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# Method of Characteristics

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. 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 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.

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Prof. Alan Hood
2000-10-30