4
1. Introduction
where 1/!(x, y , t) is the stream function for the nondivergent velocity field and
1.1.1 First-Order Hyperbolic Equations
Many waves can be mathematically described as solutions to hyperbolic partial
differential equations. One simple example of a hyperbolic partial differential
equation is the general first-order quasi-linear equation
au
au
A(x, t, u)- + B(x , t, u)- = C(x, t, u),
at
ax
(U)
where A, B, and C are real-valued functions with continuous first derivatives. This
equation is hyperbolie beeause there exists a family of real-valued eurves in the xt plane along which the solution ean be locally determined by integrating ordinary
differential equations. These eurves, ealled characteristics, may be defined with
respect to the parameter s by the relations
dt
-=A,
ds
dx
- = B .
ds
(1.2)
The identity
du
au dt
Bu dx
- = - - + - -
ds
at ds
ax ds
can then be used to express (U) as the ordinary differential equation
du
-=C.
ds
(1.3)
Given the value of u at some arbitrary point (xo, to), the coordinates of the charaeteristic eurve passing through (xo, to) ean be determined by integrating the ordinary differential equations (1.2). The solution along this charaeteristie ean be
obtained by integrating the ordinary differential equation (1.3). A unique solution to (U) ean be determined throughout some loeal region of the x -t plane by
specify ing data for u along any noneharaeteristic line.
In physical applieations where the independent variable t represents time, the
particular solution of (I.l) is generally determined by specifying initial data for u
along the line t = O. In such applications A is nonzero, and any perturbation in
the distribution of u at the point (xo, to) translates through a neighborhood of Xo
at the speed
dx
B
dt
A
The solutions to (1.1) are wave-like in the general sense that the perturbations in
u travel at well-defined velocities even though they may distort as they propagate.