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WAVERLY. 373 WAVES. WAVERLY. The county-seat of Bremer County, Iowa, 71 miles northwest of Cedar Rapids, on the Ued Cedar Kiver, and on the Clii- cago Liriat Western, the Illinois Central, and the Chicago, Hock Island and Pacific railroads (Map: Iowa, E 2). It has the Warthurg Teach- ers' Seminary and Academy, and a public library. Waverly is the commercial centre of a farming, dairying, and cattle-breeding section, and car- ries on considerable trade also in its manufac- tured products, which include condensed milk, butter, cheese, canned goods, fruit, paint, etc. The water-works are the property of the mu- nicipality. I'opulation, in 1890, 234G; in 1900, 3177. WAVERLY. A village in Tioga County, N. Y., IS miles southeast of Elraira, on the Cayuta Creek, and on the Eric, the Lehigh Val- ley, and the Delaware, Lackawanna and Western railroads (Map: New York, D 3). It is sur- rounded by a dairying and farming section, and manufactures stove pipe, furniture, baskets, but- ter packages, etc. Waverh' is also an important distributing point for the W.yoniing Valley coal fields. The government is vested in a president, chosen annually, and a board of trustees. Wa- verlv was settled about ISIO, and was incorpor- ated in 18.54. Population, in 1890, 4123; in 1900, 4405. WAVES (AS. wafimi, to wave, fluctuate, waver, MHG. irahcn, to wave. Bavarian Ger. u'diben, to waver, totter). Wave-motions are of two kinds: one is the advance of a disturbance into a medium, and the other is its advance along a surface. Illustrations of the former kind are given by waves produced in air or in the interior of water by a vibrating body such as a bell, and those produced in the ether by an electrical or •atomic vibration ; illustrations of the latter kind are given by waves on the surface of a lake or ocean. The former class of waves are due to the elasticity and inertia of the medium ; and the velocity with which the disturbance spreads out from the vibrating centre depends upon these tuo [iroperties of the fluid alone. Elastic waves in homogeneous media have a velocity given where E is by the formula velocit.y d' the coefficient of elasticity and d is the density. See Acoustics; Ether; Electkicitt. Waves on the surface of a liquid are due to the action of gravitation, which tends to make the surface of a liquid horizontal (to the sur- face-tension if the w'aves are nothing but ripples), and to the inertia of the liquid. (For a discus- sion of these 'water-waves,' see Htdeostatics.) The 'wave-front' of a train of waves is the sur- face which at any instant includes all those points of the medium which the disturbances have just reached; or, more generally, it is a surface in the medium including those points where the motion is in the same 'phase.' The Avave-front from a point source is a spliere in the case of an elastic wave in an isotropic medium ; it is a circle for waves on the surface of water. Since all waves consist in the motion of por- tions of matter, and since the medium carrying the waves is not in its natural position or condi- tion, there is both kinetic and potential energy associated with wave-motion. This energy' is lost by the vibrating source and gained by the body absorbing the waves. The 'intensity' of the waves is defined to be the energy carried in unit time through an area of one square centimeter of surface at right angles to the direction of ad- vance of the waves. Thus, if the source of waves is a point and if the energ.v emitted per unit time is E, the intensity of the waves at a dis- E tance r is I, = -; , because the area of the surface of a sphere of radius r„ inclosing the point -source as a centre, is 4Tri'. Similarly, the E intensity at a distance r, is L := -; : . Therefore ivi:/ h = 'i' or the intensity of the waves from a point-source varies inversely as the square of the distance. Since waves are due to some vibrating centre, the simplest t,vpe of train of waves will be one produced by tile simplest vibration, that is a simple harmonic vibration, such as that of a tuning fork in the case of aerial waves. A train of waves ]U'oduced by a simple harmonic vibra- tion is called a simple harmonic train. It is characterized by its 'amplitude' and its 'wave- length' or 'wave-number.' The iim|ditude is the extent of the vibration of any individual jiarticle of the medium owing to the passage of the waves. The wave-length is the distaijce from any one point in the medium to the next point, in the (lirectiou of advance of the waves, where the conditions are at any instant exactly the same — both in displacement of the particle of the me- dium and in its velocity. The wave-number is the number of complete vibrations which each particle of the medium makes in one second ; or, what is the same thing, it is the number of waves which pass any one point of the medium in one second. The velocity of the train of waves is, then, obviously the product of the wave-length and the wave-niunber. Moreover, since, in the case of waves due to the elasticity of a homo- geneous medium, their velocity depends upon the elasticity and inertia alone, it is the same for waves of all lengths. Therefore, for a given me- dium, if the wave-length is known, the wave- number ma,v be at once calcilated. If the me- dium is not homogeneous, waves of different length have difl'erent velocities. (See Light and Dispersion.) It is not difficult to prove that the energy carried by a train of waves varies as the square of the amplitude; and, since in the case of waves emitted by a point-source the in- tensity of the waves varies inversely as the square of the distance from the source, the ampli- tude of the waves must 'vary inversely as the dis- tance itself. A complex vibration, made up of several simple harmonic vibrations, will produce a com- plex train of waves which is equivalent to the superposition of several trains of simple har- monic waves. The characteristics of such a com- plex train of waves are, first, the number of the component trains, and. secondly, their ampli- tudes, wave-numbers, and relative 'phases.' By relative phases is simply meant their relative po- sitions in the medium. (See Acoustics.) As trains of waves pass through any medium some energy' is always absorbed, as is shown by the gradual" decrease in amplitude. This is called 'attenuation;' and it is found that long waves