Exercise 2.2 Prove the previous theorem
2.3 Green's Functions
2.3 Green's Functions 35
X X+AX
Time
Fig. 2.4. Velocity of a particle of mass m resulting from an impulsive force of magnitude / .
and A = I/m for small AT. Combining this result with equation 2.27 provides
v(t) =
0, if t < r. (2.28)
Equation 2.28 represents the response of the particle to a single abrupt blow (Figure 2.4). Now suppose that the particle suffers a series of blows Ik at time r^, k = 1, 2,..., N. The response of the particle to each blow should be independent of all other blows, and the velocity becomes
N
t>rN. (2.29) If the blows become sufficiently rapid, the particle is subjected to a continuous force. Then /& —-> f(r)dr and
v(t) = -
m t > T0,
which can be rewritten as
t
V(t)= J i{>(t,T)f(T)dT, (2.30)
where
(0, iit<r
Equation 2.30 is the solution to the differential equation 2.25. It pre- sumes that the response of the particle at each instant of impact is independent of all other times. Given this property, the response of the particle to f(t) is simply the sum of all the instantaneous forces, and the particle is said to be a linear system. Many mechanical and electrical systems (and, as it turns out, many potential-field problems) have this property.
The function ip(t,r) is the response of the particle at time t due to an impulse at time r; it is called the impulse response or Green's function of the linear system. The Green's function, therefore, satisfies the initial conditions and is the solution to the differential equation 2.25 subject to the initial conditions when the forcing function is an impulse.
Equation 2.26 is a heuristic description of an impulse. In the limit as AT approaches zero, the impulse of equation 2.26 becomes arbitrarily large in amplitude and short in duration while its integral over time remains the same. The limiting case is called a Dirac delta function 8(t), which has the properties
oo
J 6(t)>
f(t)S(t)dt =
— oo oo
f(t)6(T-t)dt = f(T). (2.31)
-oo
These definitions and properties are meaningless if 8(i) is viewed as an ordinary function. It should be considered rather as a "generalized function" characterized by the foregoing properties.
Green's functions are very useful tools; equation 2.30 shows that if the Green's function i\) is known for a particular linear system, then the state of the linear system due to any forcing function can be derived for any time.
2.3 Green's Functions 37 2.3.2 Green's Functions and Laplace's Equation
The previous mechanical example provides an analogy for potential the- ory. In Chapter 3, we will derive Poisson's equation
2 (2.32)
This second-order differential equation describes the Newtonian poten- tial U throughout space due to a mass distribution with density p.
Clearly V2U = 0 and U is harmonic in regions where p = 0. We seek a solution for U that satisfies the differential equation and the boundary condition that U is zero at infinity.
The density distribution in Poisson's equation is obviously the source of U and in this sense is analogous to the forcing function f(t) of the previous section. We know from the previous section that the response to an impulsive forcing function f(t) = 6(t) is the Green's function, so we could try representing the density distribution in R as an "impulse"
and see what happens to U. An impulsive source in three dimensions can be written as <5(P, Q), where
6(P,Q)dv = l,
6(P, Q)=0 if P ^ g , and where Q is the point of integration as in Section 2.2.1.
Hence, we let the density be <5(P, Q) and the potential be ip\ in equa- tion 2.32,
Then from the Helmholtz theorem and equations 2.14 and 2.15,
where r is the distance between P and Q. This is a very interesting result. We see that 7/r is the solution to Poisson's equation when p is an "impulsive" density distribution located at Q. Indeed, we will show in Chapter 3 that 7/r is the Newtonian potential at P due to a point mass at Q. Hence, 7/r is the "impulse response" for Poisson's relation;
with it, the potential due to any density distribution can be determined with an integral equation analogous to equation 2.30:
v (2.34)
-I-
p(Q)dv, (2.35) where it is understood that integration is over all space. This fundamen- tal equation relating gravitational potential to causative density distri- butions will be derived in a different way in Chapter 3. The important point to be made here is that the function ipi = 7/r is analogous to the Green's function of the mechanical example in the previous section: It satisfies the required boundary condition, that ipi is zero at infinity, and is the solution to Poisson's differential equation when the density is an"impulse."
Half-Space Regions
The representation formula, which followed from Green's third identity, shows that the value of a function harmonic in R can be found at any point within R strictly from the behavior of U and its normal derivative on the boundary of i?, that is,
dnr
where Q is the point of integration and r is the distance from P to Q. In practical situations, we are unlikely to have both the potential and its normal derivative at our disposal, and elimination of ^ would make this equation much more useful. We should expect that such a simplification is possible because earlier results have shown that the potential is uniquely determined by its boundary conditions.
To eliminate ^ from the third identity, we begin with Green's second identity. Let both U and V be harmonic in equation 2.5 so that
s
Adding this equation to equation 2.36 provides
4TT J |_ dn r r dn dn dn J
.-inu±(v + i)-(v+i)%}«.
2.3 Green's Functions 39
/ P(x,y,-Az) i V2U=0
I ^^rx, \
— Z=0
P'(x,y,Az)
Fig. 2.5. Function U is harmonic throughout the half-space z < 0 and assumes known values on the surface z = 0. Parameter r is the distance between points P and Q\ r' is the distance between P' and Q. Point P' is the image of point P such that r = rf when Q is on the surface z = 0.
If we select a harmonic V such that V + - = 0 at each point of 5, then
Hence, if for a particular geometry we can find a function V such that (1) V is harmonic throughout R and (2) V + £ = 0 at each point of 5, then £/ can be found throughout the region, and only values of U on the boundary will be required. The function V + Ms called the Green's function for Laplace's equation in restricted regions. It satisfies Laplace's equation throughout the region (except when P = Q) and is zero on the boundary.
In principle, equation 2.37 provides a simple way to solve Laplace's equation from specified boundary conditions. Unfortunately, the func- tion V is very difficult to derive analytically except for the simplest sorts of geometrical situations, such as half-spaces and spheres. As an exam- ple, consider the half-space problem, where U is harmonic for all z < 0 and is known on the planar surface z = 0 (Figure 2.5). Boundary S then consists of the z = 0 plane plus the z < 0 hemisphere, as shown in Fig- ure 2.5. We construct a point P' below the z — 0 plane that is the image of point P. The necessary properties are satisfied if we let V = — p-, where r' is the distance from P' to Q: namely, V is always harmonic
since Q is always above or on the z = 0 plane, V + ^ = 0 when Q is on the z = 0 plane, and 7 + ^ = 0 when Q is on the infinite hemisphere.
Hence, V defined in this way satisfies the necessary requirements to be used in equation 2.37; that is,
I
U7T (~ ~ ^
On \r r' s
oo oo
f f U^M
rdadp, (2.39)
J J [ (x_a) 2 + (y_/3)2 + A22]f ^ ' ^ >
27T
— oo — o o
where Az > 0. Equation 2.39 provides a way to calculate the potential at any point above a planar surface on which the potential is known.
Such calculations are called upward continuation, a subject that will be revisited at some length in Chapter 12.
Terminology
The Green's function for Poisson's equation throughout space is usually derived from a general form of Poisson's equation, V20 = —/, and thus is given by G = -^ (e.g., Duff and Naylor [81], Strauss [274]). Here, we started with the gravitational case of Poisson's equation, V2£/ = — 4TT7/?,
and derived a slightly different form for the Green's function, '01 = ^- The present approach led to equation 2.34, that is,
where integration is over all space. This simple integral expression for the potential in terms of density and the Green's function will prove useful in following chapters.
We also showed that if a function V can be found satisfying just two properties (V is harmonic throughout a region, and V + ^ is zero on the boundary of the region), then the representation formula reduces to a very simple form,
The function V + \ is the Green's function for Laplace's equation in restricted regions. In future chapters, however, we will use a somewhat