Trapped modes and scattering for oblique waves in a two-layer fluid

2014 ◽  
Vol 753 ◽  
pp. 427-447 ◽  
Author(s):  
M. I. Romero Rodríguez ◽  
P. Zhevandrov
Keyword(s):  
Scattering Problem ◽  
Total Reflection ◽  
Unperturbed System ◽  
Trapped Modes ◽  
Theoretical Understanding ◽  
Transmission Coefficients ◽  
Trapped Mode ◽  
Geometric Conditions ◽  
Small Height ◽  
Incident Waves

AbstractThe system describing time-harmonic motions of a two-layer fluid in the linearised shallow-water approximation is considered. It is assumed that the depth is constant, with a cylindrical protrusion (an underwater ridge) of small height. For obliquely incident waves, the system reduces to a pair of coupled ordinary differential equations. The values of frequency for which wave propagation in the unperturbed system is possible are bounded from below by a cutoff, to the left of which no propagating modes exist. Under the perturbation, a trapped mode appears to the left of the cutoff and, if a certain geometric requirement is imposed upon the shape of the perturbation (for example, if the ridge is a rectangular barrier of certain width), a trapped mode appears whose frequency is embedded in the continuous spectrum. When these geometric conditions are not satisfied, the embedded trapped mode transforms into a complex pole of the reflection and transmission coefficients of the corresponding scattering problem, and the phenomenon of almost total reflection is observed when the frequency coincides with the real part of the pole. Exact formulae for the trapped modes are obtained explicitly in the form of infinite series in powers of the small parameter characterising the perturbation. The results provide a theoretical understanding of analogous phenomena observed numerically in the literature for the full problem for the potentials in a two-layer fluid in the presence of submerged cylinders, and furnish explicit formulae for the frequencies at which total reflection occurs and the trapped modes exist.

Subharmonic resonance of proposed storm gates for Venice Lagoon

1994 ◽  
Vol 444 (1920) ◽  
pp. 257-265 ◽  
Keyword(s):  
Experimental Evidence ◽  
Incident Wave ◽  
Laboratory Tests ◽  
Venice Lagoon ◽  
Subharmonic Resonance ◽  
Horizontal Axis ◽  
Trapped Modes ◽  
Resonance Mechanism ◽  
Trapped Mode ◽  
Incident Waves

A recent design of storm barriers at the inlets of Venice Lagoon consists of a number of articulated inclined gates hinged on a horizontal axis on the seabed. In laboratory tests with normally incident waves the gates have been found to oscillate at half of the incident wave frequency and out of phase with their immediate neighbours. In this paper we identify the resonance mechanism by first showing the existence of trapped modes as a consequence of the articulated construction. Experimental evidence is shown for the trapped mode and its subharmonic resonance by normally incident waves.

Interaction of water waves with three-dimensional periodic topography

2001 ◽  
Vol 434 ◽  
pp. 301-335 ◽  
Author(s):  
R. PORTER ◽  
D. PORTER
Keyword(s):  
Incident Wave ◽  
Water Waves ◽  
Homogeneous Problem ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Trapped Waves ◽  
Trapped Mode ◽  
Bed Elevation ◽  
Input Mode ◽  
Incident Waves

The scattering and trapping of water waves by three-dimensional submerged topography, infinite and periodic in one horizontal coordinate and of finite extent in the other, is considered under the assumptions of linearized theory. The mild-slope approximation is used to reduce the governing boundary value problem to one involving a form of the Helmholtz equation in which the coefficient depends on the topography and is therefore spatially varying.Two problems are considered: the scattering by the topography of parallel-crested obliquely incident waves and the propagation of trapping modes along the periodic topography. Both problems are formulated in terms of ‘domain’ integral equations which are solved numerically.Trapped waves are found to exist over any periodic topography which is ‘sufficiently’ elevated above the unperturbed bed level. In particular, every periodic topography wholly elevated above that level supports trapped waves. Fundamental differences are shown to exist between these trapped waves and the analogous Rayleigh–Bloch waves which exist on periodic gratings in acoustic theory.Results computed for the scattering problem show that, remarkably, there exist zeros of transmission at discrete wavenumbers for any periodic bed elevation and for all incident wave angles. One implication of this property is that total reflection of an incident wave of a particular frequency will occur in a channel with a single symmetric elevation on the bed. The zeros of transmission in the scattering problem are shown to be related to the presence of a ‘nearly trapped’ mode in the corresponding homogeneous problem.The scattering of waves by multiple rows of periodic topography is also considered and it is shown how Bragg resonance – well-established in scattering of waves by two-dimensional ripple beds – occurs in modes other than the input mode.

scholarly journals Trapped modes in a waveguide with a long obstacle

2000 ◽  
Vol 403 ◽  
pp. 251-261 ◽  
Author(s):  
N. S. A. KHALLAF ◽  
L. PARNOVSKI ◽  
D. VASSILIEV
Keyword(s):  
Two Dimensional ◽  
Trapped Modes ◽  
Acoustic Waveguide ◽  
Trapped Mode ◽  
Rectangular Obstacle

Consider an infinite two-dimensional acoustic waveguide containing a long rectangular obstacle placed symmetrically with respect to the centreline. We search for trapped modes, i.e. modes of oscillation at particular frequencies which decay down the waveguide. We provide analytic estimates for trapped mode frequencies and prove that the number of trapped modes is asymptotically proportional to the length of the obstacle.

Wave Propagation Analysis of an Axially Strained, Rotating Timoshenko Shaft: Part II

1997 ◽  
Author(s):  
Bongsu Kang ◽  
Chin An Tan
Keyword(s):  
Boundary Conditions ◽  
Wave Reflection ◽  
Axial Strain ◽  
Free Vibration Analysis ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Propagation Analysis ◽  
Geometric Discontinuities ◽  
Bernoulli Models ◽  
Incident Waves

Abstract In this paper, the wave reflection and transmission characteristics of an axially strained, rotating Timoshenko shaft under general support and boundary conditions, and with geometric discontinuities are examined. As a continuation to Part I of this paper (Kang and Tan, 1997), the wave reflection and transmission at point supports with finite translational and rotational constraints are further discussed. The reflection and transmission matrices for incident waves upon general supports and geometric discontinuities are derived. These matrices are combined, with the aid of the transfer matrix method, to provide a concise and systematic approach for the free vibration analysis of multi-span rotating shafts with general boundary conditions. Results on the wave reflection and transmission coefficients are presented for both the Timoshenko and the Euler-Bernoulli models to investigate the effects of the axial strain, shaft rotation speed, shear and rotary inertia.

scholarly journals TRANSMISSION OF RANDOM WAVES THROUGH PILE BREAKWATERS

1986 ◽  
Vol 1 (20) ◽  
pp. 169 ◽  
Author(s):  
Clifford L. Truitt ◽  
John B. Herbich
Keyword(s):  
Random Waves ◽  
Pile Spacing ◽  
Transmission Coefficients ◽  
Pile Diameter ◽  
Model Pile ◽  
Incident Waves ◽  
Good Agreement ◽  
Monochromatic Waves

Several previous investigators have conducted experiments leading to expressions for predicting the transformation of waves passing through closely-spaced pile breakwaters. The present study extends those earlier experiments using monochromatic waves to the case of a spectrum of random waves. Records of incident waves and of waves after transmission through a model pile breakwater were compared to determine a coefficient of transmission. Results are presented for several cases of pile spacing and pile diameter. Good agreement is found between observed transmission coefficients and those predicted using the expression proposed by Hayashi et al. (1966).

Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides

1999 ◽  
Vol 386 ◽  
pp. 233-258 ◽  
Author(s):  
R. PORTER ◽  
D. V. EVANS
Keyword(s):  
Finite Number ◽  
Surface Waves ◽  
Cross Section ◽  
Electromagnetic Waves ◽  
Mode Shapes ◽  
Special Choice ◽  
Trapped Mode ◽  
Uniform Cross Section ◽  
New Type ◽  
Incident Waves

Rayleigh–Bloch surface waves are acoustic or electromagnetic waves which propagate parallel to a two-dimensional diffraction grating and which are exponentially damped with distance from the grating. In the water-wave context they describe a localized wave having dominant wavenumber β travelling along an infinite periodic array of identical bottom-mounted cylinders having uniform cross-section throughout the water depth. A numerical method is described which enables the frequencies of the Rayleigh–Bloch waves to be determined as a function of β for an arbitrary cylinder cross-section. For particular symmetric cylinders, it is shown how a special choice of β produces results for the trapped mode frequencies and mode shapes in the vicinity of any (finite) number of cylinders spanning a rectangular waveguide or channel. It is also shown how one particular choice of β gives rise to a new type of trapped mode near an unsymmetric cylinder contained within a parallel-sided waveguide with locally-distorted walls. The implications for large forces due to incident waves on a large but finite number of such cylinders in the ocean is discussed.

scholarly journals Inter-Particle Effects with a Large Population in Acoustofluidics

Actuators ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 101
Author(s):  
Kun Jia ◽  
Yulong Wang ◽  
Liqiang Li ◽  
Jian Chen ◽  
Keji Yang
Keyword(s):  
Rayleigh Scattering ◽  
Acoustic Radiation ◽  
Large Population ◽  
Scattering Problem ◽  
Relative Size ◽  
Radiation Force ◽  
Mie Scattering ◽  
Separation Distance ◽  
High Concentration ◽  
Theoretical Understanding

The ultrasonic manipulation of cells and bioparticles in a large population is a maturing technology. There is an unmet demand for improved theoretical understanding of the particle–particle interactions at a high concentration. In this study, a semi-analytical method combining the Jacobi–Anger expansion and two-dimensional finite element solution of the scattering problem is proposed to calculate the acoustic radiation forces acting on massive compressible particles. Acoustic interactions on arrangements of up to several tens of particles are investigated. The particle radius ranges from the Rayleigh scattering limit (ka«1) to the Mie scattering region (ka≈1). The results show that the oscillatory spatial distribution of the secondary radiation force is related to the relative size of co-existing particles, not the absolute value (for particles with the same radius). In addition, the acoustic interaction is non-transmissible for a group of identical particles. For a large number of equidistant particles arranged along a line, the critical separation distance for the attraction force decreases as the number of particles increases, but eventually plateaus (for 16 particles). The range of attraction for the formed cluster is stabilized when the number of aggregated particles reaches a certain value.

scholarly journals Trapped modes around a row of circular cylinders in a channel

1999 ◽  
Vol 386 ◽  
pp. 259-279 ◽  
Author(s):  
T. UTSUNOMIYA ◽  
R. EATOCK TAYLOR
Keyword(s):  
Expansion Method ◽  
Wave Theory ◽  
Multipole Expansion ◽  
Circular Cylinders ◽  
Large Radius ◽  
Trapped Modes ◽  
Resonant Phenomenon ◽  
Trapped Mode ◽  
Linear Water Wave Theory ◽  
Multipole Expansion Method

Trapped modes around a row of bottom-mounted vertical circular cylinders in a channel are examined. The cylinders are identical, and their axes equally spaced in a plane perpendicular to the channel walls. The analysis has been made by employing the multipole expansion method under the assumption of linear water wave theory. At least the same number of trapped modes is shown to exist as the number of cylinders for both Neumann and Dirichlet trapped modes, with the exception that for cylinders having large radius the mode corresponding to the Dirichlet trapped mode for one cylinder will disappear. Close similarities between the Dirichlet trapped modes around a row of cylinders in a channel and the near-resonant phenomenon in the wave diffraction around a long array of cylinders in the open sea are discussed. An analogy with a mass–spring oscillating system is also presented.

Edge waves on a gently sloping beach: uniform asymptotics

1991 ◽  
Vol 233 ◽  
pp. 483-493 ◽  
Author(s):  
Peter Zhevandrov
Keyword(s):  
Finite Number ◽  
Shallow Water Equation ◽  
Uniform Asymptotics ◽  
Trapped Modes ◽  
Edge Waves ◽  
Trapped Mode ◽  
Complete Set ◽  
Constant Slope ◽  
Sloping Beach ◽  
Uniform Asymptotic Expansions

Edge waves on a beach of gentle slope ε [Lt ] 1 are considered. For constant slope, Ursell (1952) has obtained a complete set of trapped modes and shown that there exists only a finite number n of such modes, (2n + 1)β < ½π, β = tan−1ε. For non-uniform slope the formulae for the trapped-mode frequencies were heuristically derived by Shen, Meyer & Keller (1968). For small n ∼ O(1) Miles (1989) has obtained formulae which coincide with Shen et al.'s (1968) with accuracy to O(ε) and differ from them by O(ε2). However, Miles’ formulae fail at n ∼ 1/ε. In this paper it is proved that Shen et al.'s (1968) formulae are valid for all n (including n ∼ 1/ε) with accuracy to O(ε) and corrections of any order in ε are given. Uniform asymptotic expansions are obtained for the corresponding eigenfunctions. These expansions give Miles’ (1989) result for small n. The formulae for the frequencies and the eigenfunctions have the same structure for both the full dispersion system and the shallow-water equation. For small n the frequencies for both models coincide with accuracy to O(ε2), but for n ∼ 1/ε they differ by O(1). In the last section the effect of rotation following Evans (1989) is taken into account. All the asymptotics have formal character, i.e. they satisfy the corresponding equations with accuracy to O(εN), N being arbitrarily large. The rigorous justification of these asymptotics is under way.

Explicit solution of the scattering problem involving a vertical flexible membrane

2020 ◽  
Author(s):  
Srinivasa Rao Manam ◽  
Ashok Kumar ◽  
Gunasundari Chandrasekar
Keyword(s):  
Wave Scattering ◽  
Scattering Problem ◽  
Physical Problem ◽  
Plane Boundary ◽  
Water Wave Scattering ◽  
Flexible Membrane ◽  
Quarter Plane ◽  
Integral Relations ◽  
Incident Waves ◽  
Membrane Deflection

&lt;p&gt;The problem of normally incident water wave scattering by a flexible membrane is completely solved. The physical problem in a half-plane is reduced to a couple of equivalent quarter-plane problems by allowing incident waves from either direction of the membrane. In the same way, quarter-plane boundary value problems are posed for solid wave potentials that are solutions of the scattering problem involving a rigid structure of the same geometric configuration. Then, two novel integral relations are introduced to establish a link between the required solution wave potentials and few resolvable solid wave potentials. Explicit expressions for the scattering quantities such as the reflection and the transmission wave amplitudes are obtained. Also, the deflection of the flexible vertical membrane and the solution potentials are determined analytically. Numerical results for the scattering quantities and the membrane deflection are presented.&lt;/p&gt;

Sign in / Sign up

Export Citation Format

Share Document