3.1 The Wave Equation on the Whole Line. D Alembert Formula

The simplest hyperbolic second-order equation is the wave equation

(3.1)

where x signifies the spatial variable or ( position , t the time variable, u = u ( x, t) the unknown function and c is a given positive constant. The wave equation describes vibrations of a string. Physically u ( x, t) represents the value of the normal displacement of a particle at position x and time t.

Using the theory of Section 2.2 the characteristic equation of (3.1) is

and

are two families of real characteristics. Introducing the new variables

and the function

the equation (3.1) reduces to

(3.2)

Therefore

and in the original variables u ( x, t) is of the form

(3.3)

known as the general solution of (3.1). It is the sum of the function g ( x ? ct) which presents a shape traveling without change to the right with speed c and the function f ( x + ct) - another shape, traveling to the left with speed c.

Consider the Cauchy (initial value) problem for (3.1)

where ? and ? are arbitrary functions of x. Further we denote R ⁺ = { t : t ? 0}.