Waves and First Order Equations Peter Romeo Nyarko Supervisor : Dr. J.H.M. ten Thije Boonkkamp 23rd September, 2009

Outline Introduction Continuous Solution Shock Wave Shock Structure Weak Solution Summary and Conclusions

Introduction What is a wave? Application of waves Light and sound Water waves Traffic flow Electromagnetic waves

Wave equations Introduction Linear wave equation Non-Linear wave equation

Continuous Solution Linear Wave Equation Solution of the linear wave equation

Continuous Solution Non-Linear Wave Equation If we consider and as functions of , Since remains constant is a constant on the characteristic curve and therefore the curve is a straight line in the plane

Continuous Solution We consider the initial value problem If one of the characteristics intersects Then is a solution of our equation, and the equation of the characteristics is where

Continuous Solution Characteristic diagram for nonlinear waves

Continuous Solution We check whether our solution satisfy the equation: ,

Continuous Solution

Continuous Solution Breaking , Breaking occur immediately Compression wave with overlap , Breaking occur immediately

Continuous Solution There is a perfectly continuous solution for the special case of Burgers equation if Rarefaction wave

Continuous Solution Kinematic waves We define density per unit length ,and flux per unit time , Flow velocity Integrating over an arbitrary time interval, This is equivalent to

Continuous Solution Therefore the integrand The conservation law. The relation between and is assumed to be Then

Shock Wave We introduce discontinuities into our solution by a simple jump in and as far as our conservation equation is feasible Assume and are continuous

Shock Structure where are the values of from below and above. where is the shock velocity and

Shock Waves Let Shock velocity

Traffic Flow (Example) Consider a traffic flow of cars on a highway . : the number of cars per unit length : velocity :The restriction on density. is the value at which cars are bumper to bumper From the continuity equation ,

Traffic Flow (Example) This is a simple model of the linear relation The conservative form of the traffic flow model where

Traffic Flow (Example) The characteristics speed is given by The shock speed for a jump from to

Traffic Flow (Example) Consider the following initial data Case t x characteristics

Shock structure We consider as a function of the density gradient as well as the density Assume At breaking become large and the correction term becomes crucial Then where Assume the steady profile solution is given by

Shock structure Then Integrating once gives is a constant Qualitatively we are interested in the possibility of a solution which tends to a constant state.

Shock Structure , as as If such a solution exist with as Then and must satisfy The direction of increase of depends on the sign of between the two zero’s

Shock Structure with and If with as required The breaking argument and the shock structure agree. Let for a weak shock , with where ,

Shock structure As , exponentially and as exponentially.

Weak Solution A function is called a weak solution of the conservation law if holds for all test functions

Weak solution Consider a weak solution which is continuously differentiable in the two parts and but with a simple jump discontinuity across the dividing boundary between and . Then , ,is normal to

Weak Solutions The contribution from the boundary terms of and on the line integral Weak solution ,discontinuous across S Since the equations must hold for all test functions, on This satisfy Points of discontinuities and jumps satisfy the shock conditions

Weak Solutions Non-uniqueness of weak solutions 1) Consider the Burgers’ equation, written in conservation form Subject to the piecewise constant initial conditions

Weak Solutions 2) Let

Weak Solutions Entropy conditions A discontinuity propagating with speed given by : Satisfy the entropy condition if where is the characteristics speed.

Weak Solutions a) Shock wave Characteristics go into shock in (a) and go out of the shock in (b) b) Entropy violating shock

Summary and Conclusion Explicit solution for linear wave equations. Study of characteristics for nonlinear equations. Weak solutions are not unique.

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