Finite Difference Solution as an Approximate Solution of a Boundary Value Problem

A concrete example will now illustrate the inherent difficulties of using the finite dif­ference solution to approximate the solution of a boundary value problem governed by partial differential equations.

Suppose the frequencies of the normal acoustic wave modes of a one­dimensional tube of length L as shown in Figure 1.1 is to be determined. The tube has two closed ends and is filled with air. The governing equations of motion of the air in the tube are the linearized momentum and energy equations, as follows:

Подпись: (1.12) (1.13) d u d p

P0 — + — = 0

0 dt dx

d p d U

It + Y P0 dx = °’

where p0, p0, and y are, respectively, the static density, the pressure, and the ratio of specific heats of the air inside the tube; and u is the velocity. The boundary conditions are

Подпись: (1.14)Подпись:At x = 0, L; u = 0.

Upon eliminatingp from (1.12) and (1.13), the equation for u is

d2u 2 d2u

9U – a dX2 = 0’

Подпись: Figure 1.1. A one-dimensional tube with closed ends.

where a = (yp0/p0)1/2 is the speed of sound.