conditions may be illustrated by the solution of the following statical problem*:

A portion of a circular cylinder bounded by two generators and two circular sections is held bent into a surface of revolution by forces applied along the bounding generators, the circular edges being free, in such a way that the displacement v tangential to the circular sections is proportional to the angular coordinate ; it is required to find the displacement.

We are to have v co, where c is constant, while u and w are independent of p. Hence

The stress-resultants S1, S, and the stress-couples H1, H2 vanish, and we

We seek to satisfy these equations and conditions approximately by the assumption that the extensional strains e1, e, are of the same order as the flexural strains he1, hk. When this is the case T, and T, are given with sufficient approximation by the formulæ

T1 = (3D/h2) (e, +σ€), T2=(3D/h2) (€1⁄2 +σ€1).

2

To satisfy the equation T1/dx = 0 and the condition T1 = 0 at x = ±l we must put 70, or eσe, and then we have T2 = 3D (1 − σ2) €/h. The equations of equilibrium are now reduced to the equation

=

+ = dx2 a2

0,

=

= 0.

მე3

If we take c-w to be a sum of terms of the form em, then m2 is large of the order 1/h; and the solution is found to be

w=c+ C1 cosh (qx/a) cos (qx/a) + C2 sinh (qx/a) sin (qx/a),

This is the problem solved for this purpose by H. Lamb, loc. cit. p. 477. The same point in the theory was illustrated by A. B. Basset, loc. cit. p. 505, by means of a different statical

problem.

The form of the solution shows that near the boundaries €1, €2, hê1, hê are all of the same order of magnitude, but that, at a distance from the boundaries which is at all large compared with (ah)1, e, and e, become small in comparison with hк2.

It may be shown that, in this statical problem, the potential energy due to extension is actually of the order √(h/a) of the potential energy due to bending*. In the case of vibrations we may infer that the extensional strain, which is necessary in order to secure the satisfaction of the boundary conditions, is practically confined to so narrow a region near the edge that its effect in altering the total amount of the potential energy, and therefore the periods of vibration, is negligible.

335. Vibrations of a thin spherical shell.

The case in which the middle surface is a complete spherical surface, and the shell is thin, has been investigated by H. Lamb† by means of the general equations of vibration of elastic solids. All the modes of vibration are extensional, and they fall into two classes, analogous to those of a solid sphere investigated in Article 194, and characterized respectively by the absence of a radial component of the displacement and by the absence of a radial component of the rotation. In any mode of either class the displacement is expressible in terins of spherical surface harmonics of a single integral degree. In the case of vibrations of the first class the frequency p/2π is connected with the degree n of the harmonics by the equation

p2a2p/μ = (n - 1) (n + 2),

.(54) where a is the radius of the sphere. In the case of vibrations of the second class the frequency is connected with the degree of the harmonics by the equation

If n exceeds unity there are two modes of vibration of the second class,

For further details in regard to this problem the reader is referred to the paper by H. Lamb already cited.

+ London Math. Soc. Proc., vol. 14 (1883), p. 50.

and the gravest tone belongs to the slower of those two modes of vibration of this class for which n=2. Its frequency p/2π is given by

if Poisson's ratio for the material is taken to be . The frequencies of all these modes are independent of the thickness.

In the limiting case of a plane plate the modes of vibration fall into two main classes, one inextensional, with displacement normal to the plane of the plate, and the other extensional, with displacement parallel to the plane of the plate. [See Articles 314 (d) and (e) and 333 and Note F at the end of the book.] The case of an infinite plate of finite thickness has been discussed by Lord Rayleigh*, starting from the general equations of vibration of elastic solids, and using methods akin to those described in Article 214 supra. There is a class of extensional vibrations involving displacement parallel to the plane of the plate; and the modes of this class fall into two sub-classes, in one of which there is no displacement of the middle plane. The other of these two sub-classes appears to be the analogue of the tangential vibrations of a complete thin spherical shell. There is a second class of extensional vibrations involving a component of displacement normal to the plane of the plate as well as a tangential component, and, when the plate is thin, the normal component is small compared with the tangential component. The normal component of displacement vanishes at the middle plane, and the normal component of the rotation vanishes everywhere; so that the vibrations of this class are analogous to the vibrations of the second class of a complete thin spherical shell. There is also a class of flexural vibrations involving a displacement normal to the plane of the plate, and a tangential component of displacement which is small compared with the normal component when the plate is thin. The tangential component vanishes at the middle plane, so that the displacement is approximately inextensional. In these vibrations the linear elements which are initially normal to the middle plane remain straight and normal to the middle plane throughout the motion, and the frequency is approximately proportional to the thickness. There are no inextensional vibrations of a complete thin spherical shell.

The case of an open spherical shell or bowl stands between these extreme cases. When the aperture is very small, or the spherical surface is nearly complete, the vibrations must approximate to those of a complete spherical shell. When the angular radius of the aperture, measured from the included pole, is small, and the radius of the sphere is large, the vibrations must approximate to those of a plane plate. In intermediate cases there must be vibrations of practically inextensional type and also vibrations of extensional type.

* London Math. Soc. Proc., vol. 20 (1889), p. 225, or Scientific Papers, vol. 3, p. 249.

Purely inextensional vibrations of a thin spherical shell, of which the edge-line is a circle, have been discussed in detail by Lord Rayleigh* by the methods described in Article 321 supra. In the case of a hemispherical shell the frequency p/2 of the gravest tone is given by

p=√(μ/p)(h/a2) (4·279).

When the angular radius a of the aperture is nearly equal to π, or the spherical surface is nearly complete, the frequency p/2π of the gravest mode of inextensional vibration is given by p=√(μ/p) {h/a2 (π — u)3} (5·657). By supposing a to diminish sufficiently, while h remains constant, we can make the frequency of the gravest inextensional mode as great as we please in comparison with the frequency of the gravest (extensional) mode of vibration of the complete spherical shell. Thus the general argument by which we establish the existence of practically inextensional modes breaks down in the case of a nearly complete spherical shell with a small aperture.

When the general equations of vibration are formed by the method illustrated above in the case of the cylindrical shell, the components of displacement being taken to be proportional to sines or cosines of multiples of the longitude 4, and also to a simple harmonic function of t, they are a system of linear equations of the 8th order for the determination of the components of displacement as functions of the co-latitude . The boundary conditions at the free edge require the vanishing, at a particular value of 0, of four linear combinations of the components of displacement and certain of their differential coefficients with respect to 0. The order of the system of equations is high enough to admit of the satisfaction of such conditions; and the solution of the system of equations, subject to these conditions, would lead, if it could be effected, to the determination of the types of vibration and the frequencies.

The extensional vibrations can be investigated by the method illustrated above in the case of the cylindrical shell. The system of equations is of the fourth order, and there are two boundary conditionst. In any mode of vibration the motion is compounded of two motions, one involving no radial component of displacement, and the other no radial component of rotation. Each motion is expressible in terms of a single spherical surface harmonic, but the degrees of the harmonics are not in general integers. The degree a of the harmonic by which the motion with no radial component of displacement is specified is connected with the frequency by equation (54), in which a is written for n; and the degree ẞ of the harmonic by which the motion

* London Math. Soc. Proc., vol. 13 (1881), or Scientific Papers, vol. 1, p. 551. See also Theory of Sound, 2nd edition, vol. 1, Chapter x A.

+ The equations were formed and solved by E. Mathieu, J. de l'École polytechnique, t. 51 (1883). The extensional vibrations of spherical shells are also discussed in the paper by the present writer cited in the Introduction, footnote 133.

with no radial component of rotation is specified is connected with the frequency by equation (55), in which is written for n. The two degrees a and B are connected by a transcendental equation, which is the frequency equation. The vibrations do not generally fall into classes in the same way. as those of a complete shell; but, as the open shell approaches completeness, its modes of extensional vibration tend to pass over into those of the complete shell.

The existence of modes of vibration which are practically inextensional is clearly bound up with the fact that, when the vibrations are assumed to be extensional, the order of the system of differential equations of vibration is reduced from 8 to 4. As in the case of the cylindrical shell, it may be shown that the vibrations cannot be strictly inextensional, and that the correction of the displacement required to satisfy the boundary conditions is more important than that required to satisfy the differential equations. We may conclude that, near the free edge, the extensional strains are comparable with the flexural strains, but that the extension is practically confined to a narrow region near the edge.

If we trace in imagination the gradual changes in the system of vibrations as the surface becomes more and more curved*, beginning with the case of a plane plate, and ending with that of a complete spherical shell, one class of vibrations, the practically inextensional class, appears to be totally lost. The reason of this would seem to lie in the rapid rise of frequency of all the modes of this class when the aperture in the surface is much diminished.

The theoretical problem of the vibrations of a spherical shell acquires great practical interest from the fact that an open spherical shell is the best representative of a bell which admits of analytical treatment. It may be taken as established that the vibrations of practical importance are inextensional, and the essential features of the theory of them have, as we have seen, been made out. The tones and modes of vibration of bells have been investigated experimentally by Lord Rayleight. He found that the nominal. pitch of a bell, as specified by English founders, is not that of its gravest tone, but that of the tone which stands fifth in order of increasing frequency; in this mode of vibration there are eight nodal meridians.

336. Problems of equilibrium.

When a thin plate or shell is held deformed by externally applied forces, the strained middle surface must, as we observed in Article 315, coincide very nearly with one of the surfaces applicable upon the unstrained middle surface. We may divide the problem into two parts: (i) that of determining

* The process is suggested by H. Lamb in the paper cited on p. 477.

+ Phil. Mag. (Ser. 5), vol. 29 (1890), p. 1, or Scientific Papers, vol. 3, p. 318, or Theory of Sound, 2nd edition, vol. 1, Chapter x.