In teaching the elements of transform theory to students of physics and engineering it is very useful to have available, as early as possible, the inversion theorem for the Hankel transform

The difficulty is that a valid proof for general values of v (cf. [1], p. 456) is complicated and involves a greater familiarity with the processes of analysis and the properties of Bessel functions than is possessed by most science students.

In recent years there have appeared solutions of several integral equations of the type

in which the kernel K(x) contains (as a factor) one of the classical orthogonal polynomials or a hypergeometric function.

The statical Reissner-Sagoci problem [1, 2, 3] is that of determining the components of stress and displacement in the interior of the semi-infinite homogeneous isotropic elastic solid z ≧ 0 when a circular area (0 ≦ p ≦ a, z = 0) of the boundary surface is forced to rotate through an angle a about an axis which is normal to the undeformed plane surface of the medium. It is easily shown that, if we use cylindrical coordinates (p, φ, z), the displacement vector has only one non-vanishing component uφ (p, z), and the stress tensor has only two non-vanishing components, σρπ(p, z) and σπz(p, z). The stress-strain relations reduce to the two simple equations

where μ is the shear modulus of the material. From these equations, it follows immediately that the equilibrium equation

is satisfied provided that the function uπ(ρ, z) is a solution of the partial differential equation

The boundary conditions can be written in the form

where, in the case in which we are most interested, f(p) = αρ. We also assume that, as r → ∞, uπ, σρπ and σπz all tend to zero.

1. The problem of determining the distribution of stress in the neighbourhood of a penny-shaped crack defined in terms of cylindrical coordinates (ρ, φ, z) by 0 ≤ ρ ≤ α, z = 0, has been considered by Sneddon ((2)) and Sack ((1)). In the latter paper the solution is derived only in the case in which the stress field is due to the application of constant pressure to the faces of the crack. In the former paper the analysis given applies to an axisymmetric distribution of pressure p(ρ) applied to both the upper and lower face of the penny-shaped cavity, but the calculation of the stress intensity factor

and of the energy W required to open up the crack is a complicated matter even in the case in which p(ρ) is a constant.

In this note we consider the problem of determining the stress on the boundary y = 0 of the elastic half-plane y ≡ 0 when there are prescribed body forces acting in the interior and the boundary is free from applied stress. Expressions for the components of stress at a general point of the half-plane when the imposed body force is concentrated at a single point have been derived by Melan [1], Sneddon [2] and Green [3], each author making use of a different method.

In the analysis of mixed boundary value problems in the plane, we encounter dual integral equations of the type

If we make the substitutions cos we obtain a pair of dual integral equations of the Titchmarsh type [1, p. 334] with α = − 1, v = − ½ (in Titchmarsh's notation). This is a particular case which is not covered by Busbridge's general solution [2], so that special methods have to be derived for the solution.

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sums

and

where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

When the theory of Hankel transforms is applied to the solution of certain mixed boundary value problems in mathematical physics, the problems are reduced to the solution of dual integral equations of the type

where α and ν are prescribed constants and f(ρ) is a prescribed function of ρ [1]. The formal solution of these equations was first derived by Titchmarsh [2]. The method employed by Titchmarsh in deriving the solution in the general case is difficult, involving the theory of Mellin transforms and what is essentially a Wiener-Hopf procedure. In lecturing to students on this subject one often feels the need for an elementary solution of these equations, especially in the cases α = ± 1, ν = 0. That such an elementary solution exists is suggested by Copson's solution [3] of the problem of the electrified disc which corresponds to the case α = –l, ν = 0. A systematic use of a procedure similar to Copson's has in fact been made by Noble [4] to find the solution of a pair of general dual integral equations, but again the analysis is involved and long. The object of the present note is to give a simple solution of the pairs of equations which arise most frequently in physical applications. The method of solution was suggested by a procedure used by Lebedev and Uflyand [5] in the solution of a much more general problem.

The problem of solving the equation of thermal conduction for cases in which heat is generated in the interior of the medium under consideration arises frequently in physics and engineering. It occurs, for instance, when we consider the diffusion of heat in a solid undergoing radioactive decay (1) or which is absorbing radiation (2). Complications of a similar nature arise when there is a generation or absorption of heat in the solid as a result of a chemical change-for example, the hydration of cement (3). The particular case in which the rate of generation of heat is independent of the temperature arises in the theory of the ripening of apples and has been discussed by Awberry (4).

1. The problem of determining the distribution of stress in a semi-infinite solid medium when its plane boundary is deformed by the pressure against it of a perfectly rigid cone is of considerable importance in various branches of applied mechanics. It arises in soil mechanics where the cone is the base of a conical-headed cylindrical pillar and the semi-infinite medium is the soil upon which it rests (1). In this instance the distribution of stress in the soil is known to be more or less similar to that calculated on the assumption that the soil is a perfectly elastic, isotropic and homogeneous medium, at least if the factor of safety of a mass of soil with respect to failure by plastic flow exceeds a value of about three (2). The same problem occurs in the theory of indentation tests in which a ductile material is indected by cylindrical punches with conical heads (3).

1. In a recent paper(1) expressions were found for the elastic stresses produced in a semi-infinite elastic medium when its boundary is deformed by the pressure against it of a perfectly rigid body. In deriving the solution of this problem—the ‘Boussinesq’ problem—it was assumed that the normal displacement of a point within the area of contact between the elastic medium and the rigid body is prescribed and that the distribution of pressure over that area is determined subsequently. The solutions for the special cases in which the free surface was indented by a cone, a sphere and a flat-ended cylindrical punch were derived, but no attempt was made to give a full account of the distribution of stress in the interior of the medium in any of these cases.

1. In a recent paper(1) the partial differential equation governing the symmetrical vibrations of a thin elastic plate was reduced to an ordinary differential equation by the use of the Hankel transform method. By the discussion of the solutions of the latter equation and by the use of the Hankel inversion theorem an account was given of the free and forced vibrations of the plate under symmetrical conditions.

In recent years, the theory of integral transforms—often in the guise of ‘operator calculus'—has been employed to obtain the solutions of a wide variety of problems in mathematical physics, particularly in the theory of vibrations. The most commonly employed transforms are those of Fourier, Laplace and Mellin; in problems where the range of one of the variables is (0, ∞) or (− ∞, ∞) these transforms have often been used with success to reduce a partial differential equation in n independent variables to one in n − 1 variables and hence to simplify the boundary value problem involved. In this paper we consider some simple problems in the theory of elastic vibrations in which the Bessel, or Hankel, transform is more useful than those mentioned above and is consequently employed throughout. In the first instance we consider problems in which the variable to be eliminated from the differential equation ranges from 0 to ∞ so that the theory of Hankel transforms is directly applicable; the application of finite intervals is then extended to the Hankel transform by a method similar to that used by Doetsch in his treatment of the Fourier transform.

1. In this paper an analysis is given of the distribution of stress in a semi-infinite elastic medium due to the action of an external force applied to the interior of the medium. It will be assumed throughout that the force acts in a direction perpendicular to that of the boundary of the solid; the analysis is similar when the line of action of the force is parallel to the boundary and is therefore not given here. The equations of plane strain parallel to the x-y plane are employed; physically this is equivalent to assuming that there is no component of the displacement vector in a direction normal to the x-y plane or, what is the same thing, that the external force is applied along are infinite line parallel to the axis of z.