2 Background .1 Cosmological .11 Mathematics

Exercises From mat/


1.1 Position of maxima and minima in special Fourier approximations: Show that the extrema of the Fourier approximation

are located at equal distances and, apart from the all S common extremal position x = p/2, between those of S2n+1.

1.2 Summation of certain arithmetic series: Compute the higher analogues to Leibnitz' s series (2.8) and the series S2, S4 in (2.18).

1.3 Expansion of sin x in a cos series: Expand sin x between 0 and p in a cos series.
a) continuing sin x to the left hand side for -p <x < 0 as even function or
b) by setting in (4.5) b = 0 and c = p.

1.4 Spectral decomposition of certain, simplest time histories: Compute the spectra of the time histories presented by Figs. 33a, b from their Fourier integral and show them graphically; also show the spectrum of a wave sin 2p t/t, which is truncated on both sides and extends from t = -T to t =+T, T = nt, Fig. 33c, and explain thereby that a spectral line appears to be the wider the shorter its duration. Hence, an absolutely sharp, fully mono-chromatic spectral line would have, by assumption, on both sides an infinitely long, totally undisturbed sin wave.

1.5 Examples to the method of complex integration: Justify the result of 1.4a (Dirichlet's dis-continuous factor) using the method of complex integration ; moreover, decompose the spectrum of the sin line, bounded on both sides, into those of two wave trains, bounded on one side.

1.6 Compute the first Hermite and Laguerre polynomials using the method, applied to the spherical functions, from their orthogonality conditions employing standard normalization:

Highest term of Hn(x) equal to (2x)n.
Lowest term of Ln(x) equal to n!

Cf. for the definition of these polynomials the table.

Chapter 2:

2.1 Elastic bar, open and closed pipe: Compute the transverse eigen-vibrations of a bar of length l, clamped at x = 0 and free at x = l, and compare your result with the eigen-vibrations of an open and closed pipe.

2.2 Second form of Green's theorem: Develop for the general elliptic differential expression L(u) the analogue to the theorem of potential theory

a) when L(u) has been given the normal form,
b) in the general case.
c) Examine the uniqueness for a self-adjoined differential expression L and the conditions under which the solution of the boundary value problem is unique.

You may confine yourself to two independent variables.

2.3 One-dimensional potential theory: Determine the one-dimensional Green function from the conditions

and apply it to the (obviously trivial) solution of the boundary value problem

Condition c) means: Output of the source of G located at x = x ; G+-is the branch x > x, G the branch x < x of G.

2.4 Application of Green's methods, developed for heat conduction, to the so-called laminar current about a plate by an incompressible fluid with friction: We assume that the current is plane and throughout along a straight line; this means that it is independent of the third co-ordinate z and directed along the y-axis. The velocity v has then only the single component vy=v, which due to the assumed incompressibility does not depend on z and y, so that there vanish the quadratic convection terms (v grad) v. The Navier-Stokes equation for v with kinematic viscosity k is then

the right hand side is here independent of x due to the corresponding equation for the vanishing x-component of the motion, i.e., it is a pure function of t, say, a(t).

Let the flow be bounded at x = 0 by a solid plate which was at rest until t = 0 and then was set into motion according to v0(t). Since the fluid sticks to the plate,

For the straight line Couette flow, we would have to add further boundary conditions at a plate at rest, which would be at a finite distance from x = 0, which, however, we displace, for the sake of simplicity, to infinity. The corresponding limiting case is called in the flow theory plate flow. In that case, one has in addition the condition for

Hence a(t) = 0, so that (1) becomes directly the heat conduction equation.

Hence, we have arrived at a boundary value problem which is specialized compared with that of Fig. 13 only in so far as that now and x0 = 0, and moreover is simplified compared with that one by the fact that in the initial state, when the plate and therefore also the fluid were at rest, one has

Its solution is like that in (12.18) when the principal solution in the last V is replaced by a suitable Green function. Discuss the resulting velocity profile v(x) for increasing values of t.

Chapter 3:

3.1 Linear conductor with external heat conduction according to Fourier: Let u(x, 0) f(x )be the initial temperature for x > 0. How is the function f(x) to be continued for x < 0 so that the condition

is fulfilled?

3.2 Normalization condition during an-harmonic analysis is to be derived by specialization from Green's theorem.

3.3 Experimental determination of the conditions of external and internal heat conductivity: A bar is held at its ends x = 0 and x = l at the constant temperatures u1, u2 and is in a stationary state after an arbitrary initial state has vanished. Then, the temperature history would be linear if the mantle of the bar were shielded adiabatically. Thus, at its centre x = l/2, one would have u2=(u1+u3)/2, whence one can determine by observing

the ratio of the external and internal heat conduction (in essence, our constant h). Derive the relation, required to evaluate the observation between q and h; q = 1 corresponds to h = 0.

3.4 Determination of the ratio heat conductivity k to electric conductivity s: Let a metal rod be heated electrically, when it receives by the current i per unit of length Joule heat

i/qs (q = cross-section of the bar);

let it be insulated sidewards against heat loss. Find the differential equation of the stationary state and fit it by immersion in water to the boundary conditions u = 0 for x = 0 and x = l. Observe the potential difference V at the bar's ends and the maximum temperature U at the centre of the bar. Compute from them the ratio k /s. For pure metals, it has a universal value (Law of Wiedemann-Franz).

Chapter 4:

4.1 Power series expansion of In(r): Find it from the integral representation (19.18)

a) for integer n,
b) for arbitrary n with the aid of a general definition of the G-functions.

4.2 Derive relations between H1n and H2n, especially for integer n, using the integral representations (19.22).

4.3 Compute the logarithmic singularity of H0(r) at 0 from the integral representation (19.22).

4.4 Elementary method for the asymptotic approximation of H1n(r): Verify by a mathematical, but rather doubtful method the asymptotic limiting value of H1n(r) during which you already neglect in the differential equation step by step 1/r and higher powers of 1/r.

4.5 Expansion of a function on the sphere:
a) Expand f at first trigonometrically in j, then in spherical functions in Scheme:

and as combination of the two expansions

b) Compose f out of general spherical surface functions Y and find the coefficients involved from the orthogonality condition for


and as combination of both expansions

Clarify for yourself the difference of the summation arrangement in (1) and (2) by a figure (number pattern in the m, n plane) and show that Amn and Anm in (1) and (2) mean formally the same (by interchange of the summation and integration sequence).

4.6 Mapping of the wedge arrangement in Fig. 17 in circular moon sickels: Transform Fig. 17 of the wedge with opening angle 60 by reciprocal radii for convenient location of the inversion centre C (cf. Hints); the three lines 1, -1; 2, -2; 3, -3 then become circular arcs, the angular spaces they bound become circular sickels. Check the relationship between two regions each and consider that the Green function of potential theory for each of these regions (in space, spherical sections) can be obtained by five-fold reflection.

4.7 a) Mapping of a plane-parallel plate into two contacting spheres, b) of two concentric spheres into a plane and a sphere:

a) Examine the space structure into which the plane parallel plate together with all its reflective repetitions goes by inversion. Place the inversion sphere suitably so that it touches the two bounding planes of the plate. Show that this plate then is mapped into the outside of two touching spheres, similarly their reflective repetitions into the space between two touching spheres.

b) Show that two concentric spheres can be inverted into the pair of planes: Plane + sphere. Hence, inversely, the potential of a sphere with respect to a plane can be transformed into the much simpler, corresponding boundary value problem for two concentric spheres, similarly the potential of two arbitrary spheres which do not intersect.

4.8 Computation of two aggregates of Bessel functions: Find in (5) of Appendix A4.1 the value of

and in (20b) that of

Chapter 5:

5.1 Questions of normalization: Normalize the functions In(lr) and yn(kr) of (26.3) and (26.2) to 1 for the basic interval 0 < r < a for the boundary conditions J'n(la) = 0, y 'n(ka) = 0 following (20.19).

5.2 Gauss' theorem of the arithmetic mean in the potential theory: Prove the theorem: The value of a potential function u is at each point P of its regular domain S equal to the arithmetic mean of its values on a sphere Ka with arbitrary radius a around P as long as this is entirely inside S.

5.3 Summation formulae extended over the roots of Bessel functions: Confirm the expansions (27.13) and (27.14) regarding the equality of their coefficients Anm and derive from the comparison of their factors of interesting identities for Y n which can be transformed into those for y n.

Chapter 6:

6.1 Vertical and horizontal antenna at the height of h over infinitely well conducting soil: Show that by (31.16) and (31.17) for both antennae, i.e.,

the conditions (31.15) for the vanishing of the tangential components of are fulfilled throughout the plane z = 0.Show also that by (31.19) and (31.20) for the magnetic antennae, i.e.,

the conditions (35.1) for the vanishing of the tangential electric component are fulfilled.

6.2 Shape of the electric lines of force in the case of a Zenneck-wave near Earth's surface: Show that the lines of force in the air space are inclined In a forward direction, i.e., in the sense of the direction of propagation and that they lag behind in Earth at a small angle against Earth's surface.

6.3 Simplified calculation of the power to be supplied to vertical and horizontal antennae: Prove the expressions (36.16) and (36.16a) by determining the work which in unit time is performed by on the current j on the path l of the length of the antenna.

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