Introduction .–Through the work of Bloch our understanding of the behaviour of electrons in crystal lattices has been much advanced. The principal idea of Bloch’s theory is the assumption that the interaction of a given electron with the other particles of the lattice may be replaced in first approximation by a periodic field of potential. With this model an interpretation of the specific heat, the electrical and thermal conductivity, the magnetic susceptibility, the Hall effect, and the optical properties of metals could be obtained. The advantages and limitations inherent in the assumption of Bloch will be much the same as those encountered when replacing the interaction of the electrons in an atom by a suitable central shielding of the unclear field, as in the work of Thomas and Hartree. In the papers quoted a number of general results were given regarding the behaviour of electrons in any periodic field of potential. To obtain a clearer idea of the details of this behaviour with a view to the application in special problems, however, it appeared worth while to investigate the mechanics of electrons in periodic fields of potential somewhat similar to those met with in practice and of such nature that the energy values W and eigenfunctions Ψ of the wave-equation can actually be computed. It is the purpose of this article to discuss a case where the integration is possible. In Section 1 the energy values and in Section 2 the wave-functions in their dependence on the binding introduced by the potential field are discussed for the one dimensional problem. In Section 3 the matrix elements of the linear momentum, which furnish the electric current associated with the various stationary states, are well as the probability of radiative transitions between these states, are evaluated. In Section 4 the results are extended to the three dimensional case and those features considered which one may expect to find in the case of more general periodic fields of potential. Section 5 deals with some applications to physical problems.

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The diffraction of sea waves round the end of a long straight breakwater is investigated, use being made of the solutions of mathematically analogous problems in the diffraction of light. The wave patterns and wave heights are determined on both the leeward and windward sides of the breakwater, and for points quite close to the breakwater. This involves some extension of the calculations previously made for optical phenomena. The conditions obtaining in the lee of a small island are discussed. The penetration of waves through a single gap in a long breakwater is examined, and the result is shown to depend very much on whether the gap is small or not compared with the length of the waves. The investigation was suggested by problems arising in the construction of the Mulberry harbours.

The possible existence, form and maximum height of strictly periodic finite stationary waves on the surface of a perfect liquid are discussed. A method of successive approximation to the solution of the hydrodynamical equations is formulated, and the solution is carried to the fifth order for the case of two-dimensional waves on a deep liquid. The convergence of the method has not been established, so that the existence of truly periodic stationary waves is not beyond doubt, but the calculations provide strong presumptive evidence for their existence, and for the existence of a finite stable wave of greatest height. The crest of this wave has a right-angled nodal form, in contrast with that of the greatest stable travelling wave for which the nodal angle is 120°. The maximum crest height is 0.141A, where A is the wave-length, and the maximum trough depth is 0.078 A. This means that the greatest stationary waves are greater than the maximum travelling waves, the ratio being 1.53. The motions of individual particles are studied and it is shown that particles in the surface, particularly those near the anti-nodes have large horizontal motions. For a given wave-length, the period increases with wave height. The wave pressure on a breakwater is examined, and the modification of the calculations to allow for the finite depth of water is considered. Doubly modulated oscillations in a deep rectangular tank are also briefly discussed.

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This paper considers the harmonic oscillations of several simple model atmospheres. The oscillations are of two types. In the first, the kinetic energy per unit volume tends to zero at great heights; in the second, the kinetic energy per unit volume remains finite. A large explosion at ground level excites a spectrum of both types of oscillation. The pulse ultimately separates into two parts—a train of travelling waves which can be observed at ground level at great distances, and a train of travelling waves which disappear into the upper atmosphere. The complete range of experimental observations on the pressure oscillations caused by explosions of energies varying between 10 20 and 10 24 ergs can only be interpreted with model atmospheres having one or more sound channels, i.e. having at least one minimum in the temperature-height relationship of the atmosphere. In spite of the complexity of the phenomena, the theory throws light on some of the characteristic features of the observations. The average period of the largest waves is roughly proportional to the cube root of the energy released by the explosion. The amplitudes of the waves from large explosions can be calculated. Conversely, good records enable the size of the explosion to be estimated. The energy of the Siberian meteorite of 1908 was about 10 16 cal, or 10 MT (T signifying a ton of t.n.t.).

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