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

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

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

3. Wave equations Introduction Linear wave equation
Non-Linear wave equation

4. Continuous Solution Linear Wave Equation
Solution of the linear wave equation

5. 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

6. 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

7. Continuous Solution
Characteristic diagram for nonlinear waves

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

9. Continuous Solution

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

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

12. 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

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

14. 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

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

16. Shock Waves
Let
Shock velocity

17. 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 ,

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

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

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

21. 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

22. 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.

23. 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

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

25. Shock structure As ,
exponentially and as
exponentially.

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

27. 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

28. 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

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

30. Weak Solutions
2) Let

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

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

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

34. THANK YOU