[pt] Muitos problemas da engenharia geotécnica podem ser resolvidos pela superposição de soluções singulares fundamentais das equações diferenciais governantes. Nesta dissertação, é investigado o comportamento mecânico de uma fundação flexível, em termos de tensões e distribuições de recalque, considerando algumas soluções fundamentais básicas publicadas na literatura. Estas soluções referem-se ao campo de tensões e deslocamentos gerados em um semi-espaço linearmente elástico, homogêneo e isotrópico, por uma força vertical aplicada na superfície (Problema de Boussinesq (1885)), uma força horizontal aplicada na superfície (Problema de Cerruti (1882)), uma força aplicada dentro de um semi-espaço (Problema de Mindlin (1936)) ou dentro de uma camada finita (Problema de Burmister (1945)). No caso de depósitos de solo estratificados, são poucas as soluções disponíveis, em rela ção à ocorrência comum deste tipo de solo na natureza. Uma dessas soluções foi proposta por Hisada (1995), a qual permite que a resposta da aplicação de cargas dinâmicas ou estáticas em qualquer ponto de um semi-espaço estratificado, seja numericamente avaliada. No desenvolvimento matemático, a aplicação do teorema de Green permite fazer facilmente a superposição das soluções fundamentais, transformando as integrais de área em integrais de linha ao longo dos contornos que definem a geometria de uma única fundação ou de um grupo, formado por um número qualquer de fundações superficiais. Alguns exemplos apresentados neste trabalho discutem o potencial da aplicação dessa técnica em problemas da engenharia geotécnica, com ênfase especial para aqueles da engenharia de fundações. / [en] Many problems of geotechnical engineering can be solved by superposition of the fundamental singular solutions to the governing differential equations. In the dissertation, the mechanical behavior of a flexible footing is investigated, in terms of stress and settlement distributions, considered some basic fundamental solutions published in literature. These solutions refer to the stress and displacement fields in an isotropic, homogeneous, linearly elastic half-space generated by a vertical force applied on the surface (Boussinesq´s problem (1885)), a horizontal force on the surface (Cerruti´s problem (1882)), a force within the half- space (Mindlin (1936)) or within a stratum of finite thickness (Burmister´s problem (1945)). For the case of horizontal stratified soil deposits there are very few solutions available, in spite the common occurrence of this kind of soils in nature. One of such solutions has been proposed by Hisada (1995), which permits the response of a stratified half-space to dynamic or static forces, applied at any given point of the elastic medium, to be numerically computed. In the mathematical development, the application of the Green`s theorem allows the superposition of fundamental solutions to be more easily done by transforming area integrals into line integrals along the boundaries that define the geometry of either a single footing or a group formed by any number of shallow foundations. Some examples are herein presented in order to indicate the potential use of this approach to geotechnical engineering problems, in general, but with special emphasis to those of the foundation engineering. / [es] Muchos problemas de la Ingeniería geotécnica pueden ser resueltos por la superposición de soluciones singulares fundamentales de las ecuaciones diferenciales governantes. En esta disertación, se investiga el comportamiento mecánico de una fundación flexible, en términos de tensiones y distribuciones de recalco, considerando algunas soluciones fundamentales básicas publicadas en la literatura. Estas soluciones se refieren al campo de tensiones y al deslizamiento generados en un semi-espacio linealmente elástico, homogéneo e isotrópico, por una fuerza vertical aplicada en la superfície (Problema de Bousinesq (1885)), una fuerza horizontal aplicada en la superfície (Problema de Cerruti (1882)), una fuerza aplicada dentro de un semi-espacio (Problema de Mindlin (1936)) o dentro de una camada finita (Problema de Burmister (1945)). En el caso de depósitos de suelo estratificados, son pocas las soluciones disponibles, comparados con la gran existencia de este tipo de suelo en la naturaleza. Una de esas soluciones fue propuesta por Hisada (1995) y permite que la respuesta de la aplicación de cargas dinámicas o estáticas en cualquier punto de un semi-espacio estratificado, sea evaluada numéricamente. En el desarrollo matemático, la aplicación del teorema de Green permite llegar facilmente a la superposición de las soluciones fundamentales, transformando las integrales de área en integrales de línea a lo largo de los contornos que definen la geometría de una única fundación o de un grupo, formado por un número cualquier de fundaciones superficiales. Algunos ejemplos que se presentan en este trabajo discuten el potencial de la aplicación de esa técnica en problemas de la Ingeniería geotécnica, con énfasis especial para aquellos de la Ingeniería de fundaciones.

Les océans de la Terre sont stratifiés en densité par les gradients de température et de salinité.L'interaction des courants de marée avec la topographie du fond océanique entraîne donc le rayonnement des ondes de gravité interne dans l'intérieur de l'océan. Ces ondes sont appelées marées internes et leur dissipation due à le déferlement des ondes nonlinéaires joue un rôle important dans le mélange de l'océan abyssal, et donc dans la circulation océanique à la grande échelle.Dans ce contexte, nous étudions la génération des ondes internes par l’oscillation d’objet de différentes géométries simplifiées afin de modéliser le marée barotropique sur la topographie océanique et considérons les effets linéaires et nonlinéaires sur ces ondes résultant d’interactions avec l'objet et entre ces ondes.La contribution relativement nouvelle de cette thèse est l'étude des aspects de flux tridimensionnels qui étaient accessibles avec notre approche expérimentale, et sont généralement difficiles à étudier par modélisation numérique et analytique.Nous étudions d'abord la structure des ondes fundamentale et des harmoniques supérieur pour un sphéroïde oscillant, émettant des ondes divergentes. Les harmoniques supérieures sont générées par l'instabilité non linéaire à la surface de l'objet avec des effets nonlinéaires dans la zone d'intersection des faisceaux fundamentales. Ils peuvent se croiser et se concentrer, donc augmenter d'énergie, et devenir dominant sur les ondes fundamentales. On détermine les structures horizontales des ondes fundamentale et des harmoniques supérieures.Subséquemment, nous considérons les ondes générées par un tore oscillant, qui convergent vers un point focal. En dehors de cette région focale, les résultats expérimentaux et les prédictions théoriques sont en bon accord, mais dans la région focale, l'amplitude de l'onde est deux fois plus grande que près du tore, conduisant à une amplification locale nonlinéaire et à un déferlement des onde pour les grandes amplitudes d'oscillations. En conséquence, la propagation des ondes fundamentales se trouve entravée dans la région focale. L'onde stationnaire se forme alors que de nouvelles ondes sont générées et émises de cette région focale.Un tore plus grand a été testé sur la plate-forme Coriolis pour comparer la focalisation des ondes de gravité internes, inertie-gravité et des ondes inertielles dans un régime faiblement visqueux. En raison de la complexité de la zone focale, une seconde harmonique est observée même quand l'amplitude d'oscillation est faible. Le champ de vorticité verticale des ondes de gravité interne présente une structure dipolaire dans la zone focale, qui se transforme dans le cas tournant en une structure de vortex "Yin-Yang". La structure globale des faisceaux des ondes inertiels est proche de celle pour des ondes de gravité internes, bien q'elle est relativement plus intense. / The Earth's oceans are stratified in density by temperature and salinity gradients.The interaction of tidal currents with ocean bottom topography results therefore in the radiation of internal gravity waves into the ocean interior. These waves are called internal tides and their dissipation owing to nonlinear wave breaking plays an important role in the mixing of the abyssal ocean, and hence in the large-scale ocean circulation.In this context we investigate the generation of internal waves by oscillating objects of different idealized geometries as a model of barotropic flow over ocean topography, and consider linear as well as nonlinear effects on these waves resulting from interactions with the object and from wave--wave interactions.The relatively novel contribution of this thesis is the investigation of three-dimensional flow aspects that were accessible with our experimental approach, and are generally difficult to investigate by numerical and analytical modelling.First we investigate the wave structure of the first and higher harmonics for an oscillating spheroid, emitting diverging waves. Higher harmonics are generated by nonlinear instability at the surface of the object together with nonlinear effects in the zone of intersection of the primary beams. They may intersect and focus, therefore increase in energy, and become dominant over the first harmonic. The horizontal structures of both, first and higher harmonics are determined.We then consider waves generated by an oscillating torus, that are converging to a focal point. Outside this focal region experimental results and theoretical predictions are in good agreement, but in the focal region the wave amplitude is twice as large as it is close to the torus, leading to local nonlinear wave amplification and incipient wave breaking for large oscillation amplitudes. As a result, the propagation of the first harmonic waves is found to be hindered in the focal region. A standing pattern forms, while new waves are generated and emitted away from this focal region.A larger torus has been tested at the Coriolis platform to compare the focusing of internal gravity, inertia--gravity and inertial waves in a low viscous regime. Owing to the complexity of the focal region, a second harmonic is observed even at low oscillation amplitude. The vertical vorticity field of internal gravity waves exhibits a dipolar structure in the focal zone, which transforms in the rotating case into a ``Yin--Yang-shaped'' monopolar vortex structure. The overall structure of the inertial wave beams is close to that for internal gravity waves, though relatively more intense.

Ce travail de thèse cherche à caractériser, à l'aide d’expériences de laboratoire, le mouvement d'un fluide stratifié à la fois tournant et cisaillé. Dans ce contexte on a considéré trois problèmes qui sont issus des observations géophysiques et astrophysiques où ces écoulements sont très communs. En premier nous avons observé expérimentalement et modélisé l'évolution en temps d'un anticyclone en milieu stratifié tournant, dans le but de comprendre la longévité des meddies, des vortex qui se forment à la sortie de la Méditerranée et peuvent perdurer pendant des années. Nos anticyclones modèles montrent une relaxation visqueuse anormale en raison de l’interaction entre stratification et cisaillement. Nos résultats expérimentaux ont été confirmés à l'aide d'un modèle géostrophique et de simulations numériques qui montrent aussi le rôle majeur joué par les circulations secondaires. Dans un deuxième temps nous avons considéré la stabilité linéaire de l'écoulement de Couette Plan, un des plus simple des écoulement plans cisaillés. On montre que cet écoulement devient instable lorsque l'on ajoute de la stratification verticale. Nous remarquons aussi que la structure spatiale de l'instabilité ressemble à celle de certains jets océaniques que l'on observe près de l'équateur. Enfin nous avons caractérisé une instabilité non linéaire dite des zombie vortex qui a été récemment découverte et pourrait jouer un rôle fondamental dans la déstabilisation des disques d'accrétion, une étape fondamentale de la formation des planétaire. Nous avons construit le diagramme de stabilité de la ZVI dans l'espace de trois fréquences caractéristiques et quantifié la dissipation visqueuse. / The present PhD work comes with the scope of characterizing, analysing and modelling some laboratory flows in the simultaneous presence of rotation, stratification and shear. To this aim we address three specific questions inspired by geophysics and astrophysics where these three ambient features are commonly relevant. First we characterize and model the time evolution of a compact anticyclone in a rotating and stratified laboratory flow. We aim to understand the longevity of analogous vortices known as meddies which populate the Atlantic ocean at the exit of the Mediterranean sea. We observe that viscous relaxation happens in an unusual way because of the balance between rotation and stratification. The results are confirmed by a quasi-geostrophic model and numerical simulations which show the crucial role played by secondary circulations. Secondly we consider the linear stability of one of the simplest parallel shear flow, namely the plane Couette Flow, and show that it becomes unstable when adding a vertical stratification. Interestingly the unstable pattern reminds of deep oceanic jets observed close to the equator. The signature of this instability is observed in an ad-hoc experimental flow and interpreted with the support of direct numerical simulations. Finally we characterize the behaviour of a recently disclosed finite amplitude instability, namely the zombie vortex instability or ZVI. This instability appears when rotation, stratification and shear are of the same order and may may destabilize proto-planetary disks. We construct a stability diagram for ZVI in the space of the three ambient frequencies and analyse the effect of viscous dissipation.

abstract: With the increase in computing power and availability of data, there has never been a greater need to understand data and make decisions from it. Traditional statistical techniques may not be adequate to handle the size of today's data or the complexities of the information hidden within the data. Thus knowledge discovery by machine learning techniques is necessary if we want to better understand information from data. In this dissertation, we explore the topics of asymmetric loss and asymmetric data in machine learning and propose new algorithms as solutions to some of the problems in these topics. We also studied variable selection of matched data sets and proposed a solution when there is non-linearity in the matched data. The research is divided into three parts. The first part addresses the problem of asymmetric loss. A proposed asymmetric support vector machine (aSVM) is used to predict specific classes with high accuracy. aSVM was shown to produce higher precision than a regular SVM. The second part addresses asymmetric data sets where variables are only predictive for a subset of the predictor classes. Asymmetric Random Forest (ARF) was proposed to detect these kinds of variables. The third part explores variable selection for matched data sets. Matched Random Forest (MRF) was proposed to find variables that are able to distinguish case and control without the restrictions that exists in linear models. MRF detects variables that are able to distinguish case and control even in the presence of interaction and qualitative variables. / Dissertation/Thesis / Ph.D. Industrial Engineering 2013

A urbanização é uma das ações antrópicas que mais crescem no mundo atual. Por este motivo pesquisas ecológicas são realizadas nas cidades com o objetivo de reconhecer seus impactos, e as aves são utilizadas como uma das ferramentas para diagnóstico ambiental. Assim, o presente estudo avaliou o método de levantamento de aves por ponto fixo, método amplamente utilizado em estudos com aves em diversos ambientes. Foram analisados três pontos que podem influenciar a amostragem de aves através deste método: 1) o habitat onde o levantamento é realizado, observando a composição dos elementos urbanos existentes na cidade; 2) o intervalo de tempo adotado em cada ponto fixo para a coleta de dados; 3) os fatores potencialmente prejudiciais a observação de aves, tais como o ruído sonoro urbano e a presença de conversas causadas por pessoas curiosas. Com a área de estudo estratificada a partir da quantidade de cobertura arbórea existente nos bairros abrangidos, 90 unidades amostrais foram selecionadas. Nestes, foram quantificados os elementos urbanos presentes, a riqueza, o número de contato de aves, os ruídos sonoros e a presença de conversas. Os resultados demonstraram que a reunião de um número maior de espécies e contatos pode ser favorecida pelas áreas de cobertura arbórea, enquanto áreas construídas e pisos impermeáveis podem prejudicar o número de espécies, sendo o número de contato prejudicado apenas pelas áreas de pisos impermeáveis. O número de espécies observadas não foi significativamente diferente após nove minutos de coleta de dados, entretanto o número de contatos continuou crescendo, demonstrando haver recontagens de indivíduos após este intervalo. A riqueza de espécies foi significativamente diferente entre os dados coletados no período seco e no período chuvoso. Conforme houve a maior presença do ruído sonoro urbano menor foi o número de espécies e contatos obtidos nos pontos. A incidência de conversas ocasionadas por pessoas curiosas foi baixa não prejudicando as coletas de dados. Os resultados encontrados sugerem que: o levantamento de aves no meio urbano através do ponto fixo deve considerar a composição do ambiente, já que a riqueza e o número de contato podem variar de acordo com a presença dos diferentes elementos; sejam adotados intervalos de tempo por ponto não superiores a nove minutos; quando possível diferentes épocas do ano devem ser utilizadas para as coletas de dados, visto que podem ser encontradas diferenças entre as estações; sejam escolhidos locais e momentos para as coletas de dados com baixo ruído sonoro. Por fim, o método de ponto fixo foi considerado eficaz para amostragem de aves urbanas, desde que tais cuidados sejam considerados. / The urbanization is one of the anthropic activities with the highest growth rate in the world. Due to this reason, ecological research are conducted in the cities with the goal of recognizing its impacts, using birds as one of the tools to assess the environmental diagnosis. Therefore, the present study assessed the samples by point counts method, which is broadly used for bird census in many environments. Three issues that might affect the sampling of the birds by using this method were analyzed: 1) the habitat where the sampling is performed, observing the urban elements presented in the city; 2) the period of point count duration spent in each sample; 3) the potential factors which disturb the birds detectability, as urban noise and presence of curious citizens who can talk to the researcher in the point count. The research area was stratified from the amount of tree canopies in the selected suburbs, where 90 sample units were selected. In these units, the presence of urban elements, the richness, the number of birds contacts, the noise and the presence of conversations were quantified. The results showed that the number of species and contacts can be benefited from the tree canopy area, while build up areas and impermeable grounds may harm the number of species, although the contact number is harmed only by the impermeable grounds. The number of observed species did not differ significantly after nine minutes of sample period, however the number of contacts kept increasing, demonstrating a repeated counting birds after this interval. The species richness was significantly different between the samples collected in dry and wet seasons. As the urban noise increased, a lower number of species and birds contacts was acknowledged. The incidence of conversation performed by curious people was low, not being able to harm the sample collection. The results suggest that: the bird survey inside the cities by point counts should consider the composition of environment, since the richness and the number of birds contacts can vary according to the presence of different elements; the time of interval should not exceed nine minutes; when possible, different annual seasons should be used for sampling, since differences may be found among them; places and moments for the sampling should be chosen with a low noise. Finally, the point counts method was considered efficient for the sampling of urban birds, provided that such care are considered.

There are currently no effective methods for treating and preventing chronic pain. The aim of this thesis was to study prognostic factors for health care utilization, and the long-term outcomes of tailored behavioral medicine treatment for patients with musculoskeletal pain. Another aim was to increase knowledge about physical therapists’ assessment and analysis of patients’ pain conditions and to investigate the potential of subgrouping patients based on prognostic factors. Methods: In Study I, a prospective population-based cohort was followed over 21 years. Data from three measure points were analyzed: 1995 (n=2425), 2007 (n=1582) and 2016 (n=1184). Study II was a 10-year follow-up of randomized controlled trial (n=97), comparing tailored behavioral medicine treatment and exercise-based physical therapy. In Study III, a descriptive and explorative design was applied, using data from video-recordings of 12 physical therapists. In study IV, assignment to three subgroups based on the Örebro Musculoskeletal Pain Screening Questionnaire was validated against reference instruments, and the stability between two points of measurement was investigated in patients (n=40) who were seeking primary health care due to musculoskeletal pain. Results: Chronic pain, female gender and high age predict high health care utilization over 21 years, and a trajectory of stable high health care utilization over the entire period. The differences between groups in favor for tailored behavioral medicine treatment reported at post-treatment and after two years, were not maintained at the 10-year follow-up. A majority of the physical therapists assessed factors for poor prognosis. The analyses were mainly based on biomedical assessments and none of the physical therapists included behavioral factors. Subgroup assignment according to the Örebro Musculoskeletal Pain Screening Questionnaire appears to be valid and stable over time. Conclusion: Prognostic factors such as chronic pain and female gender need to be considered when allocating health care resources and planning treatment to improve long-term outcomes. The treatment should also be tailored based on individual functional behavioral analyses of key behaviors and on patient´s biomedical and psychosocial condition, including strategies for maintenance of behavioral changes. Evidence-based methods for integrating behavioral medicine treatment into physical therapy need to be further evaluated and improved.

La thèse concerne la modélisation de deux écoulements issus de problèmes géophysiques. Un premier problème a trait à la présence de tourbillons longitudinaux apparaissant dans les cours d'eau. L'origine de ces structures reste indéterminée. On aborde ce problème par une simulation numérique d'un écoulement turbulent cisaillé dans un canal ouvert utilisant un code pseudo-spectral. On tente de voir si un cisaillement de surface même faible est capable ou non de comprendre ces observations. Le second problème est lié à la resuspension des sédiments sur les bords d'une mer ou d'un lac par des ondes internes. Ces ondes existent à cause de la stratification en densité de la colonne d'eau. Lorsque elles s'approchent de la côte, elles se déstructurent générant un cisaillement sur le fond capable de resuspendre du sédiment. On a envisagé à nouveau cette étude par le biais de la simulation numérique directe. On examine comment l'onde interne se brise sur les bords en fonction de (a) la nature de la stratification, (b) la forme de la topographie du fond et (c) l'amplitude des ondes. On calcule dans chaque cas le cisaillement sur le fond. On en déduit le flux et le transport de sédiment dans toute la colonne d'eau. / This thesis studies the modeling of two problems that take their origin from a geophysical context. A first problem is related to the presence of longitudinal vortices which have been measured in rivers. The origin of these structures remains unknown. We address this problem by numerical simulations of a sheared turbulent flows in an open channel using a pseudo-spectral code. We try to determine if the presence of an imposed shear at the surface coupled with a pressure gradient is capable or not to explain these observations. The second study focuses on the sediment resuspension on shores of seas or lakes by the action of internal waves. These waves exist because of density stratification of the water column. When waves approach the shore, their patterns evolve generating a shear on the bottom capable to resuspend sediment. By direct numerical simulations, we analyse how internal waves breaking changes according to (a) stratification, (b) bottom topography and (c) wave amplitude. We compute for each case the shear exerted on the bottom, the sediment flux and transport throughout the water column.

A numerical model has been developed for the study of stratified tidal circulation in Indian Arm - a representative inlet on the southern coast of British Columbia. Equations for horizontal velocity, salt conservation, continuity, density (calculated as a linear function of salinity), and the hydrostatic approximation govern the dynamics. All equations have been laterally integrated under the assumption of negligible cross-inlet variability. The model is time dependent and includes nonlinear advective terms, horizontal and vertical turbulent diffusion of salt and momentum, and variations in width and depth. Provisions for surface wind stress and a flux of fresh water are also included, although neither was utilized in this study. An explicit finite difference scheme centred in both time and space was used to solve for the horizontal and vertical velocity components, salinity, and surface elevation on a staggered rectangular grid. A backward Euler scheme was used to suppress the computational mode. Tests using a semi-implicit scheme to solve the finite difference equations over realistic topography led to numerical instabilities at modest values of the time step - in spite of the unconditional stability criteria - suggesting that linear stability analysis may give misleading results for strongly nonlinear systems. Surface elevations calculated from tidal harmonic analysis and salinity timeseries derived from linearly interpolated CTD casts were prescribed at the open boundary. Initial and boundary conditions based on observations in Burrard Inlet and Indian Arm during the winter of 1974-75 were used to study the inlet's response to tidal forcing and to simulate the deep-water renewal that occurred during this period. Coefficients for the horizontal turbulent diffusion of momentum and salt were set equal to 10⁶ cm² s⁻¹. Reducing this value by a factor of two was found to have little impact on the solution. A further reduction to 10³ cm² s⁻¹ led to numerical instabilities under conditions of dense water inflow. The side friction term in the momentum balance was tuned to match calculated and observed dissipation rates in Burrard Inlet; leading to good agreement between the observed and calculated barotropic tide. Contour plots of tidal amplitudes and phases for model currents and salinities revealed a standing wave pattern for the K₁ and M₂ internal tides in Indian Arm; thus allowing for the possibility of resonance. A comparison of model results with vertical amplitude and phase profiles from harmonic analysis of Cyclesonde current meter timeseries at two locations in Indian Arm was consistent with this result. A least-squares fit was made of the vertical modal structure in the model to the complex tidal amplitudes. This led to calculations of the kinetic energy contained in each of the modes along the model inlet for the M₂ and K₁ constituents. Most of the energy was found to be contained in the barotropic and first baroclinic modes, with the latter dominating in the deep basin, and the former dominating near the sill. Second mode energy was significant for the K₁ constituent at some locations in Indian Arm. There are clear indications in the model of barotropic tidal energy being radiated into the inlet basin via the internal tide. Simulations of the influx of dense water into Indian Arm yielded exchange rates that are consistent with observed values and suggest the possibility of fine-tuning the model coefficients to allow prediction of future overturning events. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

It is often desirable to study simulated turbulent flows at steady state even if the flow has no inherent source of turbulence kinetic energy. Doing so requires a numerical forcing scheme and various methods have been studied extensively for turbulence that is isotropic and homogeneous in three dimensions. A review of these existing schemes is used to form a framework for more general forcing methods. In this framework, the problem of developing a forcing scheme in Fourier space is abstracted into the two problems of (1) prescribing the spectrum of the input power and (2) specifying a force that has the desired characteristics and that adds energy to the flow with the correct spectrum. The framework is used to construct three forcing schemes for horizontally homogeneous and isotropic, vertically stratified turbulence. These schemes are implemented in large-eddy simulations and their characteristics analyzed. Which method is “best” depends on the purpose of the simulations, but the framework for specifying forcing schemes enables a systematic approach for identifying a method appropriate for a particular application.

Internal waves are generated in stratified fluids, like the ocean, where density increases with depth. Tides are one of the major generation mechanisms of internal waves. As the tides move water back and forth over underwater topography, internal waves can be generated. Topography slope and amplitude are major factors in the behavior of the generated internal wave field. In order to further understand the effects topographic shape plays, the effect of asymmetry on internal waves is investigated. This research investigates internal waves generated by dual-ridge topographies. Four cases of symmetric topographies, T, M, W, and W2, with three different peak spacings are compared to their singular ridge counter parts at three oscillation frequencies, ω = 0.6N, ω = 0.75N, and ω = 0.9N. Both subcritical and supercritical symmetric ridges were investigated. Experiments were also performed for subcritical, asymmetric dual ridges at the middle oscillation frequency. The internal wave fields were captured with synthetic schlieren and analyzed with the Hilbert transform and sum of kinetic energy in wavenumber space. It is found that for wave fields from substantially separated ridges, mixing and wave interference occurs that decreases total kinetic energy of the system.