On the linearized shallow-water waves over a sloping bottom

Summary This paper presents an investigation concerning long waves on sloping beaches. On the other hand, an attempt is made to broaden the class of the realistic forms of bottom topography,h(x), with which an exact solution of the related eigenvalue problem is possible. Under the linearizing assumption, the investigation confirms thath ^−1/2 is a factor associated with the bottom topography which is involved in wave breaking along the shoreline (Green's law). Further, the bottom topography in the neighbourhood of the shoreline is approximated byh(x)=K ₀ x ^j. There-from it is deduced that all positive real values ofj are associated with the possible structure of the bottom profile. Consequently, each can be employed in the construction of possible wave forms. Finally, an attempt is made to obtain a closed form of solution associated with the governing linear differential equation whenh(x) is arbitrary. The comparison of the results with those obtained by using the refraction diagram suggests close agreement over the beach considered.

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Riassunto In questo lavoro si presenta una ricerca che riguarda onde lunghe su spiagge in pendenza. Inoltre si fa un tentativo di ampliare la classe delle forme realistiche della topografia del fondo,h(x), con cui è possibile una soluzione esatta del problema correlato dell'autovalore. Nell'ipotesi di linearizzazione la ricerca conferma cheh ^−1/2 è un fattore associato con la topografia del fondo che è implicata nella rottura dell'onda lungo la linea di costa (legge di Green). Inoltre, la topografia del fondo in prossimità della linea di costa si approssima conh(x)=K ₀ x ^j. Da questo si deduce che tutti i valori reali positivi dij sono associati con la struttura possibile del profilo del fondo. Di conseguenza, ognuno di questi può essere utilizzato nella costruzione di possibili forme d'onda. Infine, si tenta di ottenere una forma chiusa della soluzione associata con l'equazione principale lineare differenziale quandoh(x) è arbitrario. Il confronto dei risultati con quelli ottenuti usando diagrammi di rifrazione suggerisce un accordo stretto per tutta la spiaggia considerata.

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Резюме В этой работе исследуются длинные волны на наклонных берегах. С другой стороны делается попытка расширить класс реалистических форм топографии дна,h(x), для которых возможно точное решение проблемы собственных значений. Предполагая линеаризацию, исследование подтверждает, чтоh ^−1/2 представляет фактор, связанный с топографией дна, который входит в разрушение волн вдоль береговой линии. Топографиа дна в окрестности береговой линии аппроксимируется выражениемh(x)=K ₀ x ^j. Отсюда следует, что все положительные значенияj связаны с возможной структурой профиля дна. Следовательно, каждое значение может быть использовано при конструровании возможных волновых форм. В заключение, предпринимается попытка получить замкнутую форму решения, связанного с определяющим дифференциальным уравнением, когдаh(x) является произвольной функцией. Сравнение результатов с результатами, полученными, используя диаграмму рефракции, свидетельствует о согласии вдоль всего рассмотренного берега.

     

Interaction of two sea surface waves over a sloping bottom

A nonlinear model is constructed for the motion of long surface waves over a sloping bottom in the shallow-water approximation. It is shown that if a linear solution is known, the quadratic and cubic nonlinear corrections can be expressed in terms of the derivatives of the functions determining the linear solution. The developed method is applied to the interaction of two periodic waves in the near-shore area.     

Nonlinear internal waves over the inclined bottom: Observations with the use of an acoustic profiler

Experimental data on propagation of solitary internal waves on the shelf of the Sea of Japan are presented and analyzed. The data are obtained by using an acoustic Doppler current profiler (ADCP). The process of propagation of the waves over the inclined bottom, up to their coming out to the coastal zone, is discussed for the case in which the pycnocline is closer to the surface than to the bottom. By using the approximation of an adiabatic invariant, it is shown that the height of the internal wave increases as it becomes closer to the overturning point. Qualitative agreement between the experimental data and the calculations is found.     

Effect of a sloping bottom on sound propagation

Acoustical Physics - The paper presents the results of field measurements of acoustic fields generated in autumn hydrological conditions of the Sea of Japan shelf by a TON-320Hz autonomous signal...     

Trapped modes and resonances for water waves over a slightly perturbed bottom

We construct exact solutions describing trapped water waves over an underwater ridge of small height in the shallow water approximation and in the complete formulation. Resonances (antibound states) in the case of an underwater trench are also constructed.     

On the localization of 2D nonlinear internal waves in a two-layer fluid

A 2D generalized Gardner equation is used to describe 2D nonlinear internal waves in a two-layer fluid. Unlike the previous model based on the Kadomtsev-Petviashvili equation, the model considered here allows for the instability of a plane internal solitary wave. Such a possibility causes the wave to be localized in any direction. Relationships between the thicknesses and densities of the layers under the instability conditions are obtained.     

Surface-to-volume wave conversion in shallow water with a gently sloping bottom

Marine sediments support seismoacoustic surface waves, which can propagate along the seafloor, in deep and shallow water, and even onshore. Because of the strong attenuation of compressional and especially shear waves in the sediments, the surface waves can significantly contribute to the acoustic field far from the shore only through their coupling with volume waves in the water. We theoretically study the excitation of acoustic normal modes by seismoacoustic surface waves in a shallow-water waveguide with a sloping bottom consisting of unconsolidated marine sediments. It is found that the coupling primarily occurs in the vicinity of a modal cutoff. The effects of geoacoustic parameters and stratification of soft marine sediments on the efficiency of surface-to-volume conversion are investigated. Published in Russian in Akusticheskiĭ Zhurnal, 2007, Vol. 53, No. 6, pp. 809–816. The article was translated by the author.     

An iterative method to solve a regularized model for strongly nonlinear long internal waves

We present a simple iterative scheme to solve numerically a regularized internal wave model describing the large amplitude motion of the interface between two layers of different densities. Compared with the original strongly nonlinear internal wave model of Miyata [10] and Choi and Camassa [2], the regularized model adopted here suppresses shear instability associated with a velocity jump across the interface, but the coupling between the upper and lower layers is more complicated so that an additional system of coupled linear equations must be solved at every time step after a set of nonlinear evolution equations are integrated in time. Therefore, an efficient numerical scheme is desirable. In our iterative scheme, the linear system is decoupled and simple linear operators with constant coefficients are required to be inverted. Through linear analysis, it is shown that the scheme converges fast with an optimum choice of iteration parameters. After demonstrating its effectiveness for a model problem, the iterative scheme is applied to solve the regularized internal wave model using a pseudo-spectral method for the propagation of a single internal solitary wave and the head-on collision between two solitary waves of different wave amplitudes.     

斜坡海底条件下声场垂直相关特性研究

利用数值仿真和实验研究分析和总结了不同斜坡海底条件下声场垂直相关性随垂直间隔和传播距离的变化规律,并利用楔形简正波理论对海底倾斜引起的声场垂直相关变化机理进行了解释.研究结果表明,上坡和下坡地形的声场垂直相关性具有很大的差别.与水平海底相比,下坡海底会增加声场垂直相关性,上坡海底会减小声场垂直相关性.有效简正波之间的干...     

内孤立波沿缓坡地形传播特性的实验研究

以南中国海东北部海域底部缓坡地形为背景, 在大型重力式分层流水槽中模拟了下凹型内孤立波沿缓坡地形传播过程中的浅化、破碎、分裂等现象, 利用分层染色标识方法和多点组合探头阵列技术对内孤立波沿缓坡地形演化特征进行了定性分析和定量测量. 实验表明: 浅化效应使内孤立波传播速度减小, 对大振幅内孤立波具有抑制作用, 对小振幅波具有放大效应; 浅化效应可导致内孤立波的剪切失稳及破碎, 还可导致大振幅内孤立波的分裂. 利用Miles稳定性理论可定性描述内孤立波沿缓坡地形传播时发生不稳定状态的位置, 实验结果与理论分析相符合. 关键词： 分层流 缓坡地形 内孤立波 不稳定性     

Analysis and modeling of broadband airgun data influenced by nonlinear internal waves

To investigate acoustic effects of nonlinear internal waves, the two southwest tracks of the SWARM 95 experiment are considered. An airgun source produced broadband acoustic signals while a packet of large nonlinear internal waves passed between the source and two vertical linear arrays. The broadband data and its frequency range (10-180 Hz) distinguish this study from previous work. Models are developed for the internal wave environment, the geoacoustic parameters, and the airgun source signature. Parabolic equation simulations demonstrate that observed variations in intensity and wavelet time-frequency plots can be attributed to nonlinear internal waves. Empirical tests are provided of the internal wave-acoustic resonance condition that is the apparent theoretical mechanism responsible for the variations. Peaks of the effective internal wave spectrum are shown to coincide with differences in dominant acoustic wavenumbers comprising the airgun signal. The robustness of these relationships is investigated by simulations for a variety of geoacoustic and nonlinear internal wave model parameters.     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

On internal nonlinear resonant interaction of capillary-gravitational waves on the flat surface of a viscous liquid

Mechanisms behind internal nonlinear resonant interaction of periodic capillary-gravitational waves on the uniformly charged flat surface of an infinitely deep viscous conducting liquid are considered. A mathematical procedure modifying the well-known method of many scales is proposed for constructing an asymptotically valid solution near the resonance. It is shown that the internal nonlinear resonant interaction results in effective energy transfer from long waves to shorter ones. An increase in the viscosity of the liquid diminishes the rate of energy transfer between resonantly interacting waves.     

Acoustic multipath arrivals in the horizontal plane due to approaching nonlinear internal waves

Simultaneous measurements of acoustic wave transmissions and a nonlinear internal wave packet approaching an along-shelf acoustic path during the Shallow Water 2006 experiment are reported. The incoming internal wave packet acts as a moving frontal layer reflecting (or refracting) sound in the horizontal plane. Received acoustic signals are filtered into acoustic normal mode arrivals. It is shown that a horizontal multipath interference is produced. This has previously been called a horizontal Lloyd's mirror. The interference between the direct path and the refracted path depends on the mode number and frequency of the acoustic signal. A mechanism for the multipath interference is shown. Preliminary modeling results of this dynamic interaction using vertical modes and horizontal parabolic equation models are in good agreement with the observed data.     

Energy fluctuations of high-frequency sound signals in a shallow water in the presence of nonlinear internal waves

The paper presents an analysis of energy fluctuations of high-frequency (2–4.5 kHz) sound signals propagating in a shallow water in the presence of nonlinear (soliton-like) internal waves (2006 Shallow Water experiment, US Atlantic shelf). Signals were received by three single hydrophones in different directions at distances of ～4, ～12, and ～5 km from the source. The angle between the first two acoustic tracks was ～15°. The third track was almost an extension of the first and was on the other side of the source. A relatively short (one to two solitons) nonlinear internal wave packet first moved approximately along the first two tracks and then along the third track. It is demonstrated that in the presence of solitons on the track in the frequency spectrum of energy fluctuations, there is an isolated frequency that depends, in particular, on the angle between the soliton front and the acoustic track. The experimental results agree well with the theory previously proposed by the authors, where the occurrence mechanism of fluctuations is explained using the ray approach.     

Relativistic nonlinear waves in a magnetised plasma

Relativistic propagation of a nonlinearly-coupled circularly-polarised electromagnetic wave and a Langmuir wave along the applied magnetic field is considered. Analytical solutions are given or are indicated for some special cases like purely transverse waves and purely longitudinal waves. In the presence of an applied magnetic field, the incident circularly polarised electromagnetic wave is found to propagate further into denser regions of the plasma — a result which is in accord with the so-called inverse Faraday effect. Finally, we shall consider general coupled transverse and longitudinal waves for which we give an approximate solution. We investigate whether this system of coupled waves exhibits any internal resonances and consequent energy exchange between them.     

Nonlinear internal waves in a one-dimensional channel

Summary By applying a perturbation technique to the Euler equation for a two-layer fluid we show that the evolution of the interface is described by a Korteweg-de Vries equation. This model is applied to describe the tide generation of internal waves in the Straits of Gibraltar.

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Riassunto Applicando tecniche perturbative all'equazione di Eulero per un fluido a due strati si può mostrare che l'evoluzione dell'interfaccia tra i due fluidi è descritta dall'equazione di Korteweg-de Vries. Questo modello è applicato per analizzare la generazione di onde interne di tipo solitonico nello Stretto di Gibilterra, considerando come forzante l'onda di marea.