Free Water Surface Oscillations in a Closed Rectangular Basin with Internal Barriers
|
|
- Sharleen Bennett
- 5 years ago
- Views:
Transcription
1 Scientia Iranica, Vol. 15, No. 3, pp 315{3 c Sharif University of Technology, June 008 Research Note Free Water Surface Oscillations in a Closed Rectangular Basin with Internal Barriers A.R. Kairi-Samani and B. Ataie-Ashtiani 1 Thee enclosed asin has certain natural frequencies of seiche, depending on the geometry of the water oundaries and the athymetry of water depths. Therefore, the variation in the water surface at a point ecomes irregular, as caused y the comination of several natural frequencies, which may e considered as the superposition of sinusoidal frequency components of dierent amplitude. This paper is mainly concerned with the motion of an incompressile irrotational uid in a closed rectangular asin with internal impervious arriers. An analytical solution is presented for predicting the characteristic of generated waves in these types of asin. The equations of free water surface oscillations and its oundary conditions are reduced to a system of linear equations, which is solved y applying the small amplitude water wave theory. The ow potential, wave amplitude, ow patterns and the natural period of waves generated in the asin with impervious internal arriers are found, ased on the asin geometry. It is shown that the natural period of the asin is strongly dependent on the location of the arriers and the size of the arrier opening. INTRODUCTION Standing waves are progressive waves reected ack on themselves and appear as alternating etween troughs and crests at a xed position. They occur in ocean asins, partly enclosed ays and seas and in estuaries. When a standing wave occurs, the surface alternately rises at one end and falls at the other end of the container (Figure 1). If dierent-sized containers are treated the same way, the period of oscillation increases as the length or depth of the container increase. Standing waves in natural asins are called seiches. Seiches can e generated y tectonic movements or winds. The free oscillations are characteristic of the system and are independent of the exciting force, except for the initial magnitude. The restoring force is provided y gravity, which returns the uid surface to its horizontal equilirium position after it is displaced y wind or pressure variations. In lakes and harours of small to medium size, the characteristic frequencies of free oscillations of the water surface (seiches) (Fig- ure 1), depend only on the asin geometry. Like water sloshing in a athtu, seiches are tide-like rises and drops in the coastal water levels of great lakes caused y prolonged strong winds that push the water toward one side of the lake, causing the water level to rise on the downwind side of the lake and to drop on the upwind side. When the wind stops, the water sloshes ack and forth, with the near-shore water level rising and falling in decreasingly small amounts on oth sides of the lake until it reaches equilirium. They occur commonly in enclosed or partially enclosed asins and are usually the result of a sudden change, or a series of intermittent-periodic changes, in atmospheric pressure or wind velocity. The period of oscillation of a seiche depends on the causative force that sets the water asin in motion and the natural or free oscillating period of the asin. Seiches can inict damage. If the natural period *. Corresponding Author, Department of Civil Engineering, Isfahan University of Technology, P.O. Box 84156, Isfahan, Iran. akairicc.iut.ac.ir 1. Department of Civil Engineering, Sharif University of Technology, P.O. Box , Tehran, Iran. Figure 1. Formation of seiche.
2 316 A.R. Kairi-Samani and B. Ataie-Ashtiani of a moored ship matches that of a seiche, then, considerale motion will result in the moored ship. Seiches could damage structures along the coastline and create large vertical accelerations for oshore structures, such as oats, arges and oating piers. Shoreline ooding may e caused y storm surges or seiches, often occurring simultaneously with high waves. A seiche is the free oscillation of the water in a closed or semi-enclosed asin at its natural period. Seiches are frequently oserved in harours, lakes, ays and in almost any distinct asin of moderate size. They may e caused y the passage of a pressure system over the asin or y the uild-up and susequent relaxation of a wind set-up in the asin. Following initiation of the seiche, the water sloshes ack and forth until the oscillation is damped out y friction. Seiches are not apparent in the main ocean asins, proaly ecause there is no force suciently coordinated over the ocean to set a seiche in motion. If the natural period, or seiche period, is close to the period of one of the tidal species, the constituents of that species (diurnal or semidiurnal) will e amplied y resonance more than those of other species. The constituent closest to the seiche period will e amplied most of all, ut the response is still a forced oscillation, whereas a seiche is a free oscillation. A variety of seiche periods may appear in the same water level record, ecause the main ody of water may oscillate longitudinally or laterally at dierent periods. It may also oscillate, oth in the open and closed mode, if the open end is somewhat restricted, and ays and harours o the main ody of water may oscillate locally at their particular seiche periods. Seiches generally have halflives of only a few periods, ut may e frequently regenerated. The largest amplitude seiches are usually found in shallow odies of water of large horizontal extent, proaly ecause the initiating wind set-up can e greater under these conditions. Seiching is the formation of standing waves in a water ody, due to wave formation and susequent reections from the ends. These waves may e incited y earthquake motions, impulsive winds over the surface, or due to wave motions entering the asin. The various modes of seiching correspond to the natural frequency response of the water ody. There are an innite numer of seiching modes possile, from the lowest (mode 1) to innity. Realistically, the lower modes proaly occur in nature, as frictional damping aects the higher modes preferentially (higher frequency). If the rectangular asin has signicant width as well as length, oth horizontal dimensions aect the natural period given in [1]: T nm p hg n a + m 1 ; (1) where T nm is the natural free oscillation period, n and m are the modes of oscillation in longitudinal and lateral coordinates, respectively and a and are the length and width of the rectangular asin. Forces of such wave motions have een attriuted to tsunamis, surges or seiches, instaility of wind, eddy trains caused y strong currents owing across the harour entrance, and surf-eat caused y long swell. owever, the most studied are harour oscillations caused y incident waves, which have typical periods of a few minutes. Due to strong winds or long wave energy, the water ody of a harour exhiits oscillatory resonant motions, which can damage moored ships and cause navigation hazards. A numer of theoretical and numerical investigations of such resonant oscillations have een carried out, ut most of them were limited to harours with a constant depth connected to open sea. The free oscillation in closed rectangular and circular asins was analyzed y []. These solutions claried the natural periods and modes of free surface oscillations related to these special congurations. McNown [3] studied the forced oscillation in a circular harour connected to the open sea through a narrow mouth. Since the radiation eect was ruled out, the results showed a harour resonance, as it does in a closed asin. The open-sea was important in allowing for the loss of energy radiated from a harour (see [4,5]). The study of wave motion and its characteristics on a physical model of a small-oat harour was undertaken in [6]. A weakly non-linear long internal wave in a closed asin was modelled y [7]. The free oscillation of water in a lake with an elliptic oundary was studied y [8]. Ishiguro [9] developed an analytical model for the oscillations of water in a ay or lake, using an electronic network and an electric analogue computer. The study of harour resonance has een extended to take into account the eect of ottom friction y [10] and wave nonlinearity [11,1]. Ippen [1] studied free oscillation in a simple two-dimensional closed asin. As an approach to an aritrary-shaped harour with a constant depth, a numerical model, using the oundary integral element method, was developed y [13,14]. For application to real depth-varying harours, a hyrid nite element model is provided in [15] and a nite dierence model is developed in [16]. A study of irregular wave incidence was undertaken y [17,18], viscous dissipation y [19] and porous reakwater y [0]. Lee and Park [1] developed a model for the prediction of harour resonance, using the nite dierence approach. Although the application of numerical models can e used for handling any complexities in this regard, the development of an analytical solution for practical cases would e very useful for the parametric study of the free oscillation phenomenon. In this work, an analytical solution for the calculation of free water surface oscillations
3 Free Water Surface Oscillations 317 in a rectangular asin with internal arriers will e presented. This analytical solution could e used to verify the accuracy of numerical models and calirate experimental models, applicale in cases such as the construction of a causeway in lakes and an expansion plan for harours. The water ow is considered as an ideal and irrotational ow. Therefore, the Laplace equation governs the velocity potential function of the ow domain. The free surface oundary condition is linearized to formulate a linear set of equations for solving the small amplitude water wave in the rectangular asin. The ow potential, wave amplitude and the natural period of waves generated in the asin with impervious internal arriers are found, ased on the asin geometry. These parameters are presented for variations in asin and arrier geometries. FREE OSCILLATION IN CLOSED BASINS WIT INTERNAL BARRIERS Figure illustrates a closed rectangular asin with the width of, length of a and a constant depth of h, with two internal arriers of length u and d. The asin is divided into two semi-closed asins and it is assumed that the thickness of the arrier is small, in comparison with the other dimensions. There is an opening etween the arriers that connects the suasins. To determine the natural periods of free water surface oscillations in this asin, an ideal ow eld is considered. Therefore, the governing equation for the velocity potential function is the Laplace equation (Equation ). Also, it is assumed that the amplitude of the water surface waves is small and, therefore, a linearized form of DFSBC can e applied. r 0: () The linearized kinematics and dynamic oundary conditions of the free surface for a left semi-closed asin can e written as Equations 3-1 and 3-: 1 g t on z 0; (3-1) z on z (x; y; t): (3-) t The ottom oundary condition is as follows: w 0 on z h: (3-3) z The oundary conditions at the vicinity of arriers are as follows: u 0 on x 0 for 0 y ; (3-4) x u x 0 on x a l for 0 y d ; (3-5) u x S l on xa l for d y d +c; (3-6) u v x 0 on xa l for d + c y ; (3-7) y 0 on y 0; for 0 x a l: (3-8) In a right semi-closed asin, the linearized oundary conditions can e written as follows: 1 g t on z 0; (4-1) z on z (x; y; t): (4-) t The ottom oundary condition is as follows: w 0 on z h: (4-3) z The oundary conditions in the vicinity of the arriers are as follows: u 0 on x 0 for 0 y ; (4-4) x u x 0 on x a l for 0 y d ; (4-5) u x S r on xa l for d y d + c; (4-6) u x 0 on x a l for d + c y ; (4-7) Figure. Schematic three-dimensional closed asin with internal arriers. v y 0 on y 0; for a l x a: (4-8)
4 318 A.R. Kairi-Samani and B. Ataie-Ashtiani To solve the dierential equation, the method of the separation of variales is applied, hence: X(x):Y (y):z(z):t (t): (5) Therefore: 1 g (t) X(x):Y (y):z(z):dt : (6) dt By sustituting and dividing the domain in the y direction to three su-domains, the solution is: a) for 0 y d ) d y d + c g cosh k(h + z) cosh kh a l cos my cos nx a l sin t; (7) cos my cos t; (8) Velocities S l and S r are equal and are evaluated y imposing the condition that the wave velocity at x a l for a left semi-closed asin is the same as the wave velocity for a right semi-closed asin, at this point. ence: cos my c) d + c y g cosh k(h + z) cosh kh a sin t; (9) cos nx a my cos cos t; (10) The oundary conditions of this case are the same as those presented for Case a. By applying the oundary conditions for the right asin and dividing the domain in the y direction to three su-domains, as elow, the results will e given as: a) 0 y d tan na a r g cosh k(h + z) cosh kh a r +! sin nx cos my sin t; (11) a r tan na a r ) d y d + c c) d + c y a r +! sin nx cos my cos tm; (1) a r g cosh k(h + z) cosh kh a cos my cos nx a sin t; (13) my cos cos t; (14) The oundary conditions of this case are the same as those presented for Case a. For motion in oth x and y directions, at a closed asin, which is divided into two semi-closed asins, the continuity for small amplitude water waves results from equating the change in ow in the two directions to the change in storage in the control volume (Figure 3). h( x + y ) + 0: (15) t Sustitution and simplication of results, for left and right asins, yields to: T s T cl n m 1 p + hg a for d y d + c; ( " n p + hg a l m Figure 3. Selected control volume. #) 1 otherwise; (16)
5 Free Water Surface Oscillations 319 n m 1 T s p + hg a T cr RESULTS for d y d + c; ( " n p + hg a r m #) 1 otherwise: (17) In this section, the results of the analytical solution presented aove, for a closed rectangular asin with internal arriers, are discussed. Figure 4 shows the variation of the normalized water level,, versus xa for various values of a l a. As shown, the wavelength decreases as the ratio of a l a increases. The results are the same as those for a simple closed asin without arriers under limiting conditions of a l a equals 0 and 1. Figure 5 illustrates the variation of the normalized water level,, versus y for various Figure 4. Variation of normalized water level versus xa for various values of a l a. values of c. The wavelength decreases as the values of c and u (or d ) increase. The opening size, c, has a signicant inuence on the normalized water level () at various positions in the asin. For the limiting case of c 1, the results are the same as those in the case of a simple asin. Also, for the case of c 0, when there is no opening etween arriers, the results are the same as those for a simple half asin. Increasing the value of u decreases the wavelength and its natural period, except under limiting conditions. Figure 6 illustrates the variation of the nondimensional parameter, a n, dened as a n (ug)a, versus xa for dierent a l a and mode m n 1. From this gure and the parameter, a n, dened here, the horizontal velocity at the x coordinate is otained, knowing the dimensions of the asin. As shown in this gure, the signicant eect of a l a on the parameter, a n, can e seen. Also, the results are symmetrical (the results of a l a and 1 a l a are the same). The amplitude of variation of the velocity component, u, in an x direction, decreases as the ratio, a l a. increases. As shown, the increase in the value of a l a causes a decrease in a n and u. Figure 7 shows the variation in the maximum values of a n versus a l a. As shown, the increase in the value of a l a causes a decrease in a n and u. Figure 8 shows the variation of the nondimensional parameter, n, dened as n (vg), versus y, for dierent modes. From this gure and the parameter, n, dened here, the horizontal velocity at the y coordinate is otained, knowing the dimensions of the asin. From this gure, the signicance of the modes of oscillation on parameter n can e seen. As seen, an increase in the mode numer causes an increase in the amplitude of the horizontal velocity (v). As seen, an increase in the mode numer causes an increase in the amplitude of the horizontal velocity Figure 5. Variation of normalized water level versus y for various values of c, when u d. Figure 6. Variation of parameter ua(g) y xa for dierent a l a.
6 30 A.R. Kairi-Samani and B. Ataie-Ashtiani Figure 7. Variation of maximum values of parameter ua(g) versus a l a. Figure 8. Variation of parameter n v(g) versus y for dierent modes. Figure 10. Normalized natural period of coupled asin to simple asin versus a l a or a r a for dierent a ratios and m n 1. causes an increase in the natural period of the coupled asin. When the asin ecomes square, the natural period ecomes the greatest. Figure 11 illustrates thep variation of the dimensionless parameter, S n T c hg, versus al, for dierent modes. From this gure, the natural modes y varying the position of the arrier. As seen, an increase in the value of a l, decreases the non-dimensional parameter, dened aove. Figure 1 shows the normalized water level contours for dierent modes of a simple closed asin and a closed asin with an internal arrier, where ala 0:5 and c 0:, and a simple closed asin with an internal continual arrier, where ala 0:5 for dierent modes. These gures illustrate the dierences of the ow pattern among these three cases. From these gures, the signicant inuence of internal arriers on the ow map, the wavelength and, more importantly, the natural frequency of free oscillation in the asin are shown. Figure 9. Variation of the maximum values of parameter n v(g) versus dierent modes. (v). This result clearly can e seen in Figure 9. In this gure, the variation of the maximum values of n, versus mode numer, m, is presented. Figure 10 shows the normalized natural period, T cl T s (or T cr T s ), versus a l a (or a r a) for dierent values of a. As seen, an increase in the value of a Figure 11. Non-dimensional parameter T c(hg) 0:5 () variations y a l.
7 Free Water Surface Oscillations 31 Figure 1. The normalized water level contours for closed simple asin and closed asin with internal arrier; c 0: and a l a r. SUMMARY An analytical solution for free surface oscillations in a rectangular asin with internal arriers was presented. The water ow was considered as an ideal ow. Therefore, the Laplace equation governs the velocity potential function of the ow domain. The free surface oundary condition was linearized to formulate a linear set of equation for solving the small amplitude water wave in the rectangular asin. The ow potential, wave amplitude and the natural period of waves generated in the asin with impervious internal arriers were found, ased on the asin geometry. It was shown that arrier geometry signicantly inuences the ow map, wavelength and natural frequency of free water oscillation in the asin. NOMENCLATURE T n T wave height natural free oscillation period the largest period T cl ; T cr T s S n S l ; S r X(x); Y (y) natural period for left and right semi asins, respectively natural period of simple closed asin non-dimensional p parameter, dened as S n T c hg velocities in x direction at the station of the arrier opening for left and right semi asins, respectively functions of x and y which demonstrate ow eld or amplitude functions Z(z); T (t) functions of z and t, which demonstrate ow eld or amplitude functions a; dimensions of simple closed asin a l ; a r dimensions of semi asins in x direction dened as a n (ug)a a n a n max u ; d n n max the maximum of a n the size of up and down arriers, respectively dened as n (vg) the maximum of n
8 3 A.R. Kairi-Samani and B. Ataie-Ashtiani c the opening size g gravitational acceleration h water depth k wave numer l length of closed asin along the axis m; n integers show the oscillation mode t time u; v velocity components in x and y directions x; y; z coordinates REFERENCES water level (amplitude) wave frequency velocity potential function 1. Ippen, A.T., Estuary and Coastal ydrodynamics, McGraw-ill Inc., USA (198).. Lam, S.., ydrodynamics, University of Camridge, 6th Ed. (193). 3. McNown, J.C. \Waves transmission through porous structures", Journal of Waterway arour and Coastal Engrg. Division., ASCE, 100(3), pp (195). 4. Ippen, A.T. and Goda, Y. \Wave induced oscillations in harours: The solution of a rectangular harour connected to the open sea", Report No. 59, ydrodynamics La., MIT, Camridge (1963). 5. Miles, J.W. and Munk, W.. \aror Paradox", Journal of Waterways, arors and Coastal Engrg. Division., ASCE, 87(3), pp (1961). 6. Bottin, R.R. \Physical modelling of small-oat harours: Design experience, lessons learned, and modelling guidelines", Technical Report (ADA58700), U.S. Army Engineer Waterways Experiment Station, Vicksurg, MS, USA (199). 7. orn, D.A., Imerger, J., Ivey, G.N. and Redekopp, L.G. \A weakly nonlinear model of long internal waves in closed asins", Journal of Fluid Mechanics, Camridge University Press, 467, pp (00). 8. Iida, F. \On the free oscillation of water in a lake of elliptic oundary", Journal of the Oceanographical Society of Japan, 1(3), pp (1965). 9. Ishiguro, S. \An analytical method for the oscillations of water in a ay or lake, using an electronic network and an electric analogue computer", Journal of the Oceanographical Society of Japan, 11(4) pp (1955). 10. Kostense, J.K., Meijer, K.L., Dinemans, M.W., Mynett, A.E. and Van Den Bosch, P. \Wave energy dissipation in aritrarily shaped harors of variale depth", 0th ICCE, pp (1986). 11. Lapelletier, T.G. and Raichlen, F. \aror oscillations induced y nonlinear transient long waves", Journal of Waterway, Port, Coast and Ocean Engrg., 113(4), pp (1987). 1. Zhou, C.P. and Cheung, Y.K. and Lee, J..W. \Response in harour due to incidence of second order low-frequency waves", Wave Motion, 13, pp (1991). 13. wang, L.S. and Tuck, E.O. \On the oscillation of harors of aritrary shape", J. Fluid Mech., 4, pp (1970). 14. Lee, J.J. \Wave-induced oscillations in harors of aritrary shape", J. Fluid Mech., 45, pp (1971). 15. Mei, C.C. and Chen,.S. \yrid element method for water waves", nd Annual Symp. of Waterways, arors and Coastal Engrg. Div., ASCE, 1, pp (1975). 16. Raichlen, F. and Naheer, E. \Wave induced oscillations of harors with variale depth", Proc. 15th ICCE., ASCE, pp (1976). 17. Goda, Y., Random Seas and Design of Maritime Structure, University of Tokyo Press, Tokyo, Japan (1985). 18. Ouellet, Y. and Theriault, T. \Wave grouping eect in irregular wave agitation in harour", Jour. Wtrway., Port, Coast and Ocean Engrg., 115(3), pp (1989). 19. Gemer, M. \Modeling dissipation in haror resonance", Coastal Engrg., 10, pp 11-5 (1986). 0. Yu, X. and Chwang, A.T. \Wave-induced oscillation in haror with porous reakwaters", Jour. Wtrway., Port, Coast and Ocean Engrg., 10(), pp (1994). 1. Lee, J.L. and Park, C.S. \Prediction of haror resonance y the nite dierence approach", Korean Society of Coastal and Ocean Eng., Asou University, Suwon, Korea (1998).
Investigating a pendulum
P3 3.6 Student practical sheet Investigating a pendulum The period of a pendulum is the time it takes to complete one swing. Different pendulums have different periods, so what determines the period of
More informationDynamic Green Function Solution of Beams Under a Moving Load with Dierent Boundary Conditions
Transaction B: Mechanical Engineering Vol. 16, No. 3, pp. 273{279 c Sharif University of Technology, June 2009 Research Note Dynamic Green Function Solution of Beams Under a Moving Load with Dierent Boundary
More informationIf we assume that sustituting (4) into (3), we have d H y A()e ;j (4) d +! ; Letting! ;, (5) ecomes d d + where the independent solutions are Hence, H
W.C.Chew ECE 350 Lecture Notes. Innite Parallel Plate Waveguide. y σ σ 0 We have studied TEM (transverse electromagnetic) waves etween two pieces of parallel conductors in the transmission line theory.
More informationChapter 3. Shallow Water Equations and the Ocean. 3.1 Derivation of shallow water equations
Chapter 3 Shallow Water Equations and the Ocean Over most of the globe the ocean has a rather distinctive vertical structure, with an upper layer ranging from 20 m to 200 m in thickness, consisting of
More informationRobot Position from Wheel Odometry
Root Position from Wheel Odometry Christopher Marshall 26 Fe 2008 Astract This document develops equations of motion for root position as a function of the distance traveled y each wheel as a function
More informationSection 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums Mass-Spring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More informationHeat and Mass Transfer over Cooled Horizontal Tubes 333 x-component of the velocity: y2 u = g sin x y : (4) r 2 The y-component of the velocity eld is
Scientia Iranica, Vol., No. 4, pp 332{338 c Sharif University of Technology, October 2004 Vapor Absorption into Liquid Films Flowing over a Column of Cooled Horizontal Tubes B. Farhanieh and F. Babadi
More informationAN-NAJ. J. RES., JAN.1988, SEC. II, VOL. I, NO. 5,
AN-NAJ. J. RES., JAN.1988, SEC. II, VOL. I, NO. 5, 12-27. 12 A DETAILED AND SIMPLIFIED SOLUTION TO HYDRODYNAMIC FORCES ON A SUBMERGED TANK SUBJECT TO LATERAL GROUND EXCITATION. A.H.Helou Civil Engineering
More informationComparison of Numerical Method for Forward. and Backward Time Centered Space for. Long - Term Simulation of Shoreline Evolution
Applied Mathematical Sciences, Vol. 7, 13, no. 14, 16-173 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ams.13.3736 Comparison of Numerical Method for Forward and Backward Time Centered Space for
More informationA Numerical Study of Chamber Size and Boundary Eects on CPT Tip Resistance in NC Sand
Scientia Iranica, Vol. 15, No. 5, pp 541{553 c Sharif University of Technology, October 2008 A Numerical Study of Chamber Size and Boundary Eects on CPT Tip Resistance in NC Sand M.M. Ahmadi 1; and P.K.
More informationEffect of Uniform Horizontal Magnetic Field on Thermal Instability in A Rotating Micropolar Fluid Saturating A Porous Medium
IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue Ver. III (Jan. - Fe. 06), 5-65 www.iosrjournals.org Effect of Uniform Horizontal Magnetic Field on Thermal Instaility
More informationFlow over oblique Side Weir
Damascus University Journal Vol. (8) - No. () 0 Al-Tali Flow over olique Side Weir Eng. Azza N. Al-Tali * Astract In this paper, an olique side weir was studied y nine models installed at the entrance
More informationEstimation of Hottest Spot Temperature in Power Transformer Windings with NDOF and DOF Cooling
Transactions D: Computer Science & Engineering and Electrical Engineering Vol. 16, No. 2, pp. 163{170 c Sharif University of Technology, Decemer 2009 Research Note Estimation of Hottest Spot Temperature
More informationNumerical simulation of wave overtopping using two dimensional breaking wave model
Numerical simulation of wave overtopping using two dimensional breaking wave model A. soliman', M.S. ~aslan~ & D.E. ~eeve' I Division of Environmental Fluid Mechanics, School of Civil Engineering, University
More informationA COMPUTATIONAL APPROACH TO DESIGN CODES FOR TSUNAMI- RESISTING COASTAL STRUCTURES
ISET Journal of Earthquake Technology, Paper No. 461, Vol. 42, No. 4, Decemer 25, pp. 137-145 A COMPUTATIONAL APPROACH TO DESIGN CODES FOR TSUNAMI- RESISTING COASTAL STRUCTURES Christopher Koutitas* and
More informationSloshing response of partially filled rectangular tank under periodic horizontal ground motion.
MATEC Web of Conferences 172, 15 (218) ICDAMS 218 https://doi.org/1.151/matecconf/21817215 Sloshing response of partially filled rectangular tank under periodic horizontal ground motion. Amiya Pandit 1+,
More informationThe Giant Tides of Fundy What are tides?
The Giant Tides of Fundy What are tides? The tide is the natural change in elevation of water over time and can easily be observed along any ocean coastline. Some areas of the world have one high tide
More informationUNSTEADY POISEUILLE FLOW OF SECOND GRADE FLUID IN A TUBE OF ELLIPTICAL CROSS SECTION
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume, Numer 4/0, pp. 9 95 UNSTEADY POISEUILLE FLOW OF SECOND GRADE FLUID IN A TUBE OF ELLIPTICAL CROSS SECTION
More informationP = ρ{ g a } + µ 2 V II. FLUID STATICS
II. FLUID STATICS From a force analysis on a triangular fluid element at rest, the following three concepts are easily developed: For a continuous, hydrostatic, shear free fluid: 1. Pressure is constant
More informationv t + fu = 1 p y w t = 1 p z g u x + v y + w
1 For each of the waves that we will be talking about we need to know the governing equators for the waves. The linear equations of motion are used for many types of waves, ignoring the advective terms,
More informationPrediction of changes in tidal system and deltas at Nakdong estuary due to construction of Busan new port
Prediction of changes in tidal system and deltas at Nakdong estuary due to construction of Busan new port H. Gm1 & G.-Y. park2 l Department of Civil & Environmental Engineering, Kookmin University, Korea
More informationA Continuous Vibration Theory for Beams with a Vertical Edge Crack
Transaction B: Mechanical Engineering Vol. 17, No. 3, pp. 194{4 c Sharif University of Technology, June 1 A Continuous Viration Theory for Beams with a Vertical Edge Crac Astract. M. Behzad 1;, A. Erahimi
More informationMelnikov s Method Applied to a Multi-DOF Ship Model
Proceedings of the th International Ship Staility Workshop Melnikov s Method Applied to a Multi-DOF Ship Model Wan Wu, Leigh S. McCue, Department of Aerospace and Ocean Engineering, Virginia Polytechnic
More informationCharacteristics of flow over the free overfall of triangular channel
MATEC We of Conferences 16, 03006 (018) BCEE3-017 https://doi.org/10.101/matecconf/0181603006 Characteristics of flow over the free overfall of triangular channel Raad Irzooki 1,*, Safa Hasan 1 Environmental
More informationASEISMIC DESIGN OF TALL STRUCTURES USING VARIABLE FREQUENCY PENDULUM OSCILLATOR
ASEISMIC DESIGN OF TALL STRUCTURES USING VARIABLE FREQUENCY PENDULUM OSCILLATOR M PRANESH And Ravi SINHA SUMMARY Tuned Mass Dampers (TMD) provide an effective technique for viration control of flexile
More informationMooring Model for Barge Tows in Lock Chamber
Mooring Model for Barge Tows in Lock Chamber by Richard L. Stockstill BACKGROUND: Extensive research has been conducted in the area of modeling mooring systems in sea environments where the forcing function
More informationViolent Sloshing H.Bredmose M.Brocchini D.H.Peregrine L.Thais
Violent Sloshing H.Bredmose Technical University of Denmark, M.Brocchini University of Genoa, D.H.Peregrine School of Mathematics, University of Bristol, d.h.peregrine@bris.ac.uk L.Thais Université des
More informationModeling of Water Flows around a Circular Cylinder with the SPH Method
Archives of Hydro-Engineering and Environmental Mechanics Vol. 61 (2014), No. 1 2, pp. 39 60 DOI: 10.1515/heem-2015-0003 IBW PAN, ISSN 1231 3726 Modeling of Water Flows around a Circular Cylinder with
More informationBOUSSINESQ-TYPE MOMENTUM EQUATIONS SOLUTIONS FOR STEADY RAPIDLY VARIED FLOWS. Yebegaeshet T. Zerihun 1 and John D. Fenton 2
ADVANCES IN YDRO-SCIENCE AND ENGINEERING, VOLUME VI BOUSSINESQ-TYPE MOMENTUM EQUATIONS SOLUTIONS FOR STEADY RAPIDLY VARIED FLOWS Yeegaeshet T. Zerihun and John D. Fenton ABSTRACT The depth averaged Saint-Venant
More informationExact Free Vibration of Webs Moving Axially at High Speed
Eact Free Viration of Wes Moving Aially at High Speed S. HATAMI *, M. AZHARI, MM. SAADATPOUR, P. MEMARZADEH *Department of Engineering, Yasouj University, Yasouj Department of Civil Engineering, Isfahan
More informationCHAPTER V MULTIPLE SCALES..? # w. 5?œ% 0 a?ß?ß%.?.? # %?œ!.>#.>
CHAPTER V MULTIPLE SCALES This chapter and the next concern initial value prolems of oscillatory type on long intervals of time. Until Chapter VII e ill study autonomous oscillatory second order initial
More informationEffect of Liquid Viscosity on Sloshing in A Rectangular Tank
International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 Volume 5 Issue 8 ǁ August. 2017 ǁ PP. 32-39 Effect of Liquid Viscosity on Sloshing
More informationLecture 7: Oceanographic Applications.
Lecture 7: Oceanographic Applications. Lecturer: Harvey Segur. Write-up: Daisuke Takagi June 18, 2009 1 Introduction Nonlinear waves can be studied by a number of models, which include the Korteweg de
More informationExploring the relationship between a fluid container s geometry and when it will balance on edge
Exploring the relationship eteen a fluid container s geometry and hen it ill alance on edge Ryan J. Moriarty California Polytechnic State University Contents 1 Rectangular container 1 1.1 The first geometric
More informationChapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.
Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To
More informationCHAPTER 11 VIBRATIONS AND WAVES
CHAPTER 11 VIBRATIONS AND WAVES http://www.physicsclassroom.com/class/waves/u10l1a.html UNITS Simple Harmonic Motion Energy in the Simple Harmonic Oscillator The Period and Sinusoidal Nature of SHM The
More informationEnergy-Rate Method and Stability Chart of Parametric Vibrating Systems
Reza N. Jazar Reza.Jazar@manhattan.edu Department of Mechanical Engineering Manhattan College New York, NY, A M. Mahinfalah Mahinfalah@msoe.edu Department of Mechanical Engineering Milwaukee chool of Engineering
More informationInvestigation of strong-motion duration consistency in endurance time excitation functions
Scientia Iranica A (2013) 20(4), 1085{1093 Sharif University of Technology Scientia Iranica Transactions A: Civil Engineering www.scientiairanica.com Investigation of strong-motion duration consistency
More informationSEG/New Orleans 2006 Annual Meeting. Non-orthogonal Riemannian wavefield extrapolation Jeff Shragge, Stanford University
Non-orthogonal Riemannian wavefield extrapolation Jeff Shragge, Stanford University SUMMARY Wavefield extrapolation is implemented in non-orthogonal Riemannian spaces. The key component is the development
More informationOscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance
Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum
More informationOceanography Lecture 8
Monday is an awful way to spend 1/7 of your week A A clear conscience is usually a sign of a bad memory I I used to have an open mind, but my brains kept falling out Oceanography Lecture 8 The surface
More informationTheory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018
Quiz No. 2, April 25, 2018 (1) viscous effects (2) shear stress (3) normal pressure (4) pursue (5) bear in mind (6) be denoted by (7) variation (8) adjacent surfaces (9) be composed of (10) integrand (11)
More informationThe University cannot take responsibility for any misprints or errors in the presented formulas. Please use them carefully and wisely.
Aide Mémoire Suject: Useful formulas for flow in rivers and channels The University cannot take responsiility for any misprints or errors in the presented formulas. Please use them carefully and wisely.
More informationANALYSIS OF THE DYNAMICS OF A FLOATING BODY WITH THIN SKIRTS BY USING THE DUAL BOUNDARY ELEMENT METHOD
598 Journal of Marine Science and Technology, Vol. 3, No. 5, pp. 598-67 (5) DOI:.69/JMST-5-5- ANALYSIS OF THE DYNAMICS OF A FLOATING BODY WITH THIN SKIRTS BY USING THE DUAL BOUNDARY ELEMENT METHOD Wen-Kai
More informationEarthquake Hazards. Tsunami
Earthquake Hazards Tsunami Review: What is an earthquake? Earthquake is the vibration (shaking) and/or displacement of the ground produced by the sudden release of energy. The point inside the Earth where
More informationContents. Parti Fundamentals. 1. Introduction. 2. The Coriolis Force. Preface Preface of the First Edition
Foreword Preface Preface of the First Edition xiii xv xvii Parti Fundamentals 1. Introduction 1.1 Objective 3 1.2 Importance of Geophysical Fluid Dynamics 4 1.3 Distinguishing Attributes of Geophysical
More informationEarthquakes Chapter 19
Earthquakes Chapter 19 Does not contain complete lecture notes. What is an earthquake An earthquake is the vibration of Earth produced by the rapid release of energy Energy released radiates in all directions
More informationPHY451, Spring /5
PHY451, Spring 2011 Notes on Optical Pumping Procedure & Theory Procedure 1. Turn on the electronics and wait for the cell to warm up: ~ ½ hour. The oven should already e set to 50 C don t change this
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationChaos and Dynamical Systems
Chaos and Dynamical Systems y Megan Richards Astract: In this paper, we will discuss the notion of chaos. We will start y introducing certain mathematical concepts needed in the understanding of chaos,
More informationExperimental and numerical investigation of 2D sloshing: scenarios near the critical filling depth
Experimental and numerical investigation of 2D sloshing: scenarios near the critical filling depth A. Colagrossi F. Palladino M. Greco a.colagrossi@insean.it f.palladino@insean.it m.greco@insean.it C.
More informationInvestigation of the Effect of the Circular Stands Diameters of Marine Structures and the Distances between Them on Wave Run-up and Force
Marine Science 16, 6(1): 11-15 DOI: 1.593/j.ms.1661. Investigation of the Effect of the Circular Stands Diameters of Marine Structures and the Distances between Them on Wave Run-up and Force Mohammad Ghatarband
More informationSteady-State Stresses in a Half-Space Due to Moving Wheel-Type Loads with Finite Contact Patch
Transaction A: Civil Engineering Vol. 7, No. 5, pp. 387{395 c Sharif University of Technology, October 2 Steady-State Strees in a Half-Space Due to Moving Wheel-Type Loads with Finite Contact Patch Abstract.
More informationSimple Harmonic Motion
Simple Harmonic Motion (FIZ 101E - Summer 2018) July 29, 2018 Contents 1 Introduction 2 2 The Spring-Mass System 2 3 The Energy in SHM 5 4 The Simple Pendulum 6 5 The Physical Pendulum 8 6 The Damped Oscillations
More informationEarthquakes and Earthquake Hazards Earth - Chapter 11 Stan Hatfield Southwestern Illinois College
Earthquakes and Earthquake Hazards Earth - Chapter 11 Stan Hatfield Southwestern Illinois College What Is an Earthquake? An earthquake is the vibration of Earth, produced by the rapid release of energy.
More informationA shallow water model for the propagation of tsunami via Lattice Boltzmann method
IOP Conference Series: Earth and Environmental Science OPEN ACCESS A shallow water model for the propagation of tsunami via Lattice Boltzmann method To cite this article: Sara Zergani et al IOP Conf. Ser.:
More information(a). t = 0 (b). t = 0.276
10 (a). t = 0 (b). t = 0.276 Figure 8: Snapshots of the sedimentation of 5040 disks of diameter 10/192 cm in 2D (W =8 cm). The initial lattice is disordered. from t = 0:026 to t = 0:476. In Figure 6, a
More informationPotential energy. Sunil Kumar Singh
Connexions module: m14105 1 Potential energy Sunil Kumar Singh This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract Potential energy is associated
More informationChapter 11 Vibrations and Waves
Chapter 11 Vibrations and Waves 11-1 Simple Harmonic Motion If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic.
More informationINFLUENCE OF FRICTION PENDULUM SYSTEM ON THE RESPONSE OF BASE ISOLATED STRUCTURES
Octoer 12-17, 28, Beijing, China INFLUENCE OF FRICTION PENDULUM SYSTEM ON THE RESPONSE OF BASE ISOLATED STRUCTURES M. Garevski 1 and M. Jovanovic 2 1 Director, Professor, Institute of Earthquake Engineering
More informationEngineering Thermodynamics. Chapter 3. Energy Transport by Heat, Work and Mass
Chapter 3 Energy Transport y Heat, ork and Mass 3. Energy of a System Energy can e viewed as the aility to cause change. Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential,
More informationNumerical comparison of two boundary meshless methods for water wave problems
Boundary Elements and Other Mesh Reduction Methods XXXVI 115 umerical comparison of two boundary meshless methods for water wave problems Zatianina Razafizana 1,2, Wen Chen 2 & Zhuo-Jia Fu 2 1 College
More informationANALYSIS AND MEASUREMENT OF STOKES LAYER FLOWS IN AN OSCILLATING NARROW CHANNEL *
770 008,0(6):770-775 ANALYSIS AND MEASUREMENT OF STOKES LAYER FLOWS IN AN OSCILLATING NARROW CANNEL PENG Xiao-xing China Ship Scientific search Center, Wuxi 1408, China, E-mail: me04px@alumni.ust.hk SU
More informationSTUDY ON DRIFT BEHAVIOR OF CONTAINER ON APRON DUE TO TSUNAMI-INDUCED INCOMING AND RETURN FLOW
STUDY ON DRIFT BEHAVIOR OF CONTAINER ON APRON DUE TO TSUNAMI-INDUCED INCOMING AND RETURN FLOW Tomoaki Nakamura 1, Norimi Mizutani 2 and Yasuhiro Wakamatsu 3 The drift behavior of a shipping container on
More informationChapter 14: Periodic motion
Chapter 14: Periodic motion Describing oscillations Simple harmonic motion Energy of simple harmonic motion Applications of simple harmonic motion Simple pendulum & physical pendulum Damped oscillations
More informationPhysics 41: Waves, Optics, Thermo
Physics 41: Waves, Optics, Thermo Particles & Waves Localized in Space: LOCAL Have Mass & Momentum No Superposition: Two particles cannot occupy the same space at the same time! Particles have energy.
More informationMethodology for sloshing induced slamming loads and response. Olav Rognebakke Det Norske Veritas AS
Methodology for sloshing induced slamming loads and response Olav Rognebakke Det Norske Veritas AS Post doc. CeSOS 2005-2006 1 Presentation overview Physics of sloshing and motivation Sloshing in rectangular
More informationWater is sloshing back and forth between two infinite vertical walls separated by a distance L: h(x,t) Water L
ME9a. SOLUTIONS. Nov., 29. Due Nov. 7 PROBLEM 2 Water is sloshing back and forth between two infinite vertical walls separated by a distance L: y Surface Water L h(x,t x Tank The flow is assumed to be
More informationReport Documentation Page
Modeling Swashzone Fluid Velocities Britt Raubenheimer Woods Hole Oceanographic Institution 266 Woods Hole Rd, MS # 12 Woods Hole, MA 02543 phone (508) 289-3427, fax (508) 457-2194, email britt@whoi.edu
More informationRiver science in the light of climate change
River science in the light of climate change Hui de Vriend 1 Introduction Climate change and a river s response to it are likely to e slow processes as compared to the responses to direct human interventions
More informationA NEW HYBRID TESTING PROCEDURE FOR THE LOW CYCLE FATIGUE BEHAVIOR OF STRUCTURAL ELEMENTS AND CONNECTIONS
SDSS Rio 010 STABILITY AND DUCTILITY OF STEEL STRUCTURES E. Batista, P. Vellasco, L. de Lima (Eds.) Rio de Janeiro, Brazil, Septemer 8-10, 010 A NEW HYBRID TESTING PROCEDURE FOR THE LOW CYCLE FATIGUE BEHAVIOR
More informationBuckling Behavior of Long Symmetrically Laminated Plates Subjected to Shear and Linearly Varying Axial Edge Loads
NASA Technical Paper 3659 Buckling Behavior of Long Symmetrically Laminated Plates Sujected to Shear and Linearly Varying Axial Edge Loads Michael P. Nemeth Langley Research Center Hampton, Virginia National
More informationFreak waves over nonuniform depth with different slopes. Shirin Fallahi Master s Thesis, Spring 2016
Freak waves over nonuniform depth with different slopes Shirin Fallahi Master s Thesis, Spring 206 Cover design by Martin Helsø The front page depicts a section of the root system of the exceptional Lie
More informationSTEADY CURRENTS INDUCED BY SEA WAVES PROPAGATING OVER A SLOPING BOTTOM
STEADY CURRENTS INDUCED BY SEA WAVES PROPAGATING OVER A SLOPING BOTTOM Erminia Capodicasa 1 Pietro Scandura 1 and Enrico Foti 1 A numerical model aimed at computing the mean velocity generated by a sea
More informationNewton's second law of motion
OpenStax-CNX module: m14042 1 Newton's second law of motion Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Abstract Second law of
More informationTsunamis and ocean waves
Department of Mathematics & Statistics AAAS Annual Meeting St. Louis Missouri February 19, 2006 Introduction Tsunami waves are generated relatively often, from various sources Serious tsunamis (serious
More informationFlows Induced by 1D, 2D and 3D Internal Gravity Wavepackets
Abstract Flows Induced by 1D, 2D and 3D Internal Gravity Wavepackets Bruce R. Sutherland 1 and Ton S. van den Bremer 2 1 Departments of Physics and of Earth & Atmospheric Sciences, University of Alberta
More information1/27/2010. With this method, all filed variables are separated into. from the basic state: Assumptions 1: : the basic state variables must
Lecture 5: Waves in Atmosphere Perturbation Method With this method, all filed variables are separated into two parts: (a) a basic state part and (b) a deviation from the basic state: Perturbation Method
More informationFrequency-Dependent Amplification of Unsaturated Surface Soil Layer
Frequency-Dependent Amplification of Unsaturated Surface Soil Layer J. Yang, M.ASCE 1 Abstract: This paper presents a study of the amplification of SV waves obliquely incident on a surface soil layer overlying
More information1. a) A flag waving in the breeze flaps once each s. What is the period and frequency of the flapping flag?
PHYSICS 20N UNIT 4 REVIEW NAME: Be sure to show explicit formulas and substitutions for all calculational questions, where appropriate. Round final answers correctly; give correct units. Be sure to show
More informationEarthquake Hazards. Tsunami
Earthquake Hazards Tsunami Review: What is an earthquake? Earthquake is the vibration (shaking) and/or displacement of the ground produced by the sudden release of energy. The point inside the Earth where
More information4 Tides. What causes tides? How do tides vary?
CHAPTER 14 4 Tides SECTION The Movement of Ocean Water BEFORE YOU READ After you read this section, you should be able to answer these questions: What causes tides? How do tides vary? National Science
More informationAvailable online at Eng. Math. Lett. 2014, 2014:17 ISSN: WAVE ATTENUATION OVER A SUBMERGED POROUS MEDIA I.
Available online at http://scik.org Eng. Math. Lett. 04, 04:7 ISSN: 049-9337 WAVE ATTENUATION OVER A SUBMERGED POROUS MEDIA I. MAGDALENA Industrial and Financial Mathematics Research Group, Faculty of
More informationD. J. Clarke Department of Mathematics, Wollongong University College, Wollongong, N.S.W., Australia
Resonant and nonresonant motion in a spindle-shaped basin with an entrance N. D. Thomas Department of Mathematics and Quantitative Methods, Mitchell College of Advanced Education, Bathurst, N.S.W., Australia
More informationEarthquake Hazards. Tsunami
Earthquake Hazards Tsunami Measuring Earthquakes Two measurements that describe the power or strength of an earthquake are: Intensity a measure of the degree of earthquake shaking at a given locale based
More informationLaboratory Investigations on Impulsive Waves Caused by Underwater Landslide
Laboratory Investigations on Impulsive Waves Caused by Underwater Landslide B. Ataie-Ashtiani 1 A. Najafi-Jilani Department of Civil Engineering, Sharif University of Technology, Tehran, Iran, Fax: +98
More informationMATHEMATICAL ANALYSIS OF A VIBRATING RIGID WATER TANK. Amin H. Helou * Abstract. Introduction
An-Najah J. Res. Vol. I ( 1989 ) Number 6 Amin Helou MATHEMATICAL ANALYSIS OF A VIBRATING RIGID WATER TANK Amin H. Helou * 0+2. 110 ja )311 1 la.a.11 JUJI 4, J 1 4. 4.16.7.2; J.41 4.)1 Li t *" I:LA 24-;
More informationGENERAL SOLUTIONS FOR THE INITIAL RUN-UP OF A BREAKING TSUNAMI FRONT
International Symposium Disaster Reduction on Coasts Scientific-Sustainable-Holistic-Accessible 14 16 November 2005 Monash University, Melbourne, Australia GENERAL SOLUTIONS FOR THE INITIAL RUN-UP OF A
More informationA nonlinear numerical model for the interaction of surface and internal waves with very large floating or submerged flexible platforms
A nonlinear numerical model for the interaction of surface and internal waves with very large floating or submerged flexible platforms T. Kakinuma Department of Civil Engineering, University of Tokyo,
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More information2. Theory of Small Amplitude Waves
. Theory of Small Amplitude Waves.1 General Discussion on Waves et us consider a one-dimensional (on -ais) propagating wave that retains its original shape. Assume that the wave can be epressed as a function
More informationTime / T p
Chapter 5 Single-Point QNS Measurements 5. Overview A total of 46 primary test cases were evaluated using the QNS technique. These cases duplicated the 8 studied by Ainley [] and extend the parameters
More informationTides: this is what we see
Last time Wind generated waves and Tsunami These are wind and earthquake generated waves Today: Tides These are Gravity waves How are they generated What are their intersting complexities? Low tide (decrease
More informationTravel Grouping of Evaporating Polydisperse Droplets in Oscillating Flow- Theoretical Analysis
Travel Grouping of Evaporating Polydisperse Droplets in Oscillating Flow- Theoretical Analysis DAVID KATOSHEVSKI Department of Biotechnology and Environmental Engineering Ben-Gurion niversity of the Negev
More informationTHREE-DIMENSIONAL FINITE DIFFERENCE MODEL FOR TRANSPORT OF CONSERVATIVE POLLUTANTS
Pergamon Ocean Engng, Vol. 25, No. 6, pp. 425 442, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0029 8018/98 $19.00 + 0.00 PII: S0029 8018(97)00008 5 THREE-DIMENSIONAL FINITE
More informationRenewable Energy: Ocean Wave-Energy Conversion
Renewable Energy: Ocean Wave-Energy Conversion India Institute of Science Bangalore, India 17 June 2011 David R. B. Kraemer, Ph.D. University of Wisconsin Platteville USA B.S.: My background Mechanical
More informationTopological structures and phases. in U(1) gauge theory. Abstract. We show that topological properties of minimal Dirac sheets as well as of
BUHEP-94-35 Decemer 1994 Topological structures and phases in U(1) gauge theory Werner Kerler a, Claudio Rei and Andreas Weer a a Fachereich Physik, Universitat Marurg, D-35032 Marurg, Germany Department
More informationChapter 7. 1 a The length is a function of time, so we are looking for the value of the function when t = 2:
Practice questions Solution Paper type a The length is a function of time, so we are looking for the value of the function when t = : L( ) = 0 + cos ( ) = 0 + cos ( ) = 0 + = cm We are looking for the
More informationSCATTERING CONFIGURATION SPACES
SCATTERING CONFIGURATION SPACES RICHARD MELROSE AND MICHAEL SINGER 1.0B; Revised: 14-8-2008; Run: March 17, 2009 Astract. For a compact manifold with oundary X we introduce the n-fold scattering stretched
More information