Mathematical Language


-->In mathematical language, the propagation problem is known as an initial-value problem (Fig. Mat-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions.
Fig. Mat-1

The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”).

Fig. Mat-2

Prototypes are:

1. Elliptic. Laplace’s equation

d2u/dx2 + d2u/dy2 = 0

Poisson’s equation

d2u/dx2 + d2u/dy2 =g(x,y)

These do not contain the variable t (time) explicitly; accordingly, their solutions represent equilibrium configurations. Laplace’s equation corresponds to a “natural” equilibrium, while Poisson’s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y).

2. Parabolic. The heat equation

d2u/dx2 + d2u/dy2 = du/dt

describes nonequilibrium or propagation states of diffusion as well as heat transfer.
3. Hyperbolic. The wave equation

d2u/dx2 + d2u/dy2 = d2u/dt2
describes wave propagation of all types when the assumption is made that the wave amplitude is small and that interactions are linear.

The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available.

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