Kinematic wave

In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., kinematic waves are important mathematical tools to understand the basic features of the associated wave phenomena. [1] These waves are also applied to model the motion of highway traffic flows.[2] [3]

In these flows, mass and momentum equations can be combined to yield a kinematic wave equation. Depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave celerity is a constant or a variable. Kinematic wave can be described by a simple partial differential equation with a single unknown field variable (e.g., the flow or wave height, ) in terms of the two independent variables, namely the time () and the space () with some parameters (coefficients) containing information about the physics and geometry of the flow. In general, the wave can be advecting and diffusing. However, in simple situation, the kinematic wave is mainly advecting.

Kinematic wave for debris flow

Non-linear kinematic wave for debris flow can be written as follows with complex non-linear coefficients:

where is the debris flow height, is the time, is the downstream channel position, is the pressure gradient and the depth dependent nonlinear variable wave speed, and is a flow height and pressure gradient dependent variable diffusion term. This equation can also be written in the conservative form:

where is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For , this equation reduces to the Burgers' equation.


  1. ^ Takahashi, T. (2007). Debris Flow: Mechanics, Prediction and Countermeasures. Taylor and Francis, Leiden.
  2. ^ Lighthill, M.J.; Whitham, G.B. (1955). "On kinematic waves. I: Flood movement in long rivers. II: A theory of traffic flow on long crowded roads". Proceedings of the Royal Society. 229A (4): 281–345.
  3. ^ Newell, G.F. (1993). "A simplified theory of kinematic waves in highway traffic, Part I: General theory". Transpn. Res. B. 27B (4): 281–287.
Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Bidirectional traffic

In transportation infrastructure, a bidirectional traffic system divides travelers into two streams of traffic that flow in opposite directions.In the design and construction of tunnels, bidirectional traffic can markedly affect ventilation considerations.Microscopic traffic flow models have been proposed for bidirectional automobile, pedestrian, and railway traffic. Bidirectional traffic can be observed in ant trails and this has been researched for insight into human traffic models. In a macroscopic theory proposed by Laval, the interaction between fast and slow vehicles conforms to the Newell kinematic wave model of moving bottlenecks.In air traffic control traffic is normally separated by elevation, with east bound flights at odd thousand feet elevations and west bound flights at even thousand feet elevations (1000 ft ≈ 305m). Above 28,000 ft (~8.5 km) only odd flight levels are used, with FL 290, 330, 370, etc., for eastbound flights and FL 310, 350, 390, etc., for westbound flights. Entry to and exit from airports is always one-way traffic, as runways are chosen to allow aircraft to take off and land into the wind, to reduce ground speed. Even in no wind cases, a preferred calm wind runway and direction is normally chosen and used by all flights, to avoid collisions. In uncontrolled airports, airport information can be obtained from anyone at the airport. Traffic follows a specific traffic pattern, with designated entry and exits. Radio announcements are made, whether anyone is listening or not, to allow any other traffic to be aware of other traffic in the area.In the earliest days of railways in the United Kingdom, most lines were built double tracked because of the difficulty of coordinating operations in pre-telegraphy times.

Most modern roads carry bidirectional traffic, although one-way traffic is common in dense urban centres. Bidirectional traffic flow is believed to influence the rate of traffic collisions. In an analysis of head-on collisions, rear-end collisions, and lane-changing collisions based on the Simon-Gutowitz bidirectional traffic model, it was concluded that "the risk of collisions is important when the density of cars in one lane is small and that of the other lane is high enough", and that "heavy vehicles cause an important reduction of traffic flow on the home lane and provoke an increase of the risk of car accident".Bidirectional traffic is the most common form of flow observed in trails, however, some larger pedestrian concourses exhibit multidirectional traffic.

Cell Transmission Model

Cell Transmission Model (CTM) is a popular numerical method proposed by Carlos Daganzo to solve the kinematic wave equation. Lebacque later showed that CTM is the first order discrete Godunov approximation.

Conservation law

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.

A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume.

From Noether's theorem, each conservation law is associated with a symmetry in the underlying physics.


DPHM-RS (Semi-Distributed Physically based Hydrologic Model using Remote Sensing and GIS) is a semi-distributed hydrologic model developed at University of Alberta, Canada.

Index of wave articles

This is a list of Wave topics.

Integrated Water Flow Model (IWFM)

Integrated Water Flow Model (IWFM) is a computer program for simulating water flow through the integrated land surface, surface water and groundwater flow systems. It is a rewrite of the abandoned software IGSM, which was found to have several programing errors. The IWFM programs and source code are freely available. IWFM is written in Fortran, and can be compiled and run on Microsoft Windows, Linux and Unix operating systems. The IWFM source code is released under the GNU General Public License.

Groundwater flow is simulated using the finite element method. Surface water flow can be simulated as a simple one-dimensional flow-through network or with the kinematic wave method. IWFM input data sets incorporate a time stamp, allowing users to run a model for a specified time period without editing the input files.

One of the most useful features of IWFM is the internal calculation of water demands for each land use type. IWFM simulates four land use classes: agricultural, urban, native vegetation, and riparian vegetation. Land use areas are delineated as a time series, with corresponding evapotranspiration rates and water management parameters. Each time step, the land use process applies precipitation, calculates infiltration and runoff, calculates water demands, and determines what portion of the demands are not met by soil moisture. For agricultural and urban land use classes, IWFM then applies surface water and groundwater at specified rates, and optionally adjusts surface water and groundwater to exactly meet water demands. This automatic adjustment feature is especially useful for calculating unmeasured flow components (such as groundwater withdrawals) or for simulating proposed future scenarios such as studying the impacts of potential climate change.In IWFM, the land surface, surface water and groundwater flow domains are simulated as separate processes, compiled into individual dynamic link libraries. The processes are linked by water flow terms, maintain conservation of mass and momentum between processes, and are solved simultaneously. This allows each IWFM process to be run independently as a stand-alone model, or to be linked to other programs. This functionality has been used to create a Microsoft Excel Add-in to create workbooks from IWFM output files. The IWFM land surface process has been compiled into a stand-alone program called the IWFM Demand Calculator IDC. The groundwater process is linked to the WRIMS modeling system and used in the water resources optimization model CalSim. This feature allows other models to be easily linked with IWFM, to either enhance the capabilities of the target model (for example, by adding groundwater flow to a land surface-surface water model) or to enhance the capabilities of IWFM (for example, linking an economic model to IWFM to dynamically change the crop mix based on the depth to groundwater, as the cost of pumping increases with depth to water).

Notable models developed with IWFM include the California Central Valley Groundwater-Surface Water Simulation Model (C2VSim), a model of the Walla-Walla Basin in Washington and Oregon, USA, a model of the Butte Basin, CA, USA, and several unpublished models. IWFM has also been peer reviewed.


MIKE 11 is a computer program that simulates flow and water level, water quality and sediment transport in rivers, flood plains, irrigation canals, reservoirs and other inland water bodies. MIKE 11 is a 1-dimensional river model. It was developed by DHI.

MIKE 11 has long been known as a software tool with advanced interface facilities. Since the beginning MIKE 11 was operated through an efficient interactive menu system with systematic layouts and sequencing of menus. It is within that framework where the latest ‘Classic’ version of MIKE 11 – version 3.20 was developed.

The new generation of MIKE 11 combines the features and experiences from the MIKE 11 ‘Classic’ period, with the powerful Windows based user interface including graphical editing facilities and improved computational speed gained by the full utilization of 32-bit technology.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

Shallow water equations

The shallow water equations are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). The shallow water equations in unidirectional form are also called Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant (see the related section below).

The equations are derived from depth-integrating the Navier–Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity scale of the fluid is small compared to the horizontal velocity scale. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the horizontal velocity field is constant throughout the depth of the fluid. Vertically integrating allows the vertical velocity to be removed from the equations. The shallow water equations are thus derived.

While a vertical velocity term is not present in the shallow water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be usable with the shallow water equations. Once a solution (i.e. the horizontal velocities and free surface displacement) has been found, the vertical velocity can be recovered via the continuity equation.

Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow water equations are widely applicable. They are used with Coriolis forces in atmospheric and oceanic modeling, as a simplification of the primitive equations of atmospheric flow.

Shallow water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets of shallow water equations can describe the state.

Storm Water Management Model

The United States Environmental Protection Agency (EPA) Storm Water Management Model (SWMM) is a dynamic rainfall–runoff–subsurface runoff simulation model used for single-event to long-term (continuous) simulation of the surface/subsurface hydrology quantity and quality from primarily urban/suburban areas. It can simulate the Rainfall- runoff, runoff, evaporation, infiltration and groundwater connection for roots, streets, grassed areas, rain gardens and ditches and pipes, for example. The hydrology component of SWMM operates on a collection of subcatchment areas divided into impervious and pervious areas with and without depression storage to predict runoff and pollutant loads from precipitation, evaporation and infiltration losses from each of the subcatchment. Besides, low impact development (LID) and best management practice areas on the subcatchment can be modeled to reduce the impervious and pervious runoff. The routing or hydraulics section of SWMM transports this water and possible associated water quality constituents through a system of closed pipes, open channels, storage/treatment devices, ponds, storages, pumps, orifices, weirs, outlets, outfalls and other regulators. SWMM tracks the quantity and quality of the flow generated within each subcatchment, and the flow rate, flow depth, and quality of water in each pipe and channel during a simulation period composed of multiple fixed or variable time steps. The water quality constituents such as water quality constituents can be simulated from buildup on the subcatchments through washoff to a hydraulic network with optional first order decay and linked pollutant removal, best management practice and low-impact development (LID) removal and treatment can be simulated at selected storage nodes. SWMM is one of the hydrology transport models which the EPA and other agencies have applied widely throughout North America and through consultants and universities throughout the world. The latest update notes and new features can be found on the EPA website in the download section. Recently added in November 2015 were the EPA SWMM 5.1 Hydrology Manual (Volume I) and in 2016 the EPA SWMM 5.1 Hydraulic Manual (Volume II) and EPA SWMM 5.1 Water Quality (including LID Modules) Volume (III) + Errata”

Traffic flow

In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems.

Traffic model

A traffic model is a mathematical model of real-world traffic, usually, but not restricted to, road traffic. Traffic modeling draws heavily on theoretical foundations like network theory and certain theories from physics like the kinematic wave model. The interesting quantity being modeled and measured is the traffic flow, i.e. the throughput of mobile units (e.g. vehicles) per time and transportation medium capacity (e.g. road or lane width). Models can teach researchers and engineers how to ensure an optimal flow with a minimum number of traffic jams.

Traffic models often are the basis of a traffic simulation.

Truck lane restriction

In traffic flow theory, the impact of freeway truck lane restrictions is an interesting topic. Intuitively, slow vehicles (e.g. trucks) will cause queues behind them, but how it relates to the kinematic wave theory was not revealed until Newell. Leclercq et al did a complete review of Newell's theory. In addition to the simulation models developed by Laval and Daganzo on the basis of numerical solution methods for Newell's theory to capture the impacts of slow vehicle, Laval also mathematically derived the analytical capacity formulas for bottlenecks caused by single-type of trucks for multi-lane freeway segments.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Undulatory locomotion

Undulatory locomotion is the type of motion characterized by wave-like movement patterns that act to propel an animal forward. Examples of this type of gait include crawling in snakes, or swimming in the lamprey. Although this is typically the type of gait utilized by limbless animals, some creatures with limbs, such as the salamander, choose to forgo use of their legs in certain environments and exhibit undulatory locomotion. This movement strategy is important to study in order to create novel robotic devices capable of traversing a variety of environments.


Vflo is a commercially available, physics-based distributed hydrologic model generated by Vieux & Associates, Inc. Vflo uses radar rainfall data for hydrologic input to simulate distributed runoff. Vflo employs GIS maps for parameterization via a desktop interface. The model is suited for distributed hydrologic forecasting in post-analysis and in continuous operations. Vflo output is in the form of hydrographs at selected drainage network grids, as well as distributed runoff maps covering the watershed. Model applications include civil infrastructure operations and maintenance, stormwater prediction and emergency management, continuous and short-term surface water runoff, recharge estimation, soil moisture monitoring, land use planning, water quality monitoring, and water resources management.

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Wave model (disambiguation)

Wave model is a concept of language development in historical linguistics.

A wave model is a theoretical concept comparing a phenomenon of any type to any part of a physical wave.

Wave model can refer to:

Wind wave model, a mathematical model of sea waves

Density wave model, a mathematical model of a spiral galaxy

Kinematic wave model, an explanation of traffic flow

Ocean zones
Sea level

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