7.4.1 Homogeneous equation and boundary conditions

Let us consider the Dirichlet boundary value problem (BVP) for the homogeneous wave equation

which describes the motion of the vibrating string.

Our goal is to find the solution of ( MWH) using the Fourier method.

A separable solution is a solution of the form

to the problem

Plugging the form into the wave equation, we get

or

(7.55)

By the boundary conditions

it follows

(7.56)

So X( x) satisfies the problem

As before the problem ( P) has nontrivial solutions

corresponding to

(7.57)

Plugging (7.57) into (7.55), we obtain the ODE

with general solution

Therefore functions of the form

known as norrnal modes of vibration, are solutions of the problem ( SW). In order to find a solution of ( MWH) we take a superposition of u _n( x, t). Namely, we are looking for a solution of the form

(7.58)

Formally the last function satisfies the initial conditions if

Using the Fourier-sine series for ?( x) and ?( x) we obtain that

(7.59)

(7.60)

In order to justify the formal solution we prove

Theorem 7.11

Suppose ? ? C ²[0, l], is piecewise continuous ? ? C ¹ [0, l], ? ? ?( x) is piecewise continuous, and

(7.61)

Then the function (7.58), where the coeficients A _n and B _n are determined by