relative absolute difference of the travel time of 17.9–26.7% in Anyer and 3.6–5.2% in Marina Jambu. This study also confirms the importance of bathymetry validity in wave ray tracing. Two regional bathymetry datasets with a mean difference of less than 5 m result in different wave ray tracing, in which one dataset does not produce the wave ray path towards the Panaitan Island. Based on bathymetric surveys in Anyer and Marina Jambu, the national bathymetry data (BATNAS) has shown its superiority to being used as a bathymetry in the ray-tracing process, with correlations of 81% and 93% in both areas compared to other available bathymetry datasets. We summarize that reliable bathymetry data and the non-dispersive ray tracing method can be used as an initial estimation of tsunami wave propagation efficiently.

Integrated mechanical computing for autonomous soft machines

Mechanical computing offers a new modality to formulate computational autonomy in intelligent matter or machines without any external powering or active elements. Transition (or solitary) waves, induced by nonreciprocity in mechanical metamaterials comprising a chain of bistable elements, have proven to be a key ingredient for dissipation-free transmission and computation of mechanical information. However, advanced processing of mechanical information in existing designs is hindered by its dissipation when interacting with networked logic gates. Here, we present a metamaterial design strategy that allows non-dispersive mechanical solitary waves to compute multi-level cascaded logic functions, termed ‘integrated mechanical computing’, by propagating through a network of structurally heterogeneous computing units. From a perspective of characteristic potential energy, we establish an analytical framework that helps in understanding the solitary wave-based mechanical computation, and governs the mechanical design of key determinants for realizing cascaded logic computation, such as soliton profile and logic elements. The developed integrated mechanical computing systems are shown to receive, transmit and compute mechanical information to actuate intelligent soft machine prototypes in a seamless and integrated manner. These findings would pave the way for future intelligent robots and machines that perform computational operations between various non-electrical environmental inputs.

Characterizing elastic wave reflection in a semiconducting isotropic diffusive medium with temperature rate-dependent nonlocal theory

In this paper, we explore the application of nonlocal theory to analyze the phenomenon of coupled thermoelastic wave reflection in a semiconducting diffusive medium, considering its temperature rate dependence. The governing equations are deconstructed using the Helmholtz vector rule, allowing us to delve into the behavior of the system. By calculating the dispersion relation in terms of propagation speed, we investigate four coupled longitudinal waves alongside an independent nondispersive transverse wave within the local medium. The cut-off frequencies for each wave are discussed, shedding light on their characteristics. Furthermore, we delve into the phenomenon of coupled longitudinal displacement waves at the medium’s boundary. Analytical derivations of amplitude ratios are presented, accompanied by graphical representations of their behavior, focusing on a semiconductor material such as copper. We examine the effects of physical parameters, including the nonlocal and diffusive parameters, on the obtained results. It is important to note that the existing literature primarily lacks consideration of diffusivity and plasma transportation. Lastly, we validate our findings by investigating the conservation of energy within the system.

Why Is the Westward Rossby Wave Propagation from the California Coast “Too Fast”?

Abstract Past work has shown that interannual California coastal sea level variability is mostly of equatorial origin, and decades of satellite sea surface height (SSH) and in situ dynamic height observations indicate that this interannual signal propagates westward from the California coast as nondispersive Rossby waves (RWs). These observations agree with standard linear vertical mode theory except that even when mean flow and bottom topography are considered, the fastest baroclinic vertical mode RW in each case is always much slower (1.6–2.3 cm s−1) than the observed 4.2 cm s−1. This order-1 disagreement is only resolved if the standard bottom boundary condition that the vertical velocity w′ = 0 is replaced by perturbation pressure p′ = 0. Zero p′ is an appropriate bottom boundary condition because south of San Francisco the northeastern Pacific Ocean boundary acts approximately like an impermeable vertical wall to the interannual equatorial wave signal, and therefore equatorial quasigeostrophic p′ is horizontally constant along the boundary. Thus, if equatorial p′ = 0 at the bottom, then this condition also applies off California. The large-scale equatorial ocean boundary signal is due to wind-forced eastward group velocity equatorial Kelvin waves, which at interannual and lower frequencies propagate at such a shallow angle to the horizontal that none of the baroclinic equatorial Kelvin wave signal reaches the ocean floor before striking the eastern Pacific boundary. Off California this signal can thus be approximated by a first baroclinic mode with p′ = 0 at the bottom, and hence the long RW speed there agrees with that observed (both approximately 4.2 cm s−1). Significance Statement The California Current System is one of the most biologically rich and best-documented coastal regions in the world. In this region coastal sea level propagates westward from the coast at about 110 km month−1, slow enough to enable us to make large-scale ocean climate forecasts of the California Current ecosystem using coastal sea level. Although the westward speed seems slow, theoretically it is about double what we would expect. Offered here is an explanation of why this speed is “too fast” by linking the California wave signal to the equator, El Niño, and the shallow equatorial ocean response to the wind.

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory. In semi-infinite space, a nondispersive Rayleigh wave mode exists, and the eigendisplacement decays exponentially with depth. In a layered model with increasing layer velocity, the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer. If the phase velocity is the same as the P- or S-wave velocity of the layer, which is called the critical mode or critical phase velocity of surface waves, the general solution of the wave equation is not a homogeneous (expressed by trigonometric functions) or inhomogeneous (expressed by exponential functions) plane wave, but one whose amplitude changes linearly with depth (expressed by a linear function). Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode, owing to the singularity at the critical phase velocity. In this study, based on the classical framework of generalized reflection and transmission coefficients, the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity. Therefore, the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem. The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models. In contrast to the normal mode, the eigendisplacement associated with the critical phase velocity exhibits different characteristics. If the phase velocity is equal to the S-wave velocity in the bottom half-space, the eigendisplacement remains constant with increasing depth.

Extreme wave formation past a submerged circular sill: The role of linear spatial focusing and resonant trapping

We investigate dispersive wave amplification past a submerged circular sill on an otherwise flat seabed. This phenomenon is important because it can generate large-amplitude waves near the sill, due to spatial focusing and resonant trapping in linear dispersive regime, endangering navigation. Based on the potential flow theory, the velocity potential is solved separately in the ocean region and in the sill region. Matching is then achieved by means of integral equations involving Galerkin expansion of the unknown velocity field at the border between the two regions. The model is successfully validated against known analytical expressions for long waves and a smoothed particle hydrodynamics numerical solution. Our results advance existing theories valid either for non-dispersive waves or for shallow submergence of the sill. We show that, for relatively short waves as compared to the ocean depth, the sill acts as a wave lens focusing energy behind it. Increasing the wavelength of the incident wave promotes transition from wave focusing behind the sill to partial trapping atop the sill. In intermediate water depth, the concurrence of focusing and partial trapping favors the emergence of extreme wave amplitudes that can exceed up to 6 times the amplitude of incident waves. We hypothesize that this phenomenon is the main cause of local peaks in skewness and kurtosis near a submerged circular shoal obtained in recent numerical simulations. Indications for further studies in the nonlinear regime are finally provided.

Time-domain Green’s function associated to the interface problem for the Klein–Gordon equation

In this work we present a representation formula for the Green’s function associated with the source or interface problem for the Klein–Gordon equation from a known result on solution for the non-dispersive wave equation. The results can extend to more general situations, in particular stratified media where the Cagniard-de Hoop method fails for reasons of non-homogeneity of the amplitude and the phase of the exponential in the Laplace transform. Examples and numerical applications are given in two dimensions for the interface problem associated to a dispersive wave equation where the reflected and incident fields are obtained. The calculation of the transmitted field is omitted for the sake of simplicity of the paper. Such a representation formula can be useful to study other qualitative properties of dispersive waves from those known on non-dispersive ones.

Observation of optical de Broglie–Mackinnon wave packets

de Broglie wave packets accompanying moving particles are dispersive and lack an intrinsic length scale solely dictated by the particle mass and velocity. Mackinnon proposed a localized non-dispersive wave packet constructed out of dispersive de Broglie phase waves that possess an intrinsic length scale via an inversion of the roles of particle and observer. So far, the de Broglie–Mackinnon wave packet has remained a theoretical proposal. Here we report the observation of optical de Broglie–Mackinnon wave packets using paraxial space–time-coupled pulsed laser fields in the presence of anomalous group-velocity dispersion. Crucially, the bandwidth of de Broglie–Mackinnon wave packets has an upper limit that is compatible with the wave-packet group velocity and equivalent mass. In contrast to previously observed linear-propagation-invariant wave packets whose spatio-temporal profiles at any axial plane are X-shaped, those for de Broglie–Mackinnon wave packets are uniquely O-shaped (circularly symmetric with respect to space and time). By sculpting their spatio-temporal spectral structure, we produce dispersion-free de Broglie–Mackinnon wave packets in the dispersive medium, observe their circularly symmetric spatio-temporal intensity profiles and closed-trajectory spectra, and tune the field parameters that uniquely determine the wave-packet length scale. de Broglie–Mackinnon wave packets are an extension of matter waves, but have so far remained a theoretical construct. Combining spatio-temporal light fields with anomalous dispersion has now allowed their experimental observation.

Two-way wave–vortex interactions in a Lagrangian-mean shallow water model

We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow.

A bioinspired apparatus for modeling peristaltic pumping in biophysical flows

In this study, we present a novel, bioinspired experimental apparatus, its construction, data acquisition methodology, and validation for the study of peristaltic flows. The apparatus consists of a series of stepper motor actuators, which deflect a deformable membrane to produce peristaltic flows. We show that this apparatus design has significant advantages over previous designs that have been used to study peristaltic flows by offering a much wider range of modeling capabilities. Comparisons between the capabilities of our apparatus and previous ones show our apparatus spanning a larger range of wavelength λ, wave speed c, amplitude A, and waveform (i.e. the apparatus is not constrained to nondispersive waves or to a sinusoidal shape). This large parameter range makes the apparatus a useful tool for biomimetic experimental modeling, particularly for systems that have complex waveforms, such as peristaltic flows in perivascular vessels, arteries, the cochlea, and the urethra. We provide details on the experimental design and construction for ease of reconstruction to the reader. The apparatus capabilities are validated for a large parameter range by comparing experimental measurements to analytic results from (Ibanez et al 2021 Phys. Rev. Fluids 6 103101) for high Reynolds number (Re > 1) and (Jaffrin and Shapiro 1971 Annu. Rev. Fluid Mech. 3 3–37) for low Reynolds number (Re < 1) applications. We show that the apparatus is useful for biophysical peristaltic studies and has potential applications in other types of studies.

Surface gravity waves in the presence of vertically shearing current and eddy viscosity

Conventional models of surface gravity waves usually use potential theory in their calculations where the governing equations are the inviscid Euler equations. Nevertheless, turbulence may play a significant role for short waves or highly energetic sea or river conditions and, therefore, must be taken into account in the flow's formulation. This work extends the model of linear monochromatic waves in the presence of vertically shearing currents to account for turbulence in the form of eddy viscosity varying in depth. The boundary value problem is found to be governed by an augmented Orr–Sommerfeld(–Squire) equation(s) in two-(three-)dimensional with additional terms of viscosity derivatives. The free-surface conditions are extended to account for shearing currents with turbulent viscosity and external stress for the air layer. This provides a fundamental model for investigating the influence of the turbulent viscosity on the oscillatory wave flow. Examples of semi-analytical and numerical solutions show a fundamentally different dynamical behavior with respect to the known non-viscous solutions. These differences include two regimes in the dispersion relation with non-dispersive short waves and dispersive longer waves with a reduced celerity. They also include tilted stream function contours (i.e., a phase which is a function of depth), much deeper penetration depth, and more. These preliminary results show that turbulence potentially has great importance in wave generation mechanisms as well as in highly turbulent regions such as under strong storms or cyclones.

OPTICAL SOLITON WAVE SOLUTIONS OF THE FRACTIONAL COMPLEX PARAXIAL WAVE DYNAMICAL MODEL ALONG WITH KERR MEDIA

This paper studies the optical soliton wave solutions of the fractional complex paraxial wave dynamical (FPWD) model with Kerr media. This model investigates the frequency of the beam obeying that reveals non-dispersive and non-diffractive Spatio-temporal localized wave envelopes promulgating in optical Kerr media. This model is equivalently the well-known nonlinear Schrödinger equation. These two models concentrate on the nonlinear visual medium to increase its strength and increase the nonlinear optical operation’s efficiency. Three recent computational schemes are employed to investigate the considered model’s abundant wave solutions; these solutions are then used to evaluate the initial and boundary conditions. The solutions’ accuracy is investigated by calculating the absolute error between solutions exact and approximate solutions. Additionally, the solution’s stability is checked by employing Hamiltonian system properties. The results’ physical interpretation is explained through some different sketches in 2D, 3D, and density plots.

Optimal unidirectional generation of a dispersive wave mode with dual-array transducer

Array transducers are commonly used to generate several types of waves, such as ultrasonic guided waves. They consist of a series of periodic wave source elements, and thus define a nominal or dominant wavelength of generated waves. A common approach to generating waves in a single direction is to use two identical wave sources, or two arrays of sources forming a dual-array transducer, that are separated by a quarter wavelength. Then, one source is activated a quarter of a time period after the other. In this way, constructive interference arises in one direction, whilst destructive interference arises in the opposite direction, making the total wave source unidirectional. This principle assumes that the frequency content of the generated wave is monochromatic. When there is some frequency bandwidth to the wave generated, such as with finite-length pulses, one cannot satisfy simultaneously constructive and destructive interference in opposite directions with dual-array transducers. For a non-dispersive wave mode, ideal destructive interference for pulsed waveform is obtained if one of the sources is inverted relative to the other, so that, in theory, complete cancellation can be achieved. Moreover, most guided wave modes are dispersive, meaning that, different frequencies propagate at different phase speeds. This limits the amount of wave cancellation in one direction and constructive interference in the other. Here, we describe the operation of the dual-array transducer in the frequency–wavenumber domain and use it to propose two new excitation methods, based on a frequency-dependent phase-shift strategy, that is designed through the dispersion relationship of the wave mode of interest. These provide ideal constructive or destructive interferences for dispersive waves, and achieve optimal unidirectional generation behaviour for dispersive wave modes. The new methods were experimentally assessed with shear horizontal ultrasonic waves, generated by a dual periodic permanent magnet electromagnet acoustic transducer. The optimal excitation signal yielded up to 30 dB unidirectionality, when generating the dispersive SH1 wave mode in an aluminium plate.

Study of flexural wave propagation through a cable stayed beam subjected to a moving load

A physical perspective of the propagation and attenuation of flexural waves is presented in this paper for the dynamic behaviors of cable stayed beams subjected to a moving load. Based on the method of reverberation-ray matrix (MRRM), the waveform solutions of the wave equations of a simplified beam-cable system subjected to a moving load (hereinafter referred to as a beam-cable system) are given, and the theory is verified by a numerical example. The dynamic response of cable stayed beams is decomposed into nine kinds of flexural waves, including traveling waves, near-field waves, and nondispersive waves, according to the wavenumber characteristics. Numerical examples are analyzed to demonstrate the propagation characteristics of flexural waves through cable stayed beams. Numerical results show that the flexural waves in the cable stayed beams are mainly low-frequency waves whose frequencies are less than 3 times the structural fundamental frequency, which can be used to further improve the computational efficiency of response analysis method based on MRRM, and the proportion of high-frequency components increases gradually with increasing structural stiffness. The near-field wave can be transformed into a traveling shear wave when its frequency is larger than the critical frequency, which decreases with increasing radius of gyration and decreasing elastic modulus of the beam. With the increase in the radius of gyration and the elastic modulus of the beam, the attenuation effect of the near-field wave weakens. The wave velocity and the wave dispersion effect have a positive correlation with the stiffness-related parameters of the beam-cable system. The study of the effect of the beam-cable system parameters on flexural wave propagation characteristics can be applied to achieve a better dynamic design for engineering structures.

An Efficient Damage Quantification Method for Cylindrical Structures Enhanced by a Dry-Point-Contact Torsional-Wave Transducer

Quantification of damage sizes in cylindrical structures such as pipes and rods is of paramount importance in various industries. This work proposes an efficient damage quantification method by using a dry-point-contact (DPC) transducer based on the non-dispersive torsional waves in the low-frequency range. Theoretical analyses are first carried out to investigate the torsional wave interaction with different sizes of defects in cylindrical structures. A damage quantification algorithm is designed based on the wave reflections from the defect and end. Capitalizing on multiple excitations at different frequencies, the proposed algorithm constructs a damage image that identifies the geometric parameters of the defects. Numerical simulations are conducted to validate the characteristics of the theoretically-predicted wave-damage interaction analyses as well as the feasibility of the designed damage quantification method. Using the DPC transducer, experiments are efficiently carried out with a simple physical system. The captured responses are first assessed to confirm the capability of the DPC transducer for generating and sensing torsional waves. The sizes of the defects in two representative steel rods are then quantified with the proposed method. Both numerical and experimental results demonstrate the efficacy of the proposed damage quantification method. The understandings of the wave-damage interaction and the concept of the damage quantification algorithm lay out the foundation for engineering applications.

Evaluation of the sensitivity of higher order modes cluster (HOMC) guided waves to plate defects

Higher Order Modes Cluster (HOMC) guided waves are non-dispersive wave packets consisting of higher order Lamb wave modes. This paper investigates the effect of frequency and dominant mode(s) of an HOMC guided wave on its sensitivity to plate defects. The propagation of HOMC guided waves in a 10-mm thick mild steel plate is simulated by two-dimensional finite element model. The level of contribution of Lamb wave modes to the wave cluster is analyzed by using a two-dimensional Fast Fourier Transform (2D FFT) at various frequency-thickness products ranging from 15 to 35 MHz-mm. The normal stress profiles of the HOMC guided waves are plotted and used to show the changes in normal stress as a function of frequency-thickness product. The HOMC guided waves interactions with notches of 5%, 10%, 30% and 50% depths relative to the plate thickness are examined at various frequency-thickness products ranging from 15 to 35 MHz-mm. It is observed that the reflected energy from these notches decreases with increase in frequency-thickness product. Experimental measurements are also conducted on a similar plate. The experimental results are in line with simulations and confirm the changes observed in the reflection coefficient of notches at different frequency-thickness products.

Relation between Orbital Velocities, Pressure, and Surface Elevation in Nonlinear Nearshore Water Waves

AbstractThe inability of the linear wave dispersion relation to characterize the dispersive properties of nonlinear shoaling and breaking waves in the nearshore has long been recognized. Yet, it remains widely used with linear wave theory to convert between subsurface pressure, wave orbital velocities, and the free surface elevation associated with nonlinear nearshore waves. Here, we present a nonlinear fully dispersive method for reconstructing the free surface elevation from subsurface hydrodynamic measurements. This reconstruction requires knowledge of the dispersive properties of the wave field through the dominant wavenumbers magnitude κ, representative in an energy-averaged sense of a mixed sea state composed of both free and forced components. The present approach is effective starting from intermediate water depths—where nonlinear interactions between triads intensify—up to the surf zone, where most wave components are forced and travel approximately at the speed of nondispersive shallow-water waves. In laboratory conditions, where measurements of κ are available, the nonlinear fully dispersive method successfully reconstructs sea surface energy levels at high frequencies in diverse nonlinear and dispersive conditions. In the field, we investigate the potential of a reconstruction that uses a Boussinesq approximation of κ, since such measurements are generally lacking. Overall, the proposed approach offers great potential for collecting more accurate measurements under storm conditions, both in terms of sea surface energy levels at high frequencies and wave-by-wave statistics (e.g., wave extrema). Through its control on the efficiency of nonlinear energy transfers between triads, the spectral bandwidth is shown to greatly influence nonlinear effects in the transfer functions between subsurface hydrodynamics and the sea surface elevation.

Engineering quantum wave-packet dispersion with a strong nonresonant femtosecond laser pulse

A non-dispersing wave packet has been attracting much interest from various scientific and technological viewpoints. However, most quantum systems are accompanied by anharmonicity, so that retardation of quantum wave-packet dispersion is limited to very few examples only under specific conditions and targets. Here we demonstrate a conceptually new and universal method to retard or advance the dispersion of a quantum wave packet through 'programmable time shift' induced by a strong non-resonant femtosecond laser pulse. A numerical simulation has verified that a train of such retardation pulses stops wave-packet dispersion.

Optical nondiffractive and nondispersive dark-wave dynamics in hydrodynamic origin

The primary attention of this work focuses on finding rational and generalized rational solutions for the (2 + 1)-dimensional Kadomtsev–Petviashvili equation that can be used to describe nondispersive and nondiffractive localized wave packet in power law nonlinear media. Specifically, the (2 + 1)-dimensional nonlinear Schrödinger equation with power law nonlinearity is first transformed into the Kadomtsev–Petviashvili equation. Then, through the bilinear form and symbolic computation, we derive two hierarchies of multi-lump solitary waves, composing of three, six, and eight lump waves. The obtained solutions observe a specific ‘circularity structure’ that the lump waves sit at the same circular. In addition, we illustrate that these waves are stable during propagation.

Contribution of Lamb wave modes in the formation of higher order modes cluster (HOMC) guided waves

In recent years, Higher Order Modes Cluster (HOMC) guided waves have been considered for ultrasonic testing of plates and pipes. HOMC guided waves consist of higher order Lamb wave modes that travel together as a single nondispersive wave packet. The objective of this paper is to investigate the effect of frequency-thickness value on the contribution of Lamb wave modes in an HOMC guided wave. This is an important issue that has not been thoroughly investigated before. The contribution of each Lamb wave mode in an HOMC guided wave is studied by using a two-dimensional finite element model. The level of contribution of various Lamb wave modes to the wave cluster is verified by using a 2D FFT analysis. The results show that by increasing the frequency-thickness value, the order of contributing modes in the HOMC wave packet increases. The number of modes that comprise a cluster also increases up to a specific frequency-thickness value and then it starts to decrease. Plotting of the cross-sectional displacement patterns along the HOMC guided wave paths confirms the shifting of dominant modes from lower to higher order modes with increase of frequency-thickness value. Experimental measurements conducted on a mild steel plate are used to verify the finite element simulations. The experimental results are found to be in good agreement with simulations and confirm the changes observed in the level of contribution of Lamb wave modes in a wave cluster by changing the frequency-thickness value.