**Eigen-functions and
Eigen-values**

While the content and length of Chapter IV occupy the material centre of these lectures, Chapter
V presents their conceptual centre. Indeed, we will develop the methods, started by Fourier, to their greatest generality and make
the **boundary value problems of physics**
widely accessible to mathematical treatment. The force of the methods developed in this chapter was displayed most strikingly when
Erwin Schrödinger (1887 - 1961) recognized in 1926 the quantum numbers as eigen-values of his wave equation and thus made
available to **atomic physics**
the entire armature of the more recent analysis. It was a fortunate circumstance that side by side with him was his colleague
Hermann Weyl (1885 - 1955), who, as the most outstanding student of David Hilbert (1862 - 1943) and his successor in Göttingen,
took part outstandingly in the development of the **theory
of integral equations**.We might just as well note here that, while it had
acknowledged the significance of the point of view of the **integral
equations** for the strict mathematical foundation, especially of the existence proof
of the eigen-functions and their eigen-values, also the older point of view of the **partial
differential equations** naturally also led to these concepts. To start with, we will
show this by a simple example, which was known long before the integral equations.

**5.25
Eigen-value and Eigen-functions of the oscillating membrane:**** **The
object of the following considerations is a skin without its own
elasticity stretched over a frame, which owes its ability to resist changes in shape
only to the tension acting on its edge, We think of this, like of the surface tension *T* of the theory of capillarity, to
be acting in the plane of the membrane and at right angle to its edge. As a result, the deformed membrane experiences a normal
pressure *N*, acting perpendicularly to the surface, which is *T *times the mean curvature of the membrane, i.e.,
for small deformations *u* equal to *TD**u.
*The vibration equation (7.4)
then yields pure harmonic vibrations with the frequency *w*

As always, we write

According to (2.10.6),
this is at the same time, if we do not let *k*² be constant, but, as we will do in the sequel, equal to an arbitrary
function *F*(*x*, *y*), the general linear self-adjoined, elliptic, second order differential equation in two
variables *x*, *y,* once it has been given the normal form.

The solutions of (1), which satisfy the boundary condition *u* = 0, are
called the **eigen-functions**,
the associated values of *k *the **eigen-values***
*of the problem. If *k*², i.e., *F*(*x*, *y*) are negative, there would be no eigen-values, as
has already been discussed in Exercise 2.2.
We will demonstrate by very simple examples that there exist for positive *k*² eigen-values and, indeed, infinitely many
of them.

**5.25.1 The rectangle 0 ****
****x ****
****a, ****0 ****
****y**** ****
****b****:**** **The
boundary conditions are met by

The differential equation then yields

Naturally, one can always add a constant amplitude factor, but we will not do so here, nor in
the sequel. To start with, we assume that *a* and *b *are **not
commensurable**. Then all the *k*_{n,m }differ and there
corresponds to every *k* **only one eigen-value**,
the number of which is infinite.

**5.25.2 Circle, Circular Ring,
Circular Sector:**** **For the
complete circle 0 **
***r*** ****
***a*, we set

when *k* satisfies the boundary condition

Since this equation has infinitely many roots (cf. Fig.
21), there are again **infinitely many
eigen-values ***k = k*_{n,m}. They all differ from each
other, but for *m* > 0, there belong to each of them two eigen-functions, corresponding to the double sign ()
in the exponent, which results in the same possibility, corresponding to the double possibility of
for *j *in real terms. You say
then that the problem for *m *> 0 is **degenerate**,
and indeed in our case **simply degenerate**.
By (20.4b),
the (non-degenerate) fundamental tone of the circular membrane is *k* = 2.40/*a*.

For the circular ring *b ***
***r** ***
***a*, one sets

In fact, one now requires **both**
particular solutions *I* and *N* of Bessel's differential equation (instead of which one could, of course, use *H*^{1}
and *H*^{2}) to meet the two boundary conditions

Here too, there are infinitely many different *k*_{n,m} and associated *c*_{n,m}.
Also, this problem is for *m*>*n *simply degenerate, since then there correspond, by (4), to the *k*_{n,m}
two different *u*_{n,m}.

For the circular sector 0* ***
***r** ***
***a*, 0* ***
***j ***
***a*, it is most
convenient to set

where now *k *must be determined from the condition *I**m*(*ka*)
= 0. There are infinitely many eigen-values *k* = *k*_{n,m}; the problem is not degenerate.

The most general problem which can be treated in this manner is the **sector
of a circular ring**** ***b**r**
***
***a*, 0* ***
***j ***
***a*, which is bounded by
two circular arcs and two radii.

**5.25.3
Ellipse and Elliptic-Hyperbolic Curved Quadrangle****: **The
wave equation can be separated in elliptic co-ordinates *x, h *and
leads in each of them to the so-called **Mathieu-equation**.
*x **= *const yields the
ellipses, *h *= const the
hyperbolae. For the complete ellipse, the boundary condition *u* = 0 on its circumference is joined by the continuity
condition for *x **= *0
(focal line) and the periodicity condition for . The
determination of the eigen-values *k* leads to transcendental equations with a complicated structure, which we will not
discuss here. The general area of this kind is the curved quadrangle bounded by two ellipses and two confocal hyperbolic arcs.

What we have analyzed here for several simple examples forms a generalization of
the **fundamental
theorem **of the theory of oscillating systems
with infinitely many degrees of freedom and the eigen-functions altogether: *For an arbitrary region, there exists a
non-truncating series of eigen-values k; in the case of the boundary condition u = *0* (or any other **earlier
discussed conditions**), the solution, continuous inside the region, solves the
differential equation *.(The
same theorem holds for the general self-adjoint differential equation
when *F* > 0, for its eigen-values *l*.) The
strict mathematical proof of the theorem has challenged repeatedly the skill of mathematicians, starting with a large work of
Henry Pincer (1854 - 1909) (Rend.Circa. Matthem. DI Palermo, 1894) and culminating in the Fredholm-Hilbert theory of integral
equations. We must here be satisfied to relate it to a theorem on mechanical systems with a finite number of degrees of freedom,
namely: *A system with f degrees of freedom, which is in a stable equilibrium position, is able to undergo about this position
exactly f , linearly independent, small (more accurately, infinitely small) sinus shaped oscillations.*

Using the notation of our Volume I, we write the kinetic energy for the
neighbourhood of an equilibrium position *q*_{1} = *q*_{2} = ··· = *q*_{f}*
*= 0 in the form:

We can view here the *a*_{nm}, due to the smallness of the
oscillations *q*, to be constants. In the same manner, the potential energy *V* is a quadratic form of the**
***q*_{n }with constant coefficients, since the linear terms
of *V *'s expansion in powers of the *q*_{i }is to vanish in the equilibrium position, i.e.,

Now, it is always possible to convert by a linear transformation of the co-ordinates the two
quadratic forms above simultaneously into sums of squares (**principal
axes transformation **of a surface of second degree) whence one might obtain

The co-ordinates introduced here are called the **normal
co-ordinates**** **of the system. As **Lagrange's
equations**, one has then

*T* is a **positive definite
form****,**** **just
as at a stable equilibrium *V* - *V*_{0}, whence *a*_{n} and *b*_{n}*
*are positive. We obtain for every normal co-ordinate a stable oscillation

altogether just as many modes of vibration as degrees of freedom. During the transition ,
there corresponds to every *w*_{n}
an eigen-value *k*_{n} and to all the *q*_{1}, ··· , *q*_{r},
belonging to the individual *x*_{n}, the corresponding eigen-function *u*_{n}.
Just as the *w*_{n},
the *k*_{n }are **real**.

This property of *k*
being real, as we note aside, can also be proved directly from the differential equation. Assuming that one *k* is complex,
then also the associated *u* would be complex and the conjugate function *u*^{*} should also satisfy the
conjugate equation and the boundary condition *u*^{*}
= 0. By Green's theorem,

the right hand side of which vanishes due to the boundary conditions, whence

However, *uu*^{* }is always
the integral can also not vanish, whence one must have *k*^{ }= *k*^{*}, i.e., *k* must be
real. Obviously, the meaning of *k* in physical terms is that the oscillation process in view of the initial assumptions is
free of absorption.

Hitherto, we have assumed that our problem is **non-degenerate**.
However, in the **theory of ****perturbations
of wave mechanics**, just the degenerate cases are of interest. We return to our
example of the rectangle and assume that the sides *a* and *b *are no longer incommensurable. This is very much the
case for the **square ***a* = *b*.
In fact, then, by (2a),

however, by (2), i.e.,

and, in contrast,

unless *n *= *m*. Hence, all oscillations are
(at least) **simply degenerate**, since
there belong for them to the same *k*_{n m}* *two different vibration types *u*_{n m }and
*u*_{m n}. Only the fundamental mode *k*_{11 }and its (in this special case
harmonic) overtones *k*_{n n }= *nk*_{11 }are **not
degenerate**.

Consider in more detail the cases *n* = 1, *m* = 2 and *n* = 2*, m*
= 1, so that . We characterize the associated
eigen-functions in Figs. 23 and 24 by their **nodal lines**,
which are the lines *u* = o along which during a dust experiment the applied powder accumulates. At the same time with *u*_{12
}and *u*_{21}, there belongs to the eigen-values *k*_{12 }= *k*_{21 }also
the eigen-function

where *l *is*
*an arbitrary constant. By continuously changing it, also the form of the nodal line inside the family
(7) are changed continuously. For example, we compute the linear combinations

The last line shows that there belongs as nodal line to the combination *l
*= -1 the diagonal *y = x, *and to *l
*= +1 the other diagonal *y* = *a - x. *For arbitrary values of *l*,
Fig. 24 shows the overall picture of the nodal lines.

However, in the case of the quadratic membrane, there occur under certain conditions not only simple, but also higher degeneracies. For example, let

then there belong to the eigen-value

4 linearly independent eigen-functions

Hence we are dealing here with a **threefold**
degeneracy. The higher degeneracy depends here on whether a number can be decomposed in
several ways into sums of squares, as, for example, in the case

According to Gauss' **Disquisitiones
arithmeticae**, this is the case for all numbers, among the prime factors of
which there are at least two of the form 4*n *+ 1. In fact, such prime numbers allow the complex decomposition

with integers *a*, *b*; and the different combinations of the complex factors
yield different representations of sums of squares. Thus, in our example, 65 = 5·13

whence

For any two eigen-functions *u*, *u*' with ,
one has obviously the **orthogonality theorem**

as a consequence of Green's theorem. The proof is the same as for (6), when one writes there u'
instead of *u*^{*}. But the conclusion fails, when *u* and *u' *are individuals of a degenerate state
of vibration, i.e., when *k* = *k*'.

In order to avoid troublesome distinctions between cases, is is now obviously suggested to enforce orthogonality also in the case of degeneracy. It then proves to be expedient to abbreviate the integral in (8), following Courant-Hilbert* to

We will return to the connection of this mode of representation with the scalar product of
ordinary vector analysis in 5.26.
However, already now we will call the integration defined in (8a) **scalar
multiplication****.**

* Courant-Hilbert . Methods of Mathematical Physics, Chapt. II, 2-nd. ed., Springer, Berlin, 1931.

We mention ahead of its discussion the theorem that *continuous, real, mutually orthogonal,
but otherwise arbitrary functions u*_{1}*, u*_{2}*, ··· , u*_{n}*,
··· , u*_{n}* are linearly independent*. In fact, if there were to exist an equation of the form

for all ,
then there would result by **scalar multiplication **of
the first of these equations with *u*_{m},
due to the second equation,

in contradiction to the assumption.

We now proceed step by step and treat, to start with, the case of **simple
****degeneracy**.
Let *u*_{1}, *u*_{2}* *be the two continuous, real, not mutually orthogonal functions of the
same eigen-value. As in (7), we consider the family

and seek in it the function which is orthogonal to *u*_{1}. It is given by the
condition

We meet this condition by setting

where Thus, we have
with *u*_{1} and *u *two mutually orthogonal functions of the family, which we can choose instead of *u*_{1},
*u*_{2}* *as representatives of the family. Subsequently, we can still normalize just by multiplication by a
factor in such a way that there will be

In the case of **double degeneracy**,
let *u*_{1}, *u*_{2} , according to (9) and (9a), be two orthogonalized and normalized functions, *u*_{3}*
*a function which is not orthogonal to them, but has the same eigen-value. We consider the family

and seek in it the function *u*, which is orthogonal to *u*_{1}
and *u*_{2}*,* i.e., we demand that

We meet both these conditions by setting

The three functions *u*_{1}, *u*_{2} ,* u* are, by the
above theorem, linearly independent, because they are orthogonal; moreover, *u* can be normalised subsequently so that

Thus, the required orthogonalization has here been obtained also for two-fold degeneracy.

Obviously, this method can also be continued in the case of higher degeneracy. In this manner,
the degenerate eigen-functions are made mutually orthogonal; by (8), the other eigen-functions of different *k* are anyhow
orthogonal.

Moreover, we place side by side with the orthogonality condition (8) the **normalization
condition**

This **normalization to 1**
yields a certain simplification of the above orthogonalization method, cf., for example, (10a). Also the notation (11) has, as we
will see in 5.26,
its vectorial analogue. We note yet that in the case of the complex notation of the eigen-functions, Equation (11) is to be
changed to

and that, in cases of separable problems, the normalization is done more conveniently for the individual factors. Hence, one has in (2) to add to the sin functions the factors

and in (3) to the exponential function, or, by (20.9a), to the Bessel functions, the factors

Our examples in (2) and (4) are therefore also determined with respect to the magnitude of their amplitudes.

Finally, we derive from these examples **two
theorems** relating to nodal lines, which we will now prove for any, arbitrarily
bounded membrane:

**1.** *I**f
there pass through a point several nodal lines, then they intersect at equal angles (***isogonal***);
in the case of two such nodal lines, the angle is **p**/2,
in the case of **n** nodal
lines **p**/**n**.*

**2.** *The
larger is the eigen-value k, the finer becomes the subdivision of the membrane into regions with alternating signs; for ,
the nodal lines everywhere approach each other more closely.*

With respect to** 1**, we refer, for example, to the entered angles *p*/2
and *p*/4 in Figs. 23,
24, in that obviously there the boundary has to be viewed as nodal line, or to the not
shown nodal line panorama of the full circle, at the centre of which, writing (3) in real form, there intersect *m* radial
nodal lines at the angle *p*/*m*.
Regarding **2**, it is sufficient to point out that the rectangle in the case of the eigen-value *k*_{m,n}*
*is subdivided into sub-regions with sides *a*/*n* and *b*/*n *of which, as ,
at least one becomes infinitely small.

In order to prove **Theorem 1**, we expand *u* in the neighbourhood of a
point *O *in a Fourier series. We employ a co-ordinate system *r*, *j
*with origin at *O*. For every shape of the rim of the membrane, we find in a
certain neighbourhood of *O *the convergent expansion

with certain coefficients *a*, *b*, which can be computed from the
given *u*. It follows from the differential equation (1) and the regularity of *u* at *O *that there must
enter as radial functions of the Fourier expansion the Bessel functions *I*_{n}. If there is to pass
through the point *O *(*r* = 0) **at least****
**one nodal line, then one must have, by (13),

If then *a*_{1} and *b*_{1 }are not both
simultaneously zero, then there passes indeed through *O ***only **one
nodal line, the direction of which is determined by the equation

from which follows, for *r* > 0,

Hereby, the direction of the nodal line is determined **uniquely****.**

If there are to pass more than one nodal line through *O*, one must have *a*_{1}
= *b*_{1} = 0. If then not simultaneously *a*_{2} = *b*_{2} = 0, one has by (13)

If *n*
nodal lines are to pass through *O*, one must have correspondingly that all *a*, *b *up to but not including *a*_{n}*
, b*_{n} vanish

In the last case, one must impose for *r* > 0 the condition

The right hand side of this equation is given by our Fourier expansion and could immediately be
set equal to tan* a*. We then
obtain from (13a) the general solution

Thus, the angles differ by the constant amount *p**/**n**,*
and the isogonality has been proved.

Turning to the proof of **Theorem 2**,
we consider two functions *u*, *v*, of which *u* is a solution of (1) for the given boundary curve and
condition and *v* the special solution

Assume that the common value *k *for *u* and* v *is large.
From this large *k*, we define a small length *a* by setting *ka* = *r*_{1},
where *r*_{1 }is the
first root of the equation *I*_{0}(*r*)
= 0. We cut out a circular disc with radius *a* and place it anywhere on the nodal line image of our eigen-function *u *(Fig.
25). We apply to this disk as region of integration Green's theorem:

The left hand side vanishes, since both *u* and *v *satisfy the
differential equation (1) with the same *k*. On the right hand side, one has for *r* = a

Thus, setting *ds* = *ad**j*,
it follows from (14) that

Thus, *u* is on the circle's periphery partly positive, partly negative,
whence there must be at least two zeroes of *u* on the periphery, i.e., our test disk is intersected by at least one nodal
line. The larger is *k*, the smaller becomes our test disk. Thus, the nodal lines move together arbitrarily much with
growing *k*. This result applies to each location of the image of nodal lines.

**5.26
General Remarks about the Boundary Value Problems of Acoustics and Heat Conduction:****
**The eigen-functions of the oscillating membrane can be transferred readily into
space. However, we are not thinking here of a vibrating solid body, but, in order to avoid all vectorial and tensorial extensions,
of an oscillating mass of air contained inside a closed rigid cover of finite dimensions. As in 4A1.2,
we understand by the scalar function *u* the velocity potential of the air vibration and use, like there, the boundary
condition

Hence, we must set for the parallelepiped with side lengths *a*, *b*,
*c *instead of (25.2)

with the eigen-value

The state is not degenerate, when *a*, *b*, *c *are
commensurable.

For the sphere of radius *a*, we have instead of (25.3)
as most general eigen-function

The eigen-value is for the present boundary condition

where *k*_{nl} is the *l*-th root of this equation. The state is
2*n*-fold degenerate, since *k*_{nl }does not depend on *m*, whence there belong to the same
*k*_{nl }all states , specified
by the superscript of *P*^{m}_{n}*.*

Here enter also the in 4.20.3
derived eigen-functions of the circular cylinder (0 < *r* < *a*, 0 < *z* < *h*), which,
transferred to the boundary condition , are
given by

the associated eigen-value follows from the condition *I*_{n}*'*(*l**a*)
= 0, the *l*-th root of which is denoted by *l*_{nl},
at

Due to the factor *e*^{±in}^{j}*
, *the state is simply degenerate.

Moreover, we imagine these eigen-functions to have been **normalized
to 1**, where we must take into consideration the earlier
remark on normalization and on degeneracy.
For example, one must set in (1) instead of cos *n**p **x*/*a*

and in (2) instead of *P*^{m}_{n}, by
(22.31b),

etc. (Exercise 5.1).

We will now transfer our fundamental
theorem and its (mathematically insufficient) proof to an arbitrary space region *S*.
Then, the theorem is:

*There exists an infinite system of eigen-functions*

*the individual members of which are regular within S and satisfy the equation*

*as well as a homogeneous boundary condition. The associated eigen-values, say
arranged by order of magnitude,*

*form a not truncating series to infinity; they yield a discrete spectrum, when
S is not infinite and are real, since the differential equation was assumed to lack absorption.*

This system of eigen-functions meets the **orthogonality
and normalization condition**

which, thanks to earlier work, we can abbreviate to

or, more general with the often more convenient complex notation of degenerate eigen functions,

Assuming that the system of the *u*_{n}* *is
**complete**,**
**we claim that a function of position, arbitrarily prescribed in *S*, for
example, a continuous function *f*, can be expanded in
terms of the *u*_{n}:

If this expansion is possible, one obtains by term by term integration of (5), with (4b),

Its possibility is postulated by the **Ohm-Rayleigh
Principle**, on which we base, without mathematical proof, the following work. We
note regarding the names: Georg Simon Ohm (1787-1854) was not only the discoverer of the fundamental law of **galvanic
conduction**, but also a penetrating explorer in the acoustic area. He reduced
the varying sound colours of the different **musical
instruments** to the relative strength of the mixture of the fundamental and the
overtones. Since, by (25.1),
the overtones *w*_{n }are
related to the eigen-values *k, *and they lie only in the string and the organ pipe harmonically with respect to the
fundamental, the production of an arbitrary sound colour implies the construction of an arbitrary function out of the, in general,
an-harmonic eigen-values. In the classical text on the Theory of Sound of Lord Rayleigh (1842 - 1919), this principle has been
extended in the sense of Equation (5) and exploited in all directions.

We now should explore briefly the so-called **Hilbert
space** and in doing so not only set up a foundation for the notation (*u*_{n},
*u*_{m}) in (4a, b), which recalls vector analysis, but also give the Ohm-Rayleigh principle an elegant
geometrical interpretation, which in the hands of the Hilbert-school has even been developed into a proving mechanism for this
principle.

With Courant-Hilbert, we
define in a space of *N* dimensions **base vectors****
**(corresponding
to the *i*, *j*, *k *of ordinary vector calculus) which point in the co-ordinate directions ,
and their scalar products, which are to satisfy the conditions

Moreover, we consider a vector

directed at the angles *a*_{1},*
a*_{2}*, *···
with respect to the co-ordinate axes and call it a **unit vector**,
when the scalar product of with itself, formed after the
rule (6), has the value 1:

A second such vector
with the direction angles *b*_{n}*
*is* *said to be **orthogonal **to
when the correspondingly formed scalar product of
and has the value zero:

As one sees, Equations (7a, b) are generalizations of the known formulae of analytic geometry in space.

We go now by the limit
to the **Hilbert space**. In the process,
we see a formal analogy between the base vectors and the
individual *u*_{n}* *of our system of eigen-functions. The relations between the latter, written in
the form (4a), are formally the same as Relations (6) between the .
The system of the *u*_{n}* *can, if it is complete, serve as replacement of the base units .
Of course, the same applies in the case of complex *u*_{n}* *of the system of the *u*_{n}^{*}.
Every other orthogonalized and to unity normalized system of functions can be composed in the sense of Equation (7) from the *u*_{n}*
*and be illustrated by a **vector **in
Hilbert space. Two such vectors transcend by a **rotation **in
Hilbert space into each other. However, also an arbitrary function *f* is composed, according to (5), of the *u*_{n}.
In the co-ordinate system, formed by the *u*_{n}, Equation (5) allots to the function *f* a
definite **point** in Hilbert space as
its representative. The co-ordinates of this point, measured in the system of the *u*_{n}, are the
expansion coefficients *A*_{n}. The Hilbert space thus becomes a **function
space**. The allotment between the arbitrary functions *f* and the points
of the space of infinitely many dimensions is unique and reversible. If we link the point, which represents the function *f*,
to the origin of the co-ordinate system of the *u*_{n}, then this vector (with infinite dimensions) is a
representative of the function *f*. By (5a), which we can write in the form *A = *(*f*, *u*_{n}^{*}),
the co-ordinates of the representative point are at the same time the projections of the representative vector onto the axes of
the system of the *u*_{n}^{*}.

We now return from these highly abstract generalizations to the physical
applications. We will confine ourselves for the present to the simple problems of **acoustics
and heat conduction** in the historically customary form. We will leave the
actual questions of wave mechanics until the end of this chapter.

The general problem of **acoustics **for
an arbitrarily shaped space *S* is as follows: Integrate the wave equation

with the boundary condition
so that are equal for *t* = 0 to arbitrarily
given functions *v* and *v* of a location in *S*. We solve this problem by setting

where *A* and *B *must be determined
so that

The second of these equations can be rewritten, due to the connection of the *w*_{n}*
*with the eigen-values *k*_{n} (namely the relation *c* =
*w*_{n}/*k*_{n}):

Hence, as in (5a),

We see that we are here concerned with the solution of an initial value problem,
while the solution of the actual **boundary value problem**
is passed on to the *u*_{n}.

One can solve the general heat conduction problem in the same way. The only
difference is that now **one**
arbitrarily chosen *v*_{0} is sufficient to describe the initial state, while the initial temperature change
is already fixed by the differential equation of heat conduction. We can employ as boundary conditions one of the three
homogeneous conditions a), b), c) in Chapter III,
to which then also the system of eigen-functions *u*_{n} must be subjected.

We now start with

*k* is here the **heat
transfer** **ability**,
not, as before, the thermal conductivity. The coefficients *A*_{n}* *are again determined from the
initial condition *v* = *v*_{0}:

Apart from satisfying the initial condition, (10) also satisfies the differential equation (25.1) and the imposed boundary conditions.

The potential equation *D**u
*= 0 has no eigen-functions; in fact, everyone of its regular solutions in S for the boundary condition ,
regular in a region *S* required inside*, *is equal to zero, const, respectively (Vol. II, p. 24). For this reason,
there cannot exist **nodal surfaces **.
However, we will construct below a solution of the general potential boundary value problem (given the value of *u = U* at
the boundary), which is constructed out of the eigen-functions of the wave equation.

Moreover, a regular solution *S* of the potential equation cannot have a
maximum or minimum inside S*. *In fact, extreme values of *u* can only lie on the boundary of *S*. This
follows from Gauss' theorem of the **arithmetic mean**,
which is readily proved with Green's theorem; cf. the end of the hint,
Exercise 2.2.

**5.27
Free and Forced Vibrations. Green's Function of the Vibration Equation:****
**The eigen-functions correspond to **free
**vibrations; they do not require a supply of energy from a non-absorbing medium. We
will now deal with **forced** vibrations,
which must be excited in the rhythm of their period, in order to be able to continue pure periodically. They must, like the free
oscillations of a homogeneous surface condition, satisfy, for example, the equation *u* = 0; the region *S*,
surrounded by the surface Chapter
V, will here be assumed to lie entirely in a finite part of space. The measure of the
excitation is to be at first given by a continuous function of location somewhere inside *S* and denoted by *r
*with reference to Poisson's equation of the potential theory.*
Correspondingly, we write the differential equation of forced vibrations

In the acoustic case, *k* is computed out of the circular frequency *w
*of excitation and the velocity of sound *c, *as noted by (26.9a),
i.e., *k* = *w*/*c*.
In doing so, we assume that

i.e., that it is another than everyone of the eigen-values of the domain *S* for equal
boundary conditions. We will treat **resonance**
*k* = *k*_{n} at the end of this section.

* *r* is
not, as in potential theory, a load density, but has the dimension sec^{-1}, when *u* is an acoustic velocity
potential.

According to the **Ohm-Taylor
principle**, we expand *r *in
a series of the to 1 normalized *u*_{n} as in (26.5)
and (26.5a):

and also give the solution of (1) the same form:

**.**

Entering both expansions in (1) yields, taking the differential equation *D**u*_{n}*
+ k*_{n}²* u*_{n}* = *0, which differs from (1), into account, by comparison
of the coefficients of *u*_{n }on the right and left hand sides

We now specialize *r *into
a *d*-function, i.e., concentrate
the so far imagined continuous excitation at a single source point *Q *with strength 1. Then,

if the region of integration contains the point *Q* and, in contrast, for every region
which does not contain the point *Q,*

Hence, by (3),

whence (4) yields

On the left hand side, we have replaced the notation *u*, used hitherto, by the
informative *G*(*P*, *Q*). Indeed, the
solution, thus obtained, is the *Green function of
our differential equation (1) for arbitrary location of P and the source point Q and arbitrary shape of S*. We have only
assumed here that the complete system of eigen-functions and -values of S is known.
Note, however, that the Ohm-Rayleigh principle by no means was called upon for the singular *d*-function,
but only for the continuous function *r *in
(3), which, for example, can be set up as a regular **Gauss
error function**. In other words, our derivation is not at all concerned with the
expansion of a general arbitrary function in terms of the *u*_{n}, but only with that in special,
throughout regular functions. Similarly, the term by term differentiation, undertaken for the derivation of (4), was executed on
the function (3a), regular ahead of the limit transition, not on the limit formula (5).

Our Green function is at the same time the solution of an **integral
equation**. In order to demonstrate this, we recall (10.13a)
which is valid for every self-adjoint differential expression *L*(*u*), but especially also for those of the wave
equation *D**u + k*²*u
*and for initially given boundary conditions in 3 dimensions

*G*(*P, Q*) is called the **kernel
of the integral equation**. Corresponding to the reciprocity
theorem d), which is to be rewritten in the complex notation of the eigen-functions into

*G* is specially characterized in the terminology of integral equations as **symmetric
kernel**. Indeed, one sees from the structure of (5) without difficulty that (6a)
is met by (5).

Certainly, the convergence of Series (5) is absolute only in the one-dimensional case; in two and more dimensions, it is conditioned solely by the change in sign of the eigen-functions in the case of a suitable choice of their sequence. For this reason, Equation (5) does not occur in Hilbert's general theory of integral equations, but only in an integrated form in which it converges absolutely. It has been proved strictly in the one-dimensional case by Erhardt Schmidt in his famous dissertation, Göttingen 1905.

The unsatisfactory convergence of (5) appears when one tries to prove the
fulfilment of the differential equation (1) by term by term differentiation. In fact, then the *n*-th term yields

Here, the sum on the right hand side has for *P = Q *only positive terms and
diverges, as it should; the fact that it converges for
and vanishes throughout is only caused by changing signs and cannot, of course, be proved out of this representation. By the way,
the degree of becoming infinite for follows directly
from the differential equation (1): *Describe about Q with the small radius r a sphere and integrate (1) over its interior*.
On the left hand side, by Gauss' theorem, the first term becomes

while the second term vanishes, whence,

This is the direct expression for *G*(*P*, *Q*) so that it has
a **unit source****
**at the point *P = Q.*

The preceding formulae are interpreted especially easily
in **Hilbert space**. In fact,
Equation (6) says that *D**G +
k*²*G *is in this space the scalar product of the two unit-vectors *u*(*P*) and *u*^{*}(*Q*).
The last are mutually orthogonal in as far as *u*(*P*) and *u*^{*}(*Q*) differ from each other
(); if they are mutually conjugate complex (*P = Q*),
orthogonality is, of course, impossible; the place of orthogonality is taken by the going to infinity of this product. The
expression (5) is formed out of the individual terms of the same product with the **resonance
denominator**** ***k*²
- *k*_{n}²* *as weight factor.

Despite their bad convergence, Equation (5) has proved often to be of value in wave mechanical calculations (cf. 5.30). We will use it here for filling a gap in the theory of spherical functions. We prepare for this work with a few general observations.

**1.** If the system of eigen-functions is **separable**,
for example, in orthogonal, curvilinear co-ordinates, which are fitted to the surface *s*
of *S*, then the sum in (5) decomposes into three sums, corresponding to each of the three co-ordinates. For example, in
the case of a parallelepiped, one would have with the numbers *n*, *m*, *l*, used in (26.1),

so that one can distribute the *x*, *y*, *z *components of the
eigen-functions in the sums with respect to *n*, *m*, *l.*

**2.** The Green function depends only on the position of the two points *P*
and *Q* relative to the boundary *s*
as well as their mutual distance *R*. In contrast, it must be independent of the orientation of the co-ordinates in space.
During a change of the co-ordinate system, in which the surface *s *is
transformed into itself and *R* remains unchanged, *G*(*P, Q*) is **invariant**.

**3.** If *s *is
a spherical surface, the demand for invariance is met during every rotation of the **spherical
polar co-ordinate system **with
*r* = 0 as centre of the sphere. **
**are** **the
co-ordinates of *P, *those of* Q.*

**4.** In this case, the system if eigen-functions (26.2)
is **degenerate**, since the eigen-value *k*_{nl}
does not depend on *m*. Hence, we can move from the three-fold sum, corresponding to (8), the denominator *k*² - *k*_{nl}²
outside the summation over *m *as well the radial components of the eigen-functions, whence one can write

Here, *Y*_{n}*
*is the function *y*_{n}
of (26.2), normalized to 1, *P*_{n}*
*in (9a) the equally so normalized spherical function *P*_{n}, *Y*_{n}
is a spherical surface function. In (9a), we have employed the fact that the eigen-function

**conjugate complex ****to the **eigen-function
of the argument *, *can be written, as a result
of *Y*_{n} and *P*_{n}*
*being real, in the form

valid for all positive and negative *m* between -*n* and +*n*.

From our observation (2) regarding the invariance of *G* and the representation (9), we
see now that the spherical surface function (9a) has an invariant meaning, independent of the rotation of the polar co-ordinate
system. However, this is exactly the theorem, which we have assumed earlier
axiomatically. The proof now has no gaps.

Hitherto, we have assumed that the excitation of forced oscillations **inside
***S, *especially in the case of Green's function, occurs at a single point *Q*
of it. We will now assume that is occurs from the **surface**.
This is the case when we prescribe instead of the homogeneous boundary condition *u*=0 the **inhomogeneous***
*one

The surface is then held in pulsations in the rhythm *w
*of the oscillation to be enforced with the amplitude *U* changing from point to
point, while inside *S* the differential equation (1) is to apply with *r* = 0. From (10.12),
we know that the boundary value problem is solved with the aid of Green's function *G*(*P*, *Q*) by the
formula

where on the right hand side one has to integrate after the point *P*,
ranging over the surface (*d**s*_{P}=surface
element, *d**n*_{P}*
*= normal element at the point *P*). By (5), Equation (11) then yields

This formula contains at the same time the general solution of the famous **Dirichlet
problem of potential theory**, if one sets *k* = 0, i.e.,

The strange part of this solution is that it is not expanded in terms of the
particular solutions of the differential equation *D**u
= *0, but in terms of the eigen-functions of the **wave equation**
( we know that the potential equation does not have eigen-functions!). Equation (12) is also maintained, if one prescribes instead
of the boundary condition (10) the more general condition

In the special case of the sphere of radius *a*, one obtains from (12) and Condition (10)
with the earlier meaning
of *Y*_{n} and *P*_{n}:

The additional factor 2*p* on
the left hand side of (13) arises, because, as with Bessel and spherical functions, also the two functions *
*in (13) and (13a) had to be normalized to 1.

The solution, expanded in the usual manner in terms of the particular solutions of *D**u
*= 0, is written in the same variables
of *Q,*

A comparison of the two formulae yields strange summation formulae [Summation over the complete
system of the roots *k* of the equation ] (Exercise
5.3).

Finally, we must still treat the exceptional case *k* = *k*_{m}*.
*One knows from Mechanics and Electro-dynamics of oscillating systems of the **resonance
catastrophe**: *In the case when the rhythm of the exciting force and an
eigen-frequency of the system are the same, the amplitudes grow to infinity.* The condition for this is *w
= w*_{m}, i.e.,
*k* = *k*_{m}. Then Equation (1) becomes

Thus, we are now concerned with an inhomogeneous equation, the left hand side of which agrees with the homogeneous equation of a free vibration.

For the sake of clarity, to start with, we will step back here to the
two-dimensional case of the membrane in 5.25,
which, however, now is exposed to a periodically changing, transverse pressure *r
*=* r*(*x*, *y*),
distributed in any way over the membrane. Do
there exist pressure distributions during which the resonance catastrophe is avoided, i.e., for which Equation (15) admits an
unconditional continuous solution (for the prescribed boundary condition *u* = 0)? The answer to this question is
physically obvious: The pressure must **not**
perform work on the membrane; **it must act ****Watt-freely**,
i.e., one must have

The pressure distribution must be orthogonal to the eigen-function *u* = *u*_{m},
resonating with it, for example, it must have in oppositely vibrating sections of the membrane the same magnitude; especially, the
pressure may attack along a nodal line with arbitrary strength.

More exactly: The pressure divided by the surface tension *T*, by which we divided during
the formulation of (25.1).
The dimension of *r *is not that
of a pressure dyn/cm², but: (dyn/cm²)/(dyn.cm) = cm^{-1}.

This orthogonality theorem
is a corner stone of the **theory of integral equations**
and has important applications in **wave-mechanical
perturbation theory**.

It transfers directly to the three-dimensional case, if we replace in (16) the area integration
over *d**s *by space
integration over *d**t**.
*We then see that the expansion coefficients *A*_{n} and *B*_{n} in Equations
(3) and (4) vanish for *n = m*. Moreover, by changing over the continuous distribution *r
*to a point-formed *d*-function,
we arrive at at a statements regarding the Green
function in the case of resonance. In fact, it follows from *A*_{m}
= 0, by (4a), also *u*_{m}^{*}(*Q*) = 0. In words: *The singularity to be
prescribed in the Green function must lie on a nodal surface of the critical eigen-oscillation u*_{m}*.*

For this and only for this location of the source point *Q*, there exists a Green
function which is everywhere regular. Its form, specialized for the case of resonance, follows from the general form (5) by
omission of the term in the sum, which relates to *k*_{m} and is therefore simply: