W A V W A V 421 The conditions of production of ripples by wind, or generally in a surface of separation of two fluids, each of which has any motion parallel to this surface, are given in HYDROMECHANICS. (9) Interference of Waves. While the disturbances con sidered are so small as to be superposable, i.e., inde pendent of one another, the effect of superposition is merely a kinematical question, and, as such, has been very fully treated under MECHANICS ( 56-67). See also ACOUSTICS, LIGHT, and WAVE THEORY. Thus ripple- patterns, ordinary beats of musical sounds, composition of lunar and solar ocean tides, diffraction, phenomena of polarized light in crystals or in transparent bodies in the magnetic field, &c., are all, in principle at least, simple kinematical consequences of superposition. But the pheno mena called Tartini s beats, breakers, a bore, a jabble, and (generally) cases in which a sufficient approximation cannot be obtained by omitting powers of the displacements higher than the first, are not of this simple character. As a single illustration, take one case of the first of these phenomena. The fact to be explained is that when two pure musical sounds, of frequencies 2 and 3 (say), that is, forming a " perfect fifth," are sounded together, we hear in addition to them a graver note, viz., that of which the first sound is the octave and the second the twelfth. When a resonator, carefully tuned to this graver note, is applied to the ear the note is usually not heard. Hence Helmholtz attributes its production to the fact that the drum of the ear (in consequence of the attachments of the ossicles) has different elastic properties for inward and for outward displacements. The force tending to restore the drum from a displacement x may therefore be represented approximately by Thus, when the drum is exposed to the two sounds above men tioned, its equation of motion is t + fi) . F>y successive approximations, it is found that there is a term in the value of x of the form abqcos (mt - /3) (p- - Qm 2 )(p- - 4m-)(p- m 2 ) which is the "dilFereiice-tone" referred to. This, of course, is communicated to the internal ear. Helmholtz points out, however, that such sounds may be produced objectively, provided the interfering disturbances are sufficiently great. No one seems yet to have obtained any really accurate notion of the smallness of the disturb ances of air which can be heard as sound. That they are excessively small has long been shown by many processes, but even more perfectly by the comparatively recent invention of the telephone. (10) Waves in an Elastic Solid. Some of the more elementary parts of this very difficult question have been treated from a theoretical point of view in the article ELASTICITY. From an observational and experimental point of view some are treated under EARTHQUAKE. See also LIGHT, and WAVE THEORY, for the luminiferous medium appears to behave like an elastic solid. (11) Waves of Temperature and of Electric Potential. In HEAT ( 78), and specially in the mathematical appendix to that article, will be found Fourier s treatment of heat-waves produced by periodic sources of various characters. It is sufficient to call attention here to the form of the equation for the linear motion of heat (which is the same as that for electricity and for diffu sion), viz., dv d ( ,dv where v represents temperature, c specific heat, and k thermal conductivity. When c and k are constants, this takes the very simple form dv __ d v dt =K d^ If plane harmonic waves of the type are to be transmitted, we must have simultaneously The first gives - d and the second, by means of this, gives 2/m 2 (A"* - B6 -*)= - m(A6 "* + 136 - *) . Thus A vanishes, which implies that the amplitude of the waves must continuously diminish as they progress. Also 2/<?i 2 = m , so that / u = Bg"" a: cos (Zicn-t nx). If the conductivity be not constant, tlien, even in the simple case of fc=& (l + atf), where a is small, the wave throws off others of inferior amplitude and of a different period. 12. Works and Memoirs on Waces. The literature is very extensive, so a few references only are given. The works of Lagrange, Laplace, Poisson, and Caucliy may be specially cited. Weber s Wtllenlehre, and Scott Russell s papers, contain valuable experimental details. Theoretical papers of importance by Earnshaw, Kellancl, Green, fcc., treat of long waves. The collected works of Ratikine, and particularly those of Stokes, deal in part with oscillatory waves. Stokes s "Reports on Hydrodynamics," in the Trans, of the British. Association, are of great value in tliemselves, and contain numerous references. A very complete discussion of the mathematics of the subject is given by Professor Greenhill iu the American Journal of Mathematics. Lord Rayleigh s Sound, and the memoirs of I)e St Venant and Von Helmholtz, together with a remarkable series of papers by Sir V. Thomson, in recent volumes of the Phil. Mag. and 1 roc. R. S. E., may fitly conclude the list. (P. G. T.) WAVERTREE, a township of Lancashire, partly included within the parliamentary limits of Liverpool, 3 miles south-east of Liverpool Exchange. The churches are all modern. There is a small circle of monoliths on the south-east boundary of the township, called the Calder Stones. A cemetery of the Neolithic period was opened a few years ago. The town possesses roperies and a brewery. An extensive pumping station connected with the Liverpool water-works is in the vicinity. The popu lation of the urban sanitary district (area 1838 acres) in 1871 was 7810, and in 1881 it was 11,097. WAVE THEOBY OF LIGHT si. A GENERAL statement of the principles of the
- - undulatory theory, with elementary explanations,
has already been given under LIGHT, and in the article on ETHER the arguments which point to the existence of an all-pervading medium, susceptible in its various parts of an alternating change of state, have been traced by a master hand ; but the subject is of such great importance, and is so intimately involved in recent optical investigation and discovery, that a more detailed expositionof the theory, with application to the leading phenomena, was reserved for a special article. That the subject is one of difficulty may be at once admitted. Even in the theory of sound, as conveyed by aerial vibrations, where we are well acquainted with the nature and properties of the vehicle, the fundamental conceptions are not very easy to grasp, and their development makes heavy demands upon our mathematical resources. That the situation is not im proved when the medium is hypothetical will be easily understood. For, although the evidence is overwhelming in favour of the conclusion that light is propagated as a vibration, we are almost entirely in the dark as to what it
is that vibrates and the manner of vibration. This ignor-