Reflection in a circular cylindrical or spherical mirror
With reference to Fig.8 and its notation, we advance further the problems, initiated in 1.6.
4A1.1 Circular Cylindrical Metal Mirror: Let the incident wave (electric vector perpendicular to the plane of the drawing) be (cf. 21.2b)
This representation is valid in the entire r, j -plane and defines for r = a the function -f(j), introduced in (1.6.4). The sum on the right hand side of N + 1 terms is the best possible approximation by the method of least squares to w; the fact that the coefficients of this sum are the same as in the non-truncated series (21.2b) follows from the final validity character of the Fourier series. We write for the radiation, reflected by the mirror (inflected and diffracted), the sum of N +1 particular solutions of the differential equation
for r < a
since it must be continuous for r < 0, only the In had to be taken into account; the sin terms drop out due to the symmetric character with respect to j = 0 of the incident wave. We could add the denominators In(ka) for the sake of convenience, since they only affect the meaning of the still available constants Cn.
We will describe the radiation, refracted outwards (r > a) by the mirror, by the sum of N + 1 particular solutions of the wave equation
The time dependence of the entire process is then imagined to be given by e-iwt, for which reason only H1 had to be included; H2 would correspond to incident waves. The boundary conditions (6.8 - 11) then yield Cn = Dn (as has already been discussed in 1.6) and by the method of least squares the system of linear equations (6.12). According to (6.7, 16.11a) and the preceding equations (3, 4), the constant gn is
According to a known theorem of the theory of differential equations, one can write for this more simply (details in Exercise 4.8)
With the notation
we have after a simple reorganisation
Here and in the sequel, the symbol (2) means the number 2 for n > 0 and, in contrast, the number 1 for n = 0. Then the left hand side of (6.12) becomes
and the right hand side of the same equation with f(j) = -w, where we take w from Equation (1) and set r = a:
Hence, (6.12) becomes the system of equations
which must be satisfied for all m = 0, 1, ··· N.
In order to control it, we consider first of all a = p, i.e., a closed circle (in space, a closed, completely conducting cylinder mantle). By (6), one has anm = 0 and (9) yields
This result is trivial. In fact, one finds by substituting into (3) instead of Dn the value (10) of Cn, for r>a, the strict solution v of the dispersion problem under consideration:
i.e., a radiating wave, which just erases on
the cylinder mantle the incident wave w of (1) and
therefore indeed yields for 
the strict solution of
the dispersion problem. In the same manner, one obtains for u
the expression
Also this result is trivial, because inside the closed circle r = a one must naturally have u + r =0.
We will examine whether (10) leads also for a < p to a useful solution. Thus, we set, understanding by b a correction term,
and obtain
Like (16), this is a system of N + 1 (in the limit, infinitely many) linear equations with which we practically cannot do anything. However, if we assume that p - a is small, i.e., the mantle of the cylinder has only a narrow slot, then anm becomes a small quantity and the product bnanm small to the second order. If we neglect this, (14) yields simply
i.e., an explicit expression for bm ,which, by (13), leads also to such a value for Cm.
An estimate of the appropriate size of the slot follows from physical considerations; its width must be small compared with the wave length of the incident radiation; only then, the field inside joins continuously the field zero of the closed cylinder. Thus, one must have
This condition is only met as an approximation in the case of Hertz-waves. Our approximation fails in the optical case, which is the reason why we spoke in 1.6.3 of a quasi-optical case. During Hertz's known concave mirror experiment, during which he had approximately a=p/2, l=200 cm, a = 50 cm, Equation (16) is approximately met, so that here our approximate method is more or less justified.
4A1.2 The segment of a sphere as
an elastic reflector: In
order to avoid all extensions of a vectorial kind, we will treat
in the space case instead of directed optical radiation scalar
acoustic waves.
We must then understand by w or rather u, v
the velocity
potentials of the
primary and secondary (reflected) radiation inside (r
< a) or outside (r > a) of the
sphere. Let the spherical element be given by 
Following
(24.7), we
write
by the method of
least squares, this is the best possible approximation to a plane
wave e
by
a finite number of spherical functions . Moreover, we write with
the available constants Cn, Dn
Due to the presence of the, for the sake of convenience, added denominator (cf. Equation (2)), zn is the half-integer Bessel-function, defined in (21.15), which corresponds to the Hankel-function H1n. At the element of the sphere (assumed to be rigid), one has
moreover, for reasons of continuity, one must have
These conditions lead again to D = C, as well as the analogous to (6.10) and (6.11) conditions
where 
is the value 
for r
= a,
and analogous to (5)
With the abbreviation
,
one computes
and from (20a)
The least squares method now yields, in analogy to (6.12), the system of equations for the Cn:
valid for m = 0, 1, ··· , N; due to (21a,b), it becomes
We can reorganize this, as in (9), into
We start again with the limiting case a = p of the closed sphere, for which anm = 0. Then, (23) yields
This value of Cm yields, on substitution into (19), the strict solution v of the problem under consideration for r > a, namely the reflection of an incident plane wave on a complete sphere. Inside the sphere, for r < a, one obtains, substituting Cm in (18), a field u which, as it must be, is opposite to the field w of the incident wave.
Here follows the problem of a spherical surface with a small
circular hole near 
With
the starting expression
(23) yields with neglect of the second order terms anmbn
i.e., an explicit computation of the corrections bm and thereby of the coefficients Cm. Compare this result with the analogous result in Equation (15) for the cylinder problem! Just as there, the width of the slot, here the diameter of the circular hole, must be small compared with the wave length of the incident radiation. Thus, we could also here only deal with a quasi-acoustic problem, a problem of the infra-sound, which is far away from the more interesting problems of the ultra-sound.
Our attention with the somewhat sketchy presentation of this appendix was only to show that, under certain circumstances, the method of least squares is also then successful, when our final validity demands respecting the coefficients Cm are not met.
Supplement to Riemann's problem of sound waves in 2.11
We shall fill here a gap in the work of 2.11, i.e., we shall prove that in the expression (11.10)
where F was the hyper-geometric series, the differential equation resulting from (11.2, 11.3)
is satisfied. While Riemann was able to justify this through his general transformation theory of the hypergeometric series, we must proceed here in an elementary manner so that we view in (1) F to be an unknown function of the given argument and substitute (1) into (2), in order to derive a differential equation for the function F(z). By showing that this equation is identical to the differential equation (4.24.20) of the hyper-geometric function, our starting expression (1), with all its specifications regarding the parameter of the hyper-geometric function, will be confirmed.
First of all, we evaluate, following (1),
and from there the sum
As first term of (2), we then find
The first two terms of (4) join with or cancel, respectively, the two terms of (3). Hence, (2) yields, after cancellation of a common factor,
However, the present differential quotients of z can be expressed in terms of z as follows:
Using (5) and (6) and removing a common factor, (5) becomes
This is now indeed Equation (24.20), if, as was assumed in (1), one sets
This concludes the problem of 2.11.