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## Finite Differences

There are many ways to approach the numerical integration of (5.1). If the material parameters are constant, then a simple explicit method can be obtained by applying centered difference approximations to both the time and space derivatives, yielding the scheme

 (5.3) (5.4)

where is a grid function defined for integer and , and represents an approximation to , where is a uniform grid spacing, and is the time step. This scheme is accurate to , but is stable only for , so it is effectively only first-order accurate; this is typical of explicit methods for systems with some parabolic character [176].

In order to deal more effectively with the varying-coefficient problem, we can divide (5.1) into a system of two PDEs, and differentiate with respect to time, to get

 (5.5a)

Here, is the beam transverse velocity, and can be interpreted as a bending moment. We have chosen these variables in order to make clear the relationship of the classical beam theory with the more modern Timoshenko theory (see §5.2). Applying centered differences to this system yields
 (5.6a)

where and are the grid functions corresponding to and . Similarly to the case of the transmission line (see §4.3.6), we have used

and in keeping with the literature [176], we have also defined

As for the transmission line, we can evaluate the grid functions and at alternating time steps. Due to the nature of the difference approximation, however, we cannot interleave these variables on the spatial grid. That is, we are forced to calculate both and at every grid location (at their respective time steps). We have written the difference scheme above such that temporal interleaving is evident, i.e. and , half-integer, are calculated only for (say) even and odd, respectively.

Next: Waveguide Network for the Up: Transverse Motion of the Previous: Phase and Group Velocity
Stefan Bilbao 2002-01-22