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originally educated as a naturalist, and saw much of the East |
Anglian gypsies, of whose superstitions and folk-lore he made
careful study. Abandoning natural history for the law, he
qualified as a solicitor and went to London, where he practised
for some years, giving his spare time to his chosen pursuit of
literature. He contributed regularly to the Examiner from 1874
and to the Athenaeum from 1875 until 1898, being for more than
twenty years the principal critic of poetry in the latter journal.
His article on "Poetry " in the ninth edition of the Ency. Brit.
(vol. xix., 1885) was the principal expression of his views on
the first principles of the subject, and did much to increase his
reputation, which was maintained by other articles he wrote for
the Encyclopaedia Britannica and for the chief periodicals and
reviews. Mr Watts-Dunton had considerable influence as the
friend of many of the leading men of letters of his time; he
enjoyed the confidence of Tennyson, and contributed an appre-
ciation of him to the authorized biography. He was in later
years Rossetti's most intimate friend. He was the bosom
friend of Swinburne (q.v.), who shared his home for nearly thirty
years before he died in 1909. The obituary notices and apprecia-
tions of the poets of the time, which he contributed to the
Athenaeum and other periodicals, bore testimony to his sympathy,
insight and critical acumen. It was not, however, until 1897
that he published a volume under his own name, this being his
collection of poems called The Coming of Love, portions of which
he had printed in periodicals from time to time. In the following
year his prose romance Aylwin attained immediate success, and
ran through many editions in the course of a few months.
Both The Coming of Love and Aylwin set forth, the one in poetry,
the other in prose, the romantic and passionate associations of
Romany life, and maintain the traditions of Borrow, whom Mr
Watts-Dunton had known well in his own early days. Imagina-
tive glamour and mysticism are their prominent characteristics,
and the novel in particular has had its share in restoring the
charms of pure romance to the favour of the general public.
He edited George Borrow's Lavengro (1893) and Romany Rye
(1900); in 1903 he published The Renascence of Wonder, a
treatise on the romantic movement; and his Studies of Shake-
speare appeared in 1910. But it was not only in his published
work that Mr Watts-Dunton's influence on the literary life of
his time was potent. His long and intimate association with
Rossetti and Swinburne made him, no doubt, a unique figure
in the world of letters; but his own grasp of metrical principle
and of the historic perspective of the glories of English poetry
made him, among the younger generation, the embodiment of
a great tradition of literary criticism which could never cease
to command respect. In 1905 he married. His life has been
essentially one of devotion to letters, faithfully and disinter-
estedly followed.

WAUGH, BENJAMIN (1839-1908), English social reformer, was born at Settle, Yorkshire, on the 20th of February 1839. He passed the early years of his life in business, but in 1865 entered the congregational ministry. Settling at Greenwich he threw himself with ardour into the work of social reform, devoting himself especially to the cause of the children. He served on the London School Board from 1870 to 1876. In 1884 he was responsible for the establishment of the London society for the prevention of cruelty to children, which four years later was established on a national basis. He was elected its honorary secretary, and it was largely owing to information obtained by him that the Criminal Law Amendment Act of 1885 was passed, while by his personal effort he secured the insertion of a clause giving magistrates power to take the evidence of children too young to understand the nature of an oath. In 1889 he saw the work accomplished by his society (of which he had been made director the same year) recognized by the passing of an act for the prevention of cruelty to children, the first stepping-stone to the act of 1908 (see CHILDREN, LAW RELATING TO). In 1895 a charter of incorporation was conferred on the society, but in 1897 it was the object of a serious attack on its administration. An inquiry was demanded by Waugh, and the commission of inquiry, which included Lord Herschell and others, completely vindicated the

society and its director. Waugh had given up pastoral work in 1887 to devote his whole time to the society, and he retained his post as director until 1905, when the state of his health compelled his retirement. He remained consulting director until his death at Westcliff, near Southend, Essex, on the 11th of March 1908. Waugh edited the Sunday Magazine from 1874 to 1896, but he had otherwise little leisure for literary work. His The Gaol Cradle, who rocks it? (1873) was a plea for the abolition of juvenile imprisonment.

WAUGH, EDWIN (1817-1890), known as "The Lancashire Poet," was born at Rochdale, on the 29th of January 1817, the son of a shoemaker. For several years he carned his living as a journeyman printer in various parts of the country. In 1855 he published his first book, Sketches of Lancashire Life and Localities, following this up with reprinted Poems and Songs (1859). His rendering of the Lancashire dialect was most happy, and his rude lyrics, full of humour and pathos, were great favourites with his countrymen. He died on the 30th of April 1890.

See Milner's Memoir in an edition of Waugh's selected works (1892-1893). WAUKEGAN, a city and the county-seat of Lake county, Illinois, U.S.A., on the W. shore of Lake Michigan, about 36 m. N. of Chicago. Pop. (1890) 4915; (1900) 9426, of whom 2506 were foreign-born; (1910 census) 16,069. It is served by the Elgin, Joliet & Eastern (of which it is a terminus) and the Chicago & North Western railways, by an interurban electric line, and by lake steamers. In 1880 the United States government undertook the formation of an artificial harbour with a channel 13 ft. deep, and in 1902-1904 the depth was increased to 20 ft. The main portion of the city is situated about 100 ft. above the level of the lake. There are a number of parks and mineral springs, and along the lake front a fine driveway, Sheridan Road. The city is a residential suburb of Chicago. The principal buildings are the Federal building, the Court House, a Carnegie library, the Masonic Temple and McAlister Hospital. At the village of North Chicago (pop. in 1910, 3306), about 3 m. S. of Waukegan, there is a United States Naval Training Station. Waukegan is the commercial centre of an agricultural and dairying region, and has various manufactures. The total value of the factory product in 1905 was $3,961,513. Waukegan was settled about 1835, and until 1849 was known as Little Fort, which is supposed to be the English equivalent of the Indian name Waukegan. It became the county-seat of Lake county in 1841, was incorporated as a town in 1849, and first chartered as a city in 1859.

WAUKESHA, a city and the county-seat of Waukesha county, Wisconsin, U.S.A., about 19 m. W. of Milwaukee on the Little Fox river. Pop. (1890) 6321; (1900) 7419, including 1408 foreign-born; (1905 state census) 6949; (1910) 8740. Waukesha is served by the Minneapolis, St Paul & Sault Ste Marie, the Chicago & North-Western and the Chicago, Milwaukee & St Paul railways, and by interurban electric railways connecting it with Milwaukee, Oconomowoc and Madison. The medicinal mineral springs (Bethesda, White Rock, &c.) are widely known. Among the public buildings are the county court house and the public library. Waukesha is the seat of the State Industrial School for Boys (established as a house of refuge in 1860) and of Carroll College (Presbyterian, co-educational, 1846). Waukesha was first settled in 1834, was named Prairieville in 1839, was incorporated as a village under its present name (said to be a Pottawatomi word meaning "fox ") in 1852, and chartered as a city in 1896. In 1851 the first railway in the state was com pleted between Milwaukee and Waukesha, but the village remained only a farming community until the exploitation of the mineral springs was begun about 1868. About 15 m. S. of Waukesha, near Mukwonago (pop. in 1910,615), in 1844-1845, there was an unsuccessful communistic agricultural settlement, the Utilitarian Association, composed largely of London mechanics led by Campbell Smith, a London bookbinder.

WAURIN (or WAVRIN), JEHAN (or JEAN DE) (d. c. 1474), French chronicler, belonged to a noble family of Artois, and was present at the battle of Agincourt. Afterwards he fought for

and then into brighter and more delicate colours, encouraged thereto, in his more recent work, by his adoption of pastel as a medium even for life-size portraits, mainly of ladies. His portraits, numbering over two hundred, include many of the greatest names in Belgium, France, and America (Wauters having for some years made Paris his chief home). Among these may be named the Baron Goffinet, the Baroness Goffinet, Madame Somzée (standing at a piano), Master Somzée (on horseback by the sea-shore), the Princess Clementine of Belgium (Brussels Museum), Lady Edward Sassoon, Baron de Bleichroder, Princess de Ligne, Miss Lorillard, a likeness of the artist in the Dresden Museum, and M. Schollaert (president of the Chamber of Deputies) the last named an amazing example of portraiture, instinct with character and vitality. The vigour of his male, and the grace and elegance of his female, portraits are unsurpassable, the resemblance perfect and the technical execution such as to place the artist in the front rank. Between 1889 and 1900 the painter contributed to the Royal Academy of London. Few artists have received such a succession of noteworthy distinctions and recognitions. His" Hugo van der Goes," the work of a youth of twenty-four, secured the grand medal of the Salon. He has been awarded no fewer than six "medals of honour"-at Paris in 1878 and 1889; Munich, 1879; Antwerp, 1885; Vienna, 1888; and Berlin, 1883. He is a member of the academy of Belgium, and honorary member of the Vienna, Berlin, and Munich academies, and corresponding member of the Institut de France and of that of Madrid. He has received the order of merit of Prussia, and is Commander of the order of Leopold, and of that of St Michael of Bavaria, officer of the Legion of Honour, &c.

the Burgundians at Verneuil and elsewhere, and then occupying | developing into the whole range of a brilliant, forceful palette, a high position at the court of Philip the Good, duke of Burgundy, was sent as ambassador to Rome in 1463. Jehan wrote, or rather compiled, the Recueil des croniques et anchiennes istories de la Grant Bretaigne, a collection of the sources of English history from the earliest times to 1471. For this work he borrowed from Froissart, Monstrelet and others; but for the period between 1444 and 1471 the Recueil is original and valuable, although somewhat untrustworthy with regard to affairs in England itself. From the beginning to 688 and again from 1399 to 1471 the text has been edited for the Rolls Series (5 vols., London, 1864-1891), by W. and E. L. C. P. Hardy, who have also translated the greater part of it into English. The section from 1325 to 1471 has been edited by L. M. E. Dupont (Paris, 1858-1863). WAUSAU, a city and the county-seat of Marathon county, Wisconsin, U.S.A., on both banks of the Wisconsin river, about 185 m. N.W. of Milwaukee. Pop. (1890) 9253; (1900) 12,354, of whom 3747 were foreign-born; (1910 census) 16,560. There is a large German element in the population, and two German semi-weekly newspapers are published here. Wausau is served by the Chicago, Milwaukee & St Paul and the Chicago & North-Western railways. The city is built for the most part on a level plateau above the river and extends to the top of high bluffs on either side. It has a fine city hall, a Carnegie library, the Marathon County Court House, a hospital, built by the Sisters of the Divine Saviour, and a Federal Building. In Wausau are a U.S. land office, the Marathon County Training School for Teachers, the Marathon County School of Agriculture and Domestic Science, and a County Asylum for the Chronic Insane. Valuable water-power furnished by the Big Bull Falls of the Wisconsin (in the city) is utilized for manufacturing, and in 1910 water-power sites were being developed on the Wisconsin river immediately above and below the city. In 1905 the factory products were valued at $4,644,457. Wausau had its origin in a logging-camp, established about 1838. In 1840 a saw-mill was built here, and in 1858 the village was incorporated under its present name. After 1880, when Wausau was chartered as a city, its growth was rapid.

WAUTERS, EMILE (1848- ), Belgian painter, was born in Brussels, 1848. Successively the pupil of Portaels and Gérôme, he produced in 1868 "The Battle of Hastings: the Finding of the body of Harold by Edith," a work of striking, precocious talent. A journey was made to Italy, but that the study of the old masters in no wise affected his individuality was proved by "The Great Nave of St Mark's " (purchased by the king of the Belgians). As his youth disqualified him for the medal of the Brussels Salon, which otherwise would have been bis, he was sent, by way of compensation, by the minister of fine arts, as artist-delegate to Suez for the opening of the canal a visit that was fruitful later on. In 1870, when he was yet only twenty-two years of age, Wauters exhibited his great historical picture of "Mary of Burgundy entreating the Sheriffs of Ghent to pardon the Councillors Hugonet and Humbercourt" (Liége Museum) which created a veritable furore, an impression which was confirmed the following year at the London International Exhibition. It was eclipsed by the celebrated "Madness of Hugo van der Goes" (1872, Brussels Museum), a picture which led to the commission for the two large works decorating the Lions' staircase of the Hôtel de Ville-"Mary of Burgundy swearing to respect the Communal Rights of Brussels, 1477 and "The Armed Citizens of Brussels demanding the Charta from Duke John IV. of Brabant." His other large compositions Sobieski and his Staff before Besieged Vienna " comprise (Brussels Muscum) and the harvest of a journey to Spain and Tangiers, "The Great Mosque," and "Serpent Charmers of Sokko," and a souvenir of his Egyptian travel, “ Cairo, from the Bridge of Kasr-el-Nil" (Antwerp Museum). His vast panorama -probably the noblest and most artistic work of this class ever produced-" Cairo and the Banks of the Nile" (1881), 380 ft. by 49 ft., executed in six months, was exhibited with extraordinary success in Brussels, Munich, and the Hague. Wauters is equally eminent as a portraitist, in his earliest period exhibiting, as in his pictures, sober qualities and subtle grip, but later on

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See M. H. Spielmann, Magazine of Art (1887); A. J. Wauters, Magazine of Art (1894); Joseph Anderson, Pall Mall Magazine (1896); G. Seraé ("Wauters as a Painter of Architecture") Architectural Record (1901). (M. H. S.)

WAVE.1 It is not altogether easy to frame a definition which shall be precise and at the same time cover the various physical 66 wave phenomena to which the term " is commonly applied. Speaking generally, we may say that it denotes a process in which a particular state is continually handed on without change, or with only gradual change, from one part of a medium to another. The most familiar instance is that of the waves which are observed to travel over the surface of water in consequence of a local disturbance; but, although this has suggested the name 1 since applied to all analogous phenomena, it so happens that water-waves are far from affording the simplest instance of the process in question. In the present article the principal types of wave-motion which present themselves in physics are reviewed in the order of their complexity. Only the leading features are as a rule touched upon, the reader being referred to other articles for such developments as are of interest mainly from the point of view of special subjects. The theory of waterwaves, on the other hand, will be treated in some detail.

1. Wave-Propagation in One Dimension.

The simplest and most casily apprehended case of wave-motion is that of the transverse vibrations of a uniform tense string. The axis of x being taken along the length of the string in its undisturbed This is assumed to be infinitely small; the resultant lateral force position, we denote by y the transverse displacement at any point. on any portion of the string is then equal to the tension (P, say) multiplied by the total curvature of that portion, and therefore in the case of an element &x to Pyox, where the accents denote differentiations with respect to x. Equating this to påx.j, where p is the line-density. we have

where

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occurring till the Bible of 1551 (Skeat, Etym. Dict., 1910). The proper
as a substantive, is late in English, not
O. Eng, word was wag, which became wawe in M. Eng.; it is cognate
with Ger. Woge, and is allied to " wag," to move from side to side,
and is to be referred to the root wegh, to carry, Lat. rehere, Eng.
weigh," &c. The O. Eng. wafian, M.Eng. waven,to fluctuate, to waver
in mind, cf. waefre, restless, is cognate with M.H.G. wabelen, to
move to and fro, cf. Eng. "wabble" of which the ultimate root is
seen in "whip," and in
quaver."

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The general solution of (1) was given by J. le R. d'Alembert in 1747; | 1806?). The temperature is not really constant, but rises and falls

it is

y=f(cl-x)+F(ct+x),

(3) where the functions f, F are arbitrary. The first term is unaltered in value when x and ct are increased by equal amounts; hence this term, taken by itself, represents a wave-form which is propagated without change in the direction of x-positive with the constant velocity c. The second term represents in like manner a wave-form travelling with the same velocity in the direction of x-negative; and the most general free motion of the string consists of two such wave-forms superposed. In the case of an initial disturbance confined to a finite portion of an unlimited string, the motion finally resolves itself into two waves travelling unchanged in opposite directions. In these separate waves we have .

j=cy',

(4) as appears from (3), or from simple geometrical considerations. It is to be noticed, in this as in all analogous cases, that the wavevelocity appears as the square root of the ratio of two quantities, one of which represents (in a generalized sense) the elasticity of the medium, and the other its inertia.

The expressions for the kinetic and potential energies of any portion of the string are

T=fdx, V=Pfy2dx,

(5) where the integrations extend over the portion considered. The relation (4) shows that in a single progressive wave the total energy is half kinetic and half potential.

When a point of the string (say the origin O) is fixed, the solution takes the form

y=f(ct-x)-f(cl+x).

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(6) As applied (for instance) to the portion of the string to the left of O, this indicates the superposition of a reflected wave represented by the second term on the direct wave represented by the first. The reflected wave has the same amplitudes at corresponding points as the incident wave, as is indeed required by the principle of energy, but its sign is reversed. The reflection of a wave at the junction of two strings of unequal densities p, p' is of interest on account of the optical analogy. If A, B be the ratios of the amplitudes in the reflected and transmitted waves, respectively, to the corresponding amplitudes in the incident wave, it is found that

A=-(-1)/(μ+1), B=2μ/(+1), . (7) where (pp), is the ratio of the wave-velocities. This is on the hypothesis of an abrupt change of density; if the transition be gradual there may be little or no reflection.

The theory of waves of longitudinal vibration in a uniform straight rod follows exactly the same lines. If denote the displacement of a particle whose undisturbed position is x, the length of an element of the central line is altered from dx to 8x+8%, and the elongation is therefore measured by . The tension across any section is accordingly Ew, where w is the sectional area, and E denotes Young's modulus for the material of the rod (see ELASTICITY). The rate of change of momentum of the portion included between two consecutive cross-sections is pwx., where p now stands for the volume-density. Equating this to the difference of the tensions on these sections we obtain (8) where

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c = √(E/p).

(9)

The solution and the interpretation are the same as in the case of (1). It may be noted that in an iron or steel rod the wave-velocity given by (9) amounts roughly to about five kilometres per second.

The theory of plane clastic waves in an unlimited medium, whether fluid or solid, leads to differential equations of exactly the same type. Thus in the case of a fluid medium, if the displacement & normal to the wave-fronts be a function of and x, only, the equation of motion of a thin stratum initially bounded by the planes x and +x is

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as the gas is alternately compressed and rarefied. When this is allowed for we have kypo, where y is the ratio of the two specific and the consequent value of c agrees well with the best direct heats of the gas, and therefore c=√(po/po). For air, y=1.41, terminations (332 metres per second at o° C.).

The potential energy of a system of sound waves is ks per unit volume. As in all cases of propagation in one dimension, the energy of a single progressive system is half kinetic and half potential. In the case of an unlimited isotropic elastic solid medium two types of plane waves are possible, viz. the displacement may be normal or tangential to the wave-fronts. taken in the direction of propagation, then in the case of a normal The axis of being displacement & the traction normal to the wave-front is (+2)/dx. where A, are the elastic constants of the medium, viz. μ is the "rigidity," and X-k-3, where k is the cubic elasticity. This leads to the equation

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In the case of steel (k=1.841. 1012, μ=8.19. 1011, p=7.849 C.G.S.) the wave-velocities a, b come out to be 6.1 and 3-2 kilometres per second, respectively.

If the medium be crystalline the velocity of propagation of plane given direction of the wave-normal there are in the most general waves will depend also on the aspect of the wave-front. For any case three distinct velocities of wave-propagation, each with its own direction of particle-vibration. These latter directions are perpendicular to each other, but in general oblique to the wavefront. For certain types of crystalline structure the results simplify, but it is unnecessary to enter into further details, as the matter is chiefly of interest in relation to the now abandoned elastic-solid theories of double-refraction. For the modern electric theory of light see LIGHT, and ELECTRIC WAVES.

Finally, it may be noticed that the conditions of wave-propagation without change of type may be investigated in another manner. If we impress on the whole medium a velocity equal and opposite to that of the wave we obtain a "steady or" stationary" state in Thus in the case of the vibrations of an inextensible string we may, which the circumstances at any particular point of space are constant. in the first instance, imagine the string to run through a fixed smooth tube having the form of the wave. The velocity c being constant there is no tangential acceleration, and the tension P is accordingly uniform. The resultant of the tensions on the two ends of an element ôs is Pôs/R, in the direction of the normal, where R denotes the radius of curvature. This will be exactly sufficient to produce the normal acceleration c/R in the mass pos, provided 2=P/p. Under this condition the tube, which now exerts no pressure on the string, may be abolished, and we have a free stationary wave on a moving string. This argument is due to P. G. Tait.

The method was applied to the case of air-waves by W. J. M. Rankine in 1870. When a gas flows steadily through a straight tube of unit section, the mass m which crosses any section in unit time must be the same; hence if u be the velocity we have

pum.

(18) Again, the mass which at time occupies the space between two fixed sections (which we will distinguish by suffixes) has its momentum increased in the time &t by (murmu2) dt, whence Pi-Pa=m(-us).

Combined with (18) this gives

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m2 = p2dp/dp,

(19)

(20)

Hence for absolutely steady motion it is essential that the expression +mp should have the same value throughout the wave. This condition is not accurately fulfilled by any known substance, whether subject to the "isothermal " or adiabatic" condition; but in the case of small variations of pressure and density the relation is equivalent to (21) and therefore by (18), if c denote the general velocity of the current, c2=dp/dp=k/p, (22) in agreement with (13). The fact that the condition (20) can only be satisfied approximately shows that some progressive change of type must inevitably take place in sound-waves of finite amplitude. This question has been examined by S. D. Poisson (1807), Sir G. G. Stokes (1848), B. Riemann (1858), S. Earnshaw (1858), W. J. M. Rankine (1870), Lord Rayleigh (1878) and others. It appears that

in

the more condensed portions of the wave gain continually on the less condensed, the tendency being apparently towards the production of a discontinuity, somewhat analogous to a "bore water-waves. Before this stage can be reached, however, dissipative forces (so far ignored), such as viscosity and thermal conduction, come into play. In practical acoustics the results are also modified by the diminution of amplitude due to spherical divergence § 2. Wave-Propagation in General.

We have next to consider the processes of wave-propagation in two or three dimensions. The simplest case is that of air-waves. When terms of the second order in the velocities are neglected, the dynamical equations are

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This represents two spherical waves travelling outwards and inwards, respectively, with the velocity c, but there is now a progressive change of amplitude. Thus in the case of the diverging wave represented by the first term, the condensation in any particular part of the wave continually diminishes as I/r as the wave spreads. The potential energy per unit volume [§ 1 (5)] varies as s, and so diminishes in inverse proportion to the square of the distance from O. It may be shown that as in the case of plane waves the total energy of a diverging (or a converging) wave is half potential and half kinetic.

•= ÷ SSF (c1)dw+å [÷S Sƒ(c1)dw], •

Sp=

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The solution of the general equation (5), first given by S. D. Poisson in 1819, expresses the value of s at any given point Pat time 1, in terms of the mean values of s and s at the instant =0 over a spherical surface of radius ct described with P as centre, viz. (8) where the integrations extend over the surface of the aforesaid sphere, do is the solid angle subtended at P by an element of its surface, and f(ct), F(ct) respectively denote the original values of s and s at the position of the element. Hence, if the disturbance be originally confined to a limited region, the agitation at any point P external to this region will begin after a time r/c and will cease after a time 72/c, where 71, 7 are the least and greatest distances of P from the boundary of the region in question. The region occupied by the disturbance at any instant is therefore delimited by the envelope of a family of spheres of radius ct described with the points of the original boundary as centres.

One remarkable point about waves diverging in three dimensions remains to be noticed. It easily appears from (3) that the value of the integral fsdt at any point P, taken over the whole time of transit of a wave, is independent of the position of P, and therefore equal to zero, as is seen by taking P at an infinite distance from the original seat of disturbance. This shows that a diverging wave necessarily contains both condensed and rarefied portions. If initially we have zero velocity everywhere, but a uniform condensation s, throughout a spherical space of radius a, it is found that we have ultimately a diverging wave in the form of a spherical shell of thickness 20, and that the value of s within this shell varies from soa/r at the anterior face to soa/r at the interior face, r denoting the mean radius of the shell.

The process of wave-propagation in two dimensions offers some peculiarities which are exemplified in cylindrical waves of sound, in waves on a uniform tense plane membrane, and in annular waves

on a horizontal sheet of water of (relatively) small depth. The equation of motion is in all these cases of the form a's (9)

where Vi2=02/dx2+ay2. In the case of the membrane s denotes the displacement normal to its plane; in the application to water-waves it represents the elevation of the surface above the undisturbed level. The solution of (9), even in the case of symmetry about the origin, is analytically Amuch less simple than that of (6). It appears that the wave due to a transient local disturbance, even of the simplest type, is now not sharply defined in the rear, as it is in the front, but has an B indefinitely prolonged "tail." This is illustrated by the annexed figures which represent graphically the time-variations in the condensation s at a particular point, as a wave originating in a local condensation passes over this point. The curve A represents (in a typical case) the effect of a plane wave, B that of a cylindrical wave, and C that of a spherical wave. The changes of type from A to B and from B to C are accounted for by the increasing degree of mobility of the medium.

FIG. 1.1

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as in 1. It appears then that the "dilatation "A and the "rotations",, are propagated with the velocities a, b, respectively. By formulae analogous to (8) we can calculate the values of A, .. at any instant in terms of the initial conditions. The subsequent determination of u, v, w is a merely analytical problem into which we do not enter; it is clear, however, that if the original disturbance be confined to a limited region we have ultimately two concentric spherical diverging waves. In the outer one of these, which travels with the velocity a, the rotations 7. vanish, and the wave is accordingly described as "irrotational,' or "condensational." In the inner wave, which travels with the smaller velocity b, the dilatation ▲ vanishes, and the wave is therefore characterized as "equivoluminal" or "distortional.' In the former wave the directions of vibration of the particles tend to become normal, and in the latter tangential, to the wave-front, as in the case of plane elastic waves (§ 1).

The problems of reflection and transmission which arise when a wave encounters the boundary of an elastic-solid medium, or the interface of two such media, are of interest chiefly in relation to the older theories of optics. It may, however, be worth while to remark that an irrotational or an equivoluminal wave does not in general give rise to a reflected (or transmitted) wave of single character; thus an equivoluminal wave gives rise to an irrotational as well as an equivoluminal reflected wave, and so on.

Finally, in a limited elastic solid we may also have systems of waves of a different type. These travel over the surface with a definite velocity somewhat less than that of the equivoluminal waves above referred to; thus in an incompressible solid the velocity is 9554b; in a solid such that A- it is 91946. The agitation due to these waves is confined to the immediate neighbourhood of the surface, diminishing exponentially with increasing depth. The theory of these surface waves was given by Lord Rayleigh in 1885In the modern theory of earthquakes three phases of the disturbance

1Figures 1, 2, 4, 6, 7 and 8 are from Professor Horace Lamb's Hydrodynamics, by permission of the Cambridge University Press.

at a station distant from the origin are recognized; the first corresponds to the arrival of condensational waves, the second to that of distortional waves, and the third to that of the Rayleigh waves (see ELASTICITY). The theory of waves diverging from a centre in an unlimited crystalline medium has been investigated with a view to optical theory by G. Green (1839), A. L. Cauchy (1830), E. B. Christoffel (1877) and others. The surface which represents the wave-front consists of three sheets, each of which is propagated with its own special velocity. It is hardly worth while to attempt an account here of the singularities of this surface, or of the simplifications which occur for various types of crystalline symmetry, as the subject has lost much of its physical interest now that the elastic-solid theory of light is practically abandoned.

83. Water-Waves. Theory of "Long" Waves.

The simplest type of water-waves is that in which the motion of the particles is mainly horizontal, and therefore (as will appear) sensibly the same for all particles in a vertical line. The most conspicuous example is that of the forced oscillations produced by the action of the sun and moon on the waters of the ocean, and it has therefore been proposed to designate by the term "tidal" all cases of wave-motion, whatever their scale, which have the above characteristic property.

Beginning with motion in two dimensions, let us suppose that the axis of x is drawn horizontally, and that of y vertically upwards. If we neglect the vertical acceleration, the pressure at any point will have the statical value due to the depth below the instantaneous position of the free surface, and the horizontal pressure-gradient aplar will therefore be independent of y. It follows that all particles which at any instant lie in a plane perpendicular to Ox will retain this relative configuration throughout the motion. The equation of horizontal motion, on the hypothesis that the velocity (u) is infinitely small, will be

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(6)

The solution is as in § 1, and represents two wave-systems travelling with the constant velocity (gh), which is that which would be acquired by a particle falling freely through a space equal to half the depth.

Two distinct assumptions have been made in the foregoing investigation. The meaning of these is most easily understood if we consider the case of a simple-harmonic train of waves in which

n=ẞcosk(cl-x), u= • cos k(cl-x),... (7)

"' waves.

where k is a constant such that 27/k is the wave-length A. The first assumption, viz. that the vertical acceleration may be neglected in comparison with the horizontal, is fulfilled if kh be small, i.e. if the wave-length be large compared with the depth. It is in this sense that the theory is regarded as applicable only to "long, The second assumption, which neglects terms of the second order in forming the equation (1), implies that the ratio n/h of the surfaceelevation to the depth of the fluid must be small. The formulae (7) indicate also that in a progressive wave a particle moves forwards or backwards according as the water-surface above it is elevated or depressed relatively to the mean level. It may also be proved that the expressions (8) for the kinetic and potential energies per unit breadth are equal in the case of a progressive wave. It will be noticed that there is a very close correspondence between the theory of "long" water-waves and that of plane waves of sound, e.g. the ratio /h corresponds exactly to the condensation " in the case of air-waves. The theory can be adapted, with very slight adjustment, to the case of waves propagated along a canal of any uniform section, provided the breadth, as well as the depth,

T={phfu2dx, V =}gofn2dx,

be small compared with the wave-length. The principal change is that in (6) h must be understood to denote the mean depth. The theory was further extended by G. Green (1837) and by Lord Rayleigh to the case where the dimensions of the cross-section are variable. If the variation be sufficiently gradual there is no sensible reflection, a progressive wave travelling always with the velocity appropriate to the local mean depth. There is, however, a variation of amplitude; the constancy of the energy, combined with the equation of continuity, require that the elevation in any particular part of the wave should vary as b--, where b is the breadth of the water surface and h is the mean depth.

Owing to its mathematical simplicity the theory of long waves in canals has been largely used to illustrate the dynamical theory of the tides. In the case of forced waves in a uniform canal, the equation (1) is replaced by

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The coefficient in the former of these equations is negative unless the ratio h/a exceed oa/g, which is about 1/311. Hence unless the

depth of our imagined canal be much greater than such depths as are actually met with in the sea the tides in it would be inverted, 1.e. there would be low water beneath the moon and at the antipodaĺ point, and high water on the meridian distant 90° from the moon. This is an instance of a familiar result in the theory of vibrations, viz. that in a forced oscillation of a body under a periodic force the phase is opposite to that of the force if the imposed frequency exceed that of the corresponding free vibration (see MECHANICS). In the present case the period of the free oscillation in an equatorial canal 11,250 ft. deep would be about 30 hours.

When the ratio 7/h of the elevation to the depth is no longer treated as infinitely small, it is found that a progressive wavesystem must undergo a continual change of type as it proceeds, even in a uniform canal. It was shown by Sir G. B. Airy (1845) that the more elevated portions of the wave travel with the greater velocities, the expression for the velocity of propagation being

c(1+In/h)

approximately. Hence the slopes will become continually steeper in front and more gradual behind, until a stage is reached at which the vertical acceleration is no longer negligible, and the theory inwards in shallow water near the shore. ceases to apply. The process is exemplified by sea-waves running periodic waves of finite (as distinguished from infinitely small) The theory of forced amplitude was also discussed by Airy. It has an application in tides" (see TIDE). tidal theory, in the explanation of "overtides" and compound

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