Rear Slope Research Articles

Modelling secondary instability of co-current a thin gas-sheared film

Recent experimental works found the existence of two types of waves on the surface of gas-sheared thin films. Slower short-living ‘secondary waves’ appear due to the instability of the rear slopes of faster long-living ‘primary waves’. In this paper, modelling of spatiotemporal evolution of liquid film in such kind of flows is performed using relatively simple theoretical models. The modelling results are directly compared with the experimental data. It is found that the phenomenon of secondary waves generation at the rear slopes of the primary waves is reproduced by the model. This allows us to reduce the number of hypotheses which explain the mechanism responsible for such instability. Recommendations for future theoretical investigations are proposed.

Research of Usability Test on Disabled Welfare Vehicle for Guardians and Passengers of Disabled People

본 연구는 17명의 장애인 보호자 및 장애인 탑승자를 대상으로 레이(Ray)의 실험 복지차량 사용 시 문제점을 예측하여 개선하기 위한 연구로써, 실험 복지차량 이용 시 18개의 주요 사용성 평가항목을 통해 실제 소비자인 장애를 고려하기 위해 장애인 보호자 및 장애인 탑승자 관점에 기초한 인간공학적 설계를 위한 기준을 제시하고자 한다. 본 연구 결과를 보면 다음과 같다. 첫째, 레이(Ray) 실험 복지차량의 만족도와 자기 소유차량 만족도 비교에서 장애인보호자군의 자가 소유차량만족도가 3.20점보다 실험복지차량 만족도가 3.88점으로 더 높았으며, 또한 장애 탑승자군에서 장애인 자기소유차량만족도가 3.00점보다 실험 복지차량 만족도가 3.31점으로 더 높은 것으로 나타났다. 이는 장애인 보호자 및 장애인 탑승자를 위한 실험복지차량이 이들의 장애특성에 대한 불편사항을 해소한 복지차량개발이라고 볼 수 있다. 둘째, 장애인 보호자 및 장애인탑승자에게 적합한 복지차량을 개발하려면 조수석의 경우 수전 동회전업시트가 필요하거나 아니면 차량 후석 슬로프를 통해 장애인을 수․전동 휠체어 탄 상태로 휠체어잠금장치, 휠체어 안전벨트, 휠체어 안전 손잡이, 하이루프 등이 차량 설계부터 장애를 고려하는 것이 필요하였다. 셋째, 후석 슬로프의 경우 차량의 전고높이(하이루프)와 수동 및 전동휠체어의 전폭을 고려하여야 하며, 휠체어 잠금장치의 경우 사용자의 휠체어를 안전하게 고정하여 견인할 수 있는 전동장치가 고려되어야 할 것이다. 또한 전동슬라이드업시트와 후석 슬로프는 휠체어 장애인과 고령자 등 대상자에 맞게 선택할 수 있을 것이다.This research is to anticipate problems of Ray’s disabled experiment welfare vehicles for 17 drivers with disability, improving them. In addition it is to provide the criteria for ergonomic design based on perspective of drivers with disability by investigating 18 items of usability test. The results are as follows. First, satisfaction degree of Ray, disabled experiment welfare vehicle, was shown to be 3.88 which is higher than normal vehicles whose degree is 3.20. This showed that the disabled experiment welfare vehicle is the one with much improvement. Second, so as to develop a welfare vehicle it needs to take into account some factors including rear slope, wheelchair locker, seat belt, safety grip, and high roof. Third, in case of rear slope, high roof and width of manual or automatic wheel chairs should be considered and motor-operated device should also be taken into account for safety lockers, which make a wheelchair firmly fixed. Moreover, motor-operated seat and rear slope can be chosen for either of the disabled and the elderly.

Numerical simulation of the propagation of ship-induced Riemann waves of depression into the Venice Lagoon

Large in situ measured ship-induced depression waves (Bernoulli wakes) in the Malamocco-Marghera industrial channel of the Venice Lagoon are interpreted as long-living strongly nonlinear Riemann (simple) waves of depression. The properties of these depressions are numerically replicated using nonlinear shallow water theory and the CLAWPACK software. The further behaviour of measured depressions is analysed by means of replicating the vessel-induced disturbances with the propagation of initially smooth free waves. It is demonstrated that vessel-driven depressions of substantial height (> 0.3 m) often propagate for more than 1 km from the navigation channel into areas of the lagoon of approximately 2 m water depth. As a depression wave propagates into the lagoon, its front slope becomes gradually less steep, but the rear slope preserves an extremely steep bore-like appearance and the amplitude becomes almost independent of the initial properties of the disturbance. Analysis suggests that even modest ships in terms of their size, sailing speed, and blocking coefficient may generate deep depressions that travel as compact and steep entities resembling asymmetric solitary waves over substantial distances into shallow water adjacent to navigation channels. Their impact may substantially increase the environmental impact of ship wakes in this and similar water bodies.

CFD Simulation of a Simplified Automotive Model for Various Rear Slant Angle

In the early design phase of automotive sector, the flow field around the vehicle is important in decision making on design changes. It would consume a lot of money and time for multiple prototypes development if adopt traditional testing method which is wind tunnel test. Thus, CFD Simulation plays an important role here. In this paper, a CFD simulation study was conducted on a simplified automotive model called Davis model with constant velocity of 40 m/s. Modification of rear slant angle bring significant effects on the wakes produced which also affect the drag performance. Many configurations of body designs can be produced by a single rear slant angle. So that, fixed rear slope and fixed rear height configuration have been chosen for investigations for various rear slant angles. In this paper, the flow of Davis model especially on the rear slanted surface is discussed. Pressure coefficient contour, pressure coefficient plot and vorticity structures are presented. This work shows that the drag coefficient value vary between fixed rear slope model and fixed rear height model even for the same rear slant angle under a range of yaw angle.

Special Issue on the 2011 off the Pacific Coast of Tohoku Earthquake Tsunami

An unprecedented M9.0 earthquake occurring at 14:46 local time on March 11, 2011, off of northeast Japan’s Pacific Ocean generated a huge tsunami which had a run-up of over 40 m at the highest point and nearly 20,000 lives were lost. The tsunami demonstrated the need to drastically readdress current tsunami countermeasures. “Guidebook for Tsunami Preparedness in Local Hazard Mitigation Planning” published prior to the March 11 tsunami had already estimated, as one of the cases of tsunami assumptions, that the tsunami could be generated by the largest earthquake near off the Sanriku Coast predicted by the recent seismology. The seismotectonics had predicted that off the Sanriku Coast consisted of three independent blocks, which could conceivably cause an M8.6 earthquake at the largest. However, three blocks were not independent and they moved continuously to yield an earthquake of M9.0. The Guidebook had recommended a combination of three approaches for handling such a tsunami; Construction of defense structures, Tsunami-resilient town development, and Disaster prevention systems – defense structures were not expected to completely prevent every tsunami but only reduce its effect. Caissons forming part of Kamaishi Port’s tsunami breakwaters and registered in Guinness World Records, were overturned but reduced the tsunami height from 14 m outside the port to 8 m inside. Many coastal dikes were also destroyed, even though three surfaces – fore slope, top slope, and rear slope – had been protected using concrete and other means. Such phenomena pinpoint the importance of toe protection against erosion. Since 2004, tsunami inundation hazard maps have been distributed to communities in Japan as an aid to public education and as part of the country’s nationwide disaster prevention system. Unexpectedly, these maps had a negative effect in many places where residents living outside inundation areas mentioned on the hazard maps believed they were safe under all condition. Many did not in fact keep track of the actual tsunami rising in front of their very eyes and not evacuate, thus losing their lives. The tsunami hitting the coast of the Fukushima Prefecture had a run-up height almost double that designed in defense plans. The Fukushima No.1 Nuclear Power Plants of the Tokyo Electric Power Company (TEPCO) located on ground 4.8 m above sea level were immerged and a concurrent electric system failure led to total plant shutdown. The Fukushima nuclear disaster itself has become well known worldwide. The effects of the tsunami, however, are less so, despite damage such as fires, railroad destruction and drifting ships caused by the tsunami. With the nuclear incident overshadowing such effects, we are concerned that these results might be overlooked. To better prepare against potential future tsunami disasters, we must understand clearly what sort and how such diverse damage has been generated by the 2011 tsunami. This special issue focuses on the various types of tsunami-induced damages, emphasizing the valuable data and modeling obtained from field investigations in the tsunami-devastated areas. It will be more than worth publication if this special issue contributes in whatever way to furthering tsunami disaster research. Finally, we extend our sincere thanks to all of the contributors and reviewers involved with these articles. (written by Nobuo Shuto and Tomoyuki Takahashi)

Role of ion beam induced solid flow in surface patterning of Si (1 0 0) using Ar ion beam irradiation

Variation in the thicknesses of amorphous layer during ripple evolution on Si surface by 50keV Ar+ ion bombardment has been studied as a function of ion fluence. Atomic force microscopy demonstrates the formation of ripples with ion beam irradiation. Cross-sectional transmission electron microscopy (XTEM) study shows that the ripple microstructure consists of thicker amorphous layer on the front slope, whereas comparatively thinner amorphous layer on the rear slope. This observation of a thicker amorphous layer on the front of ripples as compared to their rear sides is quite common. However, the fluence dependence study on the thickness variation is not reported so far. A detailed analysis of the stationary cross-sectional area of amorphous layer shows that the incompressible solid flow plays a dynamical role in the thickness variation of amorphous layer during ion beam induced patterning of the solid surfaces. The variation in thickness of amorphous layer is also verified by Micro-Raman investigations. Rutherford back scattering experiment was performed to confirm the argon (Ar) incorporation in the near surface region of silicon samples.

STABILITY OF INTERLOCKING ARMOUR UNITS ON A BREAKWATER CREST

The hydraulic stability of single layer, interlocking armour units on low crested and submerged breakwaters was investigated in 2D hydraulic model tests. Displacements of armour units and rocking were monitored and have been applied as indicators for the armour layer stability on the crest, front and rear slope. The effect of freeboard, packing density and wave steepness on the armour layer stability have been investigated. The stability of interlocking concrete armour units on low crested and submerged structures is qualitatively different from rock armour. About 40% to 50% larger armour units are required on the seaward slope and crest of low crested structures (as compared to conventional high crested breakwaters). About 35% larger armour units are required on the rear slope. Larger armour units are not required on submerged breakwaters if the water depth on the crest exceeds 50% of design wave height.

An experimental observation of a solitary wave impingement, run-up and overtopping on a seawall

A sequence of laboratory experiments using solitary waves was performed to model the effect of leading form of three types of tsunamis (a bore, an impinging wave and an overtopping wave) on a seawall on a sloping beach. The wave evolution process, impinging pressure along the seawall surface, total overtopping discharge behind the seawall and the maximum run-up height on the rear slope were measured and compared. Laboratory data were employed to re-examine relevant empirical formulae in the literature. The effect of the presence of the seawall in reducing maximum run-up height using the present setup was briefly discussed. The present data can be used for calibrating numerical and mathematical models.

Core made of geotextile sand containers for rubble mound breakwaters and seawalls: Effect on armour stability and hydraulic performance

A systematic armour stability and the hydraulic performance, including wave reflection, wave transmission, experimental study in the twin-wave flumes of Leichtweiss-Institute (LWI) is performed on a geocore breakwater and a conventional rubble mound breakwater in order to comparatively determine the wave run-up and wave overtopping. The geocore breakwater consists of a core made of sand-filled geotextile containers (GSC) covered by an armour made of rock. The geocore is more than an order of magnitude less permeable than the quarry run core of a conventional breakwater. As expected, the core permeability substantially affects the armour stability on the seaside slope, the wave transmission and the wave overtopping performance. Surprisingly, however, wave reflection and hydraulic stability of the rear slope are less affected. Formulae for the armour stability and hydraulic performance of the geocore breakwater are proposed, including wave reflection, transmission, run-up and overtopping.

Advanced Experimenting on Wave Interaction with Low-Crested Breakwaters

Two-dimensional scaled physical tests were carried out in the Hydraulics Laboratory wave tank of the University of Porto in order to more accurately understand the effect of submerged breakwaters on wave-induced velocities and pressure fields. The slow-varying pressures acting normally on the slope were simultaneously measured with the flow-induced velocities at the surface, and an analysis of their distribution along the whole structure-on the front slopes, rear slopes and along the crown-was done as this information is considered potentially useful for hydraulic design. Different points on the breakwater model were considered and diverse wave conditions were tested, allowing study of the dependency of pressures/velocities on model characteristics. Maximum, minimum and mean phase-averaged values of these variables were calculated, allowing the detection of potentially sensitive areas in terms of pressure and velocities at the surface. A spectral analysis of these physical variables was also perfomed in order to check the results obtained with the regular wave tests.

Slope stability modelling with SINMAP in a settlement area of the Swabian Alb

During the last decades, damages to houses caused by landslides have been consistently occurring in a residential area in Oschingen/Germany. The residential area is located in ductile Callovian clays (Jurassic). Furthermore, in the back slope of the research area, a large Pleistocene slide mass negatively influences the slope stability. Through an integrative approach, the maximum data available for the study area was compiled in order to create a susceptibility map for landslide hazard. Detailed geomorphological field survey provided a valuable base for the assessment of slope stability using Stability Index Mapping (SINMAP). In the framework of long-term studies, consolidated results concerning mass movements and climatic-driven Pleistocene slope evolution, as well as recent slope dynamics, could be gained. These outcomes are compared to the results provided by slope stability modelling with SINMAP. The calculations outline some parameters responsible for higher risks. In general, the interaction of topography, water balances and substrate at the Schonberger Kapf can be designated to cause instability in wide areas. Hydrological parameters are essential for destabilisation of slope and they cause at least temporally destabilisation along channel structures, which presumably are influenced by seasonally increased spring discharge and a reduction in the underground shear strength. The exceptional dimension of the rotational block in connection with the specific slope hydrologic conditions and the intensive anthropogenic impact in the rear slope of the building area Auchtert in Oschingen has to be termed very problematic on the basis of the studies carried out.

Photoluminescence from Si: Effect of ripple microstructures induced by argon ion irradiation

We performed photoluminescence (PL) measurements on Si surface irradiated with 60 keV Ar + at a fixed ion fluence of 10 18 ions/cm 2 for two angles of ion incidence, namely 0° (with respect to surface normal of the sample) and 60°. Periodically modulated ripple morphology is observed for a 60° angle of ion incidence. The ripple microstructure consists of amorphous structure on the rear slope and a comparatively thicker amorphous layer with Ar bubbles on the front slope, whereas a uniformly thick amorphous layer with relatively large bubbles is created under normal bombardment. Room temperature PL of the rippled Si shows a visible band with a peak at ~ 700 nm and a strong infrared (IR) band having a peak at ~ 1000 nm. However, the visible PL was very weak and no IR emission was observed for normally irradiated Si.

Spatial distribution of Ar on the Ar-ion-induced rippled surface of Si

We have measured spatial distribution of Ar atoms on the rippled surface generated on Si undergoing 60 keV Ar bombardment at a 60° angle of ion incidence. Elemental mapping and line scans using energy dispersive x-ray spectrometry attached in a scanning electron microscope confirmed that subsequent to the interpeak shadowing of incident ion flux, most of the argon atoms are incorporated around the middle part of the front slope of ripple facing the ion beam as compared to the rear slope. The spatial extension of the argon rich phase amounts about half of the ripple wavelength. The experimentally observed compositional heterogeneity between the two faces of the ripples agrees reasonably good to the well-known Monte Carlo ion simulator TRIM based theoretical calculations.

Structural investigation of keV Ar-ion-induced surface ripples in Si by cross-sectional transmission electron microscopy

Using cross-section transmission electron microscopy (XTEM) we have studied the surface and subsurface structure of individual ripples having submicron scale wavelength and nanometer scale amplitude, generated by obliquely incident (50--120 keV) Ar ion bombardment of Si. The XTEM results reveal that the front slopes of ion-induced ripples have amorphous layers containing bubbles with sizes ranging from about 3 to 15 nm facing the ion beam direction. A thinner amorphous layer without bubbles, on the other hand, persists on the rear slope of ripples. We also observe an irregular interface between $a\ensuremath{-}\mathrm{Si}$ and $c\ensuremath{-}\mathrm{Si},$ which is due to the direct impact amorphization mechanism prevalent near the end-of-range during heavy ion irradiation.

Design of tsunami-protective embankments with a complex front surface

The most effective and reliable means of protecting coasts from tsunamis is to construct special engineering structures. Shore embankments and walls are constructed for protection against tsunamis. The height of the embarkrnents is determined on the basis of the maximum overwash of the design tsunami with a frequency of once in 100 years. The embankments are calculated for a hydrostatic load with reflection of the tsunami; the seismic stability of the slopes is checked without fail. The necessary stability of the embankments is provided by making the slope more gentle or by reinforcing the body of the embankment. The steepness of the slopes is usually 1:3. The front slope of the embankment is reinforced with stone or concrete slabs. For protection from possible erosion in the case of brief overflow of water, the crest and rear slope of the embankment are faced with stone, concrete, or polymer materials [1, 2]. When constructing tsunami-protective embankments, it is expedient to construct the fi'ont (i.e., seaward) side of the embankment in the form of a system of terraces. This provides better dissipation of the energy of the tsunami due to increased friction losses by the water flow forming during run-up of the wave on the embankment. The size of these terraces can be determined form the following considerations. It is known that when a fluid flow with velocity v strikes a plane at angle/3, a jet stream, the velocity of which (u) is severalfold greater than the velocity of the incident flow, "beats" along the plane. The velocity of this jet stream is determined by the relation [5, 6]

Interpretación geodinámica de la vertiente centro-occidental surpirenaica (cuencas de Jaca-Tremp)

Durante el Eoceno y Oligoceno en las cuencas de Jaca y Tremp se produce un sistema de cabalgamientos imbricados ("piggy-back sequence") contemporáneos con el relleno sedimentario de las mismas. Toda la sedimentación eocena está condicionada por las rampas anticlinales formadas, originándose una facies de talud sintectónica en los flancos suaves o traseros de cada cabalgamiento (calizas y margas de Burgui), Entre estas rampas anticlinales se crean una serie de surcos en los cuales se depositan diferentes sistemas turbidíticos, En estos sistema se inercalan conjuntos de megacapas carbonatadas que se canalizan a lo largo de dichos surcos. Estos conjuntos migran claramente hacia el sur, observándose también, en las megacapas que los componen, un desplazamiento oeste a este. Esta disposición es un reflejo del movimiento de los cabalgamientos, al emplazarse antes en el oeste de la cuenca de Jaca para irse desplazando en el tiempo hacia el sur y este de la misma. El material carbonatado que forma las megacapas procedería de una plataforma eocena septentrional, localizada en el borde activo de la cuenca, así como de la erosión de las rampas anticlinales creadas simultáneamente. El sistema cabalgante imbricado, que afecta a la cobertera, está ligado y superpuesto a un gran sistema "dúplex" en el basamento. Este último buza hacia el noroeste, es de tipo "hinterland dipping" y su raíz o "branch line" se localiza en una falla intrapaleozoica en la actual zona axial. El sistema tectónico formado se origina por una compresión NNW-SSE, debida, posiblemente, al movimiento dextrógiro relativo entre las placas Europea e Ibérica.

Action of a Variable Stress at the Surface of Water Waves

A boundary-layer argument shows that, paradoxically, a variable tangential stress which is greatest at the wave crests and least in the wave troughs produces a thickening of the boundary layer on the rear slopes of the waves and a thinning on the forward slopes. In deep water, a variable tangential stress ⊥ is precisely equivalent to a normal stress i⊥ in quadrature with the tangential stress. The corresponding rate of growth of the waves is calculated.

Height, structure and displacements of noctilucent clouds

Observations of noctilucent clouds have been carried out during the summer of 1958 at Torsta (63.3° N; 14.6° E) in Central Sweden as part of the IGY working program of the Institute of Meteorology, University of Stockholm. Simultaneous pairs of cloud photographs have been taken with accurate phototheodolite cameras from the end-points of a geodetically determined base-line of length 51.5 km. The picture pairs were subsequently analyzed in stereo instruments (autographs) by which Cartesian space coordinates were obtained for various points in the cloud system. These coordinates, duly corrected for atmospheric refraction, were used for determination of the height of the individual features. Through the stereoscopic effect, measurements could be made on diffuse parts of the cloud system as well as on marked details. Additional information about movements of the cloud system was obtained from a time-lapse film in Kodachrome. The results were plotted and analyzed by conventional methods and maps of the cloud topography at consecutive time intervals could be prepared. In addition to these maps, vertical cross-sections through the cloud system were made as well as detailed studies of particularly interesting cloud features. This paper gives a presentation and interpretation of the results obtained so far and a brief description of the photogrammetric technique applied. The results presented below were obtained during a very bright cloud display with good visibility conditions on August 10–11th, 1958. Thirty pairs of pictures were taken of various parts of the cloud system, which covered the entire northern horizon. Eight of these have been analyzed so far. The results can be summarized as follows. The cloud system moved in a direction north-east to south-west with velocities of the order of 50 to 100 m/s. It consisted of a continuous diffuse layer interchanging with regions of sharply defined features such as systems of parallel billows and bands, blobs and other smaller-scale irregularities of various shapes. The measured heights varied between 81.5 and 85.5 km. The long parallel bands were identified as a system of waves with wavelengths of the order of 50 km and amplitudes up to 4 km which propagated in a direction nearly opposite to that of the cloud system with absolute velocities of the order of 10 to 20 m/s. The wave crests were oriented nearly perpendicular to the main air flow and were continuous over distances of hundreds of kilometers and exhibited local refraction effects. The smaller billows had wavelengths of the order of 5–10 km and amplitudes about 0.5–1.0 km; they moved with the cloud sysem. The billows showed no preferred orientation and were observed to pass through the crests of the longer waves. It is indicated by the analysis the regular changes in the brightness of these clouds are due to changes of the optical thickness of the cloud layer, particularly at the rear slope of the long waves. The measured details had well-defined lower boundarie. DOI: 10.1111/j.2153-3490.1962.tb00115.x

Height, structure and displacements of noctilucent clouds

Observations of noctilucent clouds have been carried out during the summer of 1968 at Torsta (63.3° N ; 14.6° E) in Central Sweden as part of the IGY working program of the Institute of Meteorology, University of Stockholm. Simultaneous pairs of cloud photographs have been taken with accurate phototheodolite cameras from the endpoints of a geodetically determined base-line of length 51.6 km. The picture pairs were subsequently analyzed in stereo instruments (autograph) by which Cartesian space coordinates were obtained for various points in the cloud system. These coordinates, duly corrected for atmospheric refraction, were used for determination of the height of the individual features. Through the stereoscopic effect, measurements could be made on diffuse parts of the cloud system as well as on marked details. Additional information about movements of the cloud system was obtained from a time-lapse film in Kodachrome. The results were plotted and analyzed by conventional methods and maps of the cloud topography at consecutive time intervals could be prepared. In addition to these maps, vertical cross-sections through the cloud system were made as well as detailed studies of particularly interesting cloud features. This paper gives a presentation and interpretation of the results obtained so far and a brief description of the photogrammetric technique applied.
 The results presented below were obtained during a very bright cloud display with good visibility conditions on August 10-11th, 1958. Thirty pairs of pictures were taken of various parts of the cloud system, which covered the entire northern horizon. Eight of these have been analyzed 80 far. The results can be summarized as follows. The cloud system moved in a direction north-east to south-west with velocities of the order of 50 to 100 m/s. It consisted of a continuous diffuse layer interchanging with regions of sharply defined features such as systems of parallel billows and bands, blobs and other smaller-scale irregularities of various shapes. The measured heights varied between 81.6 and 86.6 km. The long parallel bands were identified as a system of waves with wavelengths of the order of 60 km and amplitudes up to 4 km which propagated in a direction nearly opposite to that of the cloud system with absolute velocities of the order of 10 to 20 m/s. The wave crests were oriented nearly perpendicular to the main air flow and were continuous over distances of hundreds of kilometers and exhibited local refraction effects. The smaller billows had wavelengths of the order of 6-10 km and amplitudes about 0.5-1.0 km; they moved with the cloud sysem. The billows showed no preferred orientation and were observed to pass through the crests of the longer waves. It is indicated by the analysis the regular changes in the brightness of these clouds are due to changes of the optical thickness of the cloud layer, particularly at the rear slope of the long waves. The measured details had well-defined lower boundaries.

On the formation of water waves by wind (second paper)

In a previous paper I investigated the problem of the formation of waves on deep water by wind, and found that the available data were consistent with the hypothesis that the growth of the waves is due principally to a systematic difference between the pressures of the air on the front and rear slopes. Lamb had already discussed the maintenance of waves against viscosity by an approximate method, but without obtaining numerical results. Being under the incorrect impression that Lamb’s approximation would not hold for the short waves I was chiefly considering, I proceeded on more elaborate lines. It now appears, however, that Lamb’s method is not only applicable to the problem of waves on deep water, but is readily extended to cover the case when the water is comparatively shallow, and to allow for surface tension. The fundamental approximations are first, the usual one that squares of the displacements from the steady state can be neglected, and second, that viscosity modifies the motion of the water to only a small extent. The motion of the water can then, to a first approximation, be considered as irrotational. With the previous notation, let ζ be the elevation of the free surface x, y, z the position co-ordinates, t the time, U the undisturbed velocity of the water, h the depth, and φ the velocity potential. Also let σ, p, q , and ϑ denote respectively ∂/∂ t , ∂/∂ x , ∂/∂ y , and ∂/∂ z , and write p 2 + q 2 = - r 2 .