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.
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These waves are also applied to model the motion of highway
traffic flows
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 devi ...
.
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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
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
is a constant or a variable. Kinematic wave can be described by a simple
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
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 situations, 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
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, ...
:
:
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
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and tr ...
.
References
Further reading
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{{physical oceanography
Fluid dynamics
Oceanographical terminology