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The first order wave equation

(1.3) 
describes the movement of a wave in one direction with no change of
shape. A solution is shown in Figure 1.3 as a surface plot
and a contour plot.
Figure 1.3:
The left figure shows a surface plot and the right figure
shows a contour plot. In the contourplot the lines are for a given
value of .

In the contour plot, in Figure 1.3, the straight lines are
given by the lines that have the same value of . is constant
along these lines.
The aim of the method of characteristics is to solve the PDE by
finding curves in the plane that reduce the equation to an ODE.
In general, any curve in the plane can be expressed in
parametric form by
where the parameter, , gives a measure of the distance along the
curve. The curve starts at the initial point, , ,
when . Assuming that we can solve the resulting ODE means that
is known everywhere along this curve, i.e. along the curve picked
out by the value of . Another choice for gives another
curve and the value of is determined along this curve. In this
manner, can be determined at any point in the plane by
choosing the curve, defined by , that passes through this point
and taking the correct value of , the distance along the curve.
Hence, we can ervaluate . These curves are illustrated in
Figure 1.4
Figure 1.4:
The parametric representation of a curve in the
plane. The solid curve starts at , and . Choosing
another value of gives the dashed curve.

Therefore, we have
and so is a function
of . Hence, the derivative of with respect to to is

(1.4) 
Compare (1.4) with the left hand side of (1.3) and we
can convert (1.3) into an ordinary derivative of with
respect to , i.e.

(1.5) 
provided the parametric representation of the curve satisfies

(1.6) 
and

(1.7) 
(1.6) and (1.7) give the characteristic
curves. (1.5) shows that

(1.8) 
along a characteristic curve but the constant may be
different on different characteristic curves. As gives us a
different characteristic curve, this implies that .
Solving (1.7), we have
and (1.6) gives

(1.9) 
defines which characteristic curve you are on and from
(1.8), is constant on a characteristic curve that depends
on . Hence,

(1.10) 
as shown before, where the arbitrary function, , is determined by
the initial condition.
Subsections
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Up: Partial Differential Equations of
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Prof. Alan Hood
20001030