Partial (as well as ordinary) differential equations such as those in Section 3.1 generally have an infinite number of solutions. Thus, when a physical
3.2. Boundary and Initial Conditions 117
problem is described by such an equation, additional conditions are needed to find the unique solution defining the behavior of the system. These additional conditions are determined by the physical nature of the system and should obey the following demands:
1. They should guarantee the uniqueness of the solution (i.e., there should not be two different functions satisfying the equation and ad- ditional conditions).
2. They should guarantee the stability of the solution (i.e., any small variations of these additional conditions or the coefficients of the differential equation result in only insignificant variations in the so- lution). In other words, the solution should depend continuously on additional conditions and the coefficients of the equation.
These additional conditions may be classified as two distinct types; initial conditions and boundary conditions.
Initial conditions characterize the function satisfying the equation at
the initial moment t = 0. Equations that are second order in time have two initial conditions. For example, in the problem of transverse oscilla- tions of a string, the initial conditions define the string’s shape and speed distribution at zero time:
u(x, 0) = ϕ(x) and ∂u
∂t(x, 0) = ψ (x), (3.7)
whereϕ(x) and ψ (x) are specified functions of x.
Boundary conditions characterize the behavior of the function satisfy-
ing the equation at the boundary of the physical region of interest for all moments of timet. In most cases, the boundary conditions for partial dif- ferential equations give the functionu(x, t) and/or the normal component of its gradient along the boundary.
Let us consider various boundary conditions for transverse oscillations of a string over the finite interval 0≤ x ≤ l from a physical point of view. 1. If the left end of the string, located at x = 0, is rigidly fixed, the
boundary condition atx = 0 is
u(0, t) = 0.
A similar condition exists for the right end of the string, located at x = l, if it is fixed. These are called fixed end boundary conditions.
2. If the motion of left end of the string is driven with the functiong(t), then
u(0, t) = g(t),
in which case we have driven end boundary conditions.
3. If the end at x = 0 can move and experiences a force, f (t), that varies with time (e.g., a string attached to a ring that is driven up and down on a vertical rod), then from Equation (3.1), we have
−T0ux(0, t) = f (t).
If this boundary condition is applied, instead, to the right end of the string located atx = l, the left-hand side of this formula will have a positive sign. These are called forced end boundary conditions and are different from driven end conditions because the slope rather than the position is specified as the initial condition.
4. If the end at x = 0 moves freely, but is still attached (e.g., a string attached to a ring that can slide up and down on a vertical rod with no friction), then the slope at the end will be zero. In this case, the last equation gives
ux(0, t) = 0
with similar equations for the right end. The conditions in this case are called free end boundary conditions.
5. If the left end is attached to a surface that can stretch, we have an
elastic boundary, in which case we add a vertical component of elas-
tic force−ku(0, t), to the left-hand side of Equation (3.1). This re- sults in the boundary condition
ux(0, t) − hu(0, t) = 0, h = k T0
. For the right end we have
ux(l, t) + hu(l, t) = 0.
If the point to which the string is elastically attached is also moving and its deviation from the initial position is described by the function ζ (t), the boundary condition becomes
ux(0, t) − h [u(0, t) − ξ(t)] = 0. (3.8)
3.2. Boundary and Initial Conditions 119
It can be seen that for stiff attachment (largek) when even a small shift of the end causes strong tension, the boundary condition of Equation (3.8) becomesu(0, t) = g(t)(k = ∞) with g(t) = ξ(t). For weak attachment (smallk, weak tensions), this condition (3.8) becomes the condition for a free endux(0, t) = 0(k = 0).
In general, for one-dimensional problems, the boundary conditions at the endsx = 0 and x = l can be summarized in the form
α1ux+β1u|x=0=g1(t), α2ux+β2u|x=l =g2(t), (3.9)
whereg1(t) and g2(t) are known functions, and α1,β1,α2,β2 are (real)
constants. As discussed in Chapter 2, due to physical constraints the nor- mal restrictions on these constants areβ1/α1< 0 and β2/α2> 0.
When functions on the right-hand sides of Equation (3.9) are zero (i.e., g1,2(t) ≡ 0), the boundary conditions are said to be homogeneous. In this case, if u1(x, t), u2(x, t),. . . , un(x, t) satisfy these boundary conditions,
then any linear combination of these functions
C1u1(x, t) + C2u2(x, t) + . . . + Cnun(x, t)
(whereC1, . . . , Cnare constants) also satisfies these conditions. This prop- erty will be used frequently in the following discussion.
We may classify the above physical notions of boundary conditions as formally belonging to one of three main types:
1. Boundary conditions of the first kind (Dirichlet boundary condi-
tions). For this case, we are given u|x=a = g(t), where here and
belowa = 0 or l. This describes a given boundary regime; for ex- ample, ifg(t) = 0 we have fixed ends.
2. Boundary conditions of the second kind (Neumann boundary condi-
tions). In this case, we are given ux|x=a = g(t), which describes a given force acting at the ends of the string; for example, ifg(t) = 0 we have free ends.
3. Boundary conditions of the third kind (mixed boundary conditions). Here we have ux± hu|x=a =g(t) (minus sign for a = 0, plus sign fora = l); for example, an elastic attachment for the case h = const.
Applying these three conditions alternately to the two ends of the string result in nine types of boundary problems. A list classifying all possible combinations of boundary conditions can be found in Appendix A.
As mentioned previously, the initial and boundary conditions com- pletely determine the solution of the wave equation. It can be proved that under certain conditions of smoothness of the functionsϕ(x), ψ (x), g1(t),
andg2(t) defined in Equations (3.7) and (3.9), a unique solution always
exists; therefore, these conditions are, in general, necessary. The following sections investigate many examples of the dependence of the solutions on the boundary conditions.
In some physical situations, either the initial conditions or the bound- ary conditions may be ignored, leaving only one condition to determine the solution. For instance, suppose the pointM0is rather distant from the
boundary and the boundary conditions are given such that the influence of these conditions atM0is exposed after a rather long time interval. In such
cases, if we investigate the situation for a relatively short time interval, in- stead of a complete problem, we can ignore the boundaries and study the
initial value problem (or the Cauchy problem). Formally, these solutions
are for an infinite region, but they apply to a finite string for times short enough that the boundary conditions have not had time to have an effect. For instance, in the one-dimensional case for short time periods, we may ignore the boundary conditions and search for the solution of the equation
utt=a2uxx+f (x, t) for − ∞ < x < ∞, t > 0, with the initial conditions
u(x, 0) = ϕ(x), ut(x, 0) = ψ (x)
for − ∞ < x < ∞.
Similarly, if we study a process close enough to one boundary (at one end for the one-dimensional case) and rather far from the other boundary, for some characteristic time of that process the boundary condition at the distant end may be insignificant. For the one-dimensional case, we arrive at a boundary value problem for a semi-infinite region, 0≤ x < ∞, where in addition to the differential equation we have additional conditions
u(0, t) = g(t), t > 0, u(x, 0) = ϕ(x), ut(x, 0) = ψ (x) 0< x < ∞.