University Mathematics Handbook (2015)
XIV. Partial Differential Equations (PDE)
Chapter 5. One-Dimensional Wave Equation
(*)
when is the velocity of wave propagation.
If , then it is a homogeneous equations.
5.1 General Solution of Homogeneous Equation
a. , are characteristic lines.
b. By change of variables , we get the canonical form of a wave equation . Its solution is , when functions are arbitrary, continuous, and have partial derivatives continuous up to second order.
c. is a general solution of (*).
d. describes a (rightwards) forward wave at speed .
describes a (leftwards) backward wave at speed .
General solution is a superposition of an forward wave and a backward wave with speed .
5.2 Vibrations of an Infinite String. D'Alembert's
Formula
a. Equations system
(*)
(**)
describes the amplitude of an (ideal) elastic infinite string vibration. Initial conditions and are given functions describing amplitude and vibration velocity at time .
b. (*),(**) is called a Cauchy problem.
c. If and , then Cauchy problem (*),(**) has unique solution given by D'Alembert's formula
(***)
d. If is continuous, , are piecewise continuous, and and/or , then there are points where the first and second order derivatives of do not necessarily exist, and therefore, function given by (**) is not a solution of problem/ (*), (**).
In any point except these ones, is a solution of the wave problem. In this case, we construct a generalized solution of the wave problem, the following way:
Choose two sequences , , when , , uniformly converging to and , respectively, in .
Let the sequence be a solution of (*) holding.
Then, the sequence uniformly converges to the generalized solution, .
5.3 Non-Homogeneous Wave Equation
a. The equations system
(*)
(**)
describes a vibration of an infinite string constrained by an external force . String amplitude is dependent of initial conditions and initial velocity at time .
b. The characteristic triangle
Through point , we draw 2 characteristic lines:
,
forming, together with the -axis, a characteristic triangle (see illustration).
c. If, in (**), and , then, the solution of (*) is
d. If , and functions and are continuous on , , then the unique solution of Cauchy problem (*),(**) is
5.4 Vibrations of Semi-Infinite String
a. Homogeneous cauchy problem:
(*)
Initial condition:
(**)
Boundary condition:
(***)
The solution is
b. The solutions of non-homogeneous problem
with the same initial condition (**) or boundary condition (***) are
when is the solution of homogeneous problem (see a.) and
c. If, in a Cauchy problem, the boundary condition is non-homogeneous, that is, (***) is replaced by , then, by substituting , we get the non-homogeneous problem
5.5 Vibrationof a Finite String (Both Ends Fixed)
The finite string wave equation is
Initial condition is:
Boundary condition is:
To use D'Alembert’s formula (see 5.3), we extend functions , , on all -axis, through interval ends and , to odd and periodic functions with a period, that is
Similarly, we construct functions and .
Using the D'Alembert’s formula mentioned in 5.3 to find the solution of the problem.
The reduction of this solution to , is the solution of the given problem.
If, in addition, satisfy the compatibility condition
and , then Cauchy problem has a unique solution .