Hyperbolic Equations -- Wave equations

The classical example of a hyperbolic equation is the wave equation

$${\partial^2 y\over\partial t^2} - c^2{\partial^2 y\over\partial x^2} =0.$$ (2.5)

The wave equation can be rewritten in the form

$$\left({\partial\over\partial t} + c {\partial\over\partial ... ...l\over\partial t} - c {\partial\over\partial x}\right) y = 0$$ (2.6)

or as a system of 2 equations

$${\partial z\over\partial t} + c {\partial z\over\partial x} = 0$$ (2.7)

$${\partial y\over\partial t} - c {\partial y\over\partial x} = z.$$ (2.8)

Note that the first of these equations (2.3a) is independent of $y$ and can be solved on it's own. The second equation (2.3b) can then be solved by using the known solution of the first. Note that we could equally have chosen the equations the other way round, with the signs of the velocity $c$ interchanged. As the 2 equations (2.3) are so similar we expect the stability properties to be the same. We therefore concentrate on (2.3a) which is known as the Advective equation and is in fact the conservation of mass equation of an incompressible fluid

$${\partial u\over\partial t} + c {\partial u\over\partial x} = 0.$$ (2.9)

Note also that the boundary conditions will usually be specified in the form

$$u(x,t) = u_0(x) \mbox{ at } t=0.$$ (2.10)

which gives the value of $u(x,t)$ for all $x$ at a particular time.