I. G. PETROVSKY LECTURES ON PARTIAL DIFFERENTIAL EQUATIONS FIRST ENGLISH EDITION 954 CONTENTS Forewor b R. Coran Translaor's Noe b Abe Shenzer Preface Chaper I. Inrocon. Classfcaon of eqaons. Defnons. Eamples.. The Cach problem. The Cach-Kowalewsk heorem. The generalze Cach problem. Characerscs 4. Unqeness of he solon of he Cach problem n he class of non-analc fncons 5. Recon o canoncal form a a pon an classfcaon of eqaons of he secon orer n one nknown fncon 6. Recon o canoncal form n a regon of a paral fferenal eqaon of he secon orer n wo nepenen varables 7. Recon o canoncal form of a ssem of lnear paral fferenal eqaons of he frs orer n wo nepenen varables Chaper II. Hperbolc eqaons I. The Cach problem for non-analc fncons
8. The reasonableness of he Cach problem 9. The noon of generalze solons. The Cach problem for hperbolc ssems n wo nepenen varables. The Cach problem for he wave eqaon. Unqeness of he solon. Formlas gvng he solon of he Cach problem for he wave eqaon. Eamnaon of he formlas whch gve he solon of he Cach problem 4. The Lorenz ransformaon 5. The mahemacal fonaons of he specal prncple of relav 6. Srve of he fnamenal facs of he heor of he Cach problem for general hperbolc ssems II. Vbraons of bone boes 7. Inrocon 8. Unqeness of he me nal an bonar-vale problem 9. Connos epenence of he solon on he nal aa. The Forer meho for he eqaon of a vbrang srng page. The general Forer meho nrocor conseraons. General properes of egenfncons an egenvales. Jsfcaon of he Forer meho 4. Anoher jsfcaon of he Forer meho 5. Invesgaon of he vbraon of a membrane
6. Spplemenar nformaon concernng egenfncons Chaper III. Ellpc eqaons 7. Inrocon 8. The mnmm-mamm proper an s conseqences 9. Solon of he Drchle problem for a crcle. Theorems on he fnamenal properes of harmonc fncons. Proof of he esence of a solon of Drchle's problem. The eeror Drchle problem. The Nemann problem he secon bonar-vale problem 4. Poenal heor 5. Applcaon of poenal heor o he solon of bonar-vale problems 6. Appromae solon of he Drchle problem b he meho of fne fferences 7. Srve of he mos mporan resls for general ellpc eqaons Chaper IV. Parabolc eqaons 8. Concon of hea n a bone srp he frs bonar-vale problem 9. Concon of hea n an nfne srp he Cach problem 4. Srve of some frher nvesgaons of eqaons of he parabolc pe
CHAPTER I INTRODUCTION. CLASSIFICATION OF EQUATIONS. Defnons. Eamples. An eqaon conanng paral ervaves of nknown fncons K N s sa o be of he nh orer f conans a leas one paral ervave of he nh orer an no paral ervaves of orer hgher han n. B he orer of a ssem of eqaons conanng paral ervaves we mean he orer of he hghes orer eqaon of he ssem. A paral fferenal eqaon s calle lnear f s lnear n he nknown fncons an n her ervaves; s calle qas-lnear f s lnear n he hghes orer ervaves of he nknown fncons. Ths for nsance he eqaon s qas-lnear of he secon orer wh respec o he nknown fncon. The eqaon a s lnear of he secon orer wh respec o. The eqaon s neher lnear nor qas-lnear wh respec o. B a solon of an eqaon conanng paral ervaves we mean a 4
5 ssem of fncons whch when p n he eqaon n place of he nknown fncons rns he eqaon no an en n he nepenen varables. A solon of a ssem of eqaons s efne n an analogos manner. We shall be prmarl nerese n lnear eqaons of he secon orer n one nknown fncon. The followng are eamples of eqaons of hs pe: he hea eqaon he wave eqaon he Laplace eqaon. Man problems n phscs rece hemselves o paral fferenal eqaons n parclar o he paral fferenal eqaons lse above... The Cach problem. The Cach-Kowalewsk heorem. Formlaon of he Cach problem. Le here be gven he followng ssem of eqaons n he paral ervaves of nknown fncons N K wh respec o he nepenen varables n K : ; ; j j n kn n k k j k N n n n n k n k k k k N j F < L K K L K K
. I s clear from or eqaons ha for each of he nknown fncons j here ess a nmber ervaves of n whch gves he orer of he hghes orer n appearng n he ssem. The nepenen varable s seen o be sngshe n wo respecs. Frs he ervave ms appear among he ervaves of he hghes orer fncon n / n n of he...n an seconl he ssem s solve for hese ervaves. In phscal problems me occpes hs sngshe poson an he oher varables K enoe space coornaes. The nmber of eqaons s eqal o he nmber of nknown fncons. For some vale of we prescrbe he vales 'nal vales' n of he nknown fncons an of her ervaves wh respec o of orers p o n. In oher wors for k k k k K n k K n. All fncons K are prescrbe n he same regon G of n he space K. B he ervave of orer zero of he fncon n we mean he fncon self. The Cach problem consss n fnng a solon of he ssem. sasfng he nal conons.. CAUCHY-KOWALEWSKI THEOREM. If all he fncons F are analc n some neghborhoo of he pon 6
n j k k K kn K K an f all he fncons k j are n analc n some neghborhoo of he pon K hen he Cach problem has a nqe analc solon n some neghborhoo of n he pon K.. The generalze Cach problem. Characerscs.. The generalze Cach problem. Le here be gven a ssem of N eqaons wh N nknown fncons K N k j Φ ; ; K n K N K K k k kn Ln For each fncon ervaves of here ess a hghes orer j K N. n of he paral wh respec o he nepenen varables K n whch appear n he ssem.. In he regon of pons n K ner conseraon here s gven a sffcenl smooh n-mensonal srface S an hrogh each pon of S here passes a crve l no angen o S whch s assme o var sffcenl smoohl as we go from pon o pon on S. The vales of he fncons an of her ervaves of orers p o n n he recon of he crves l are prescrbe on he srface S. These conons as prescrbe on S are a generalzaon of he Cach conons nal aa consere n he 7
prevos secon. We are reqre o fn n some neghborhoo of he srface S a solon conons prescrbe on S. K N of he ssem. sasfng he Eample. For he Laplace eqaon he relaon.7 assmes he form L n n L. Ths relaon ogeher wh. mpl ha he Laplace eqaon has no real characerscs. Eample. For he wave eqaon he relaon.7 akes he form. Snce accorng o. we ms also have follows ha or ±/ Ths means ha he angen planes o all characersc srfaces form an angle of 45 eg wh he -as. Usng hs proper of he characersc srfaces one can easl pcre he form of he characersc srfaces 8
gong hrogh efne crves n he plane cons. Ths for eample f l s a sragh lne n ha plane he characersc srface gong hrogh l s he plane hrogh l nclne a 45 eg wh respec o he plane cons. Agan f K s he crcmference of a crcle n he plane cons. he characersc srfaces hrogh K wll be crclar cones whose aes are parallel o he -as an whose generaors form a 45 eg angle wh he plane cons. or whch s he same hng wh he -as. Eample. For he hea eqaon he relaon 7 akes he form n n L L. Usng. we concle ha. Therefore he characersc srfaces are he hperplanes cons. Eample 4. For he eqaon a K n a K n K an K n.7 akes he form a K n a K n L an K n n Hence all hperplanes gong hrogh he pon K an he n vecor a K a emanang from K are characersc a n K. n n Eample 5. For he ssem of eqaons n wo nknowns n. 9
n j j n j j a bj cj j j j n he relaon.7 assmes he form a b. j j In hs case he characersc crves are he crves along whch / / : / s eqal o a roo k of he eqaon ka j bj. Here we are assmng ha s he eqaon of he characersc crve. n 5. Recon o canoncal form a a pon an classfcaon of eqaons of he secon orer n one nknown fncon. We conser he lnear secon-orer eqaon j A K j F K n n j B K n one nknown fncon. We assme ha n j C K j A A n 5. B C an F are all real-vale an efne n some regon of he space K n. We now efne a coornae ransformaon b png
where n a k n 5. k k a k are ceran consans. We assme ha he ransformaon 5. s non-snglar.e. ha he eermnan a k oes no vansh. Then he ransformaon 5. s one-o-one. In he new coornaes K n eqaon 5. wll ake he form n n A a j k l j k a lj L. 5. k l Here we have wren o onl erms wh secon-orer ervaves of he nknown fncon. I s clear from he eqal 5. ha ner he ransformaon 5. of he nepenen varables he coeffcens of he secon-orer ervaves of change n eacl he same manner n whch he coeffcens of he qarac form change when he n j j We regar he coeffcens A k are replace b he n j 5.4 k accorng o he formlas k a k k..n 5.5 A j of he form 5.4 as consans whose vales are he same as hose of he vales of he coeffcens A K of eqaon 5. a some pon K n G. j n n I s prove n algebra ha here ess a real non-snglar
ransformaon 5.5 whch reces an form 4.5 wh real coeffcens A j o he form m ± where m n. 5.6 There are man non-snglar real ransformaons 5.5 whch rece he form 5.4 o he form 5.6 b he nmber of erms wh posve sgns an he nmber of erms wh negave sgns n he form 5.6 s eermne enrel b he form 5.4 an s nepenen of he choce of he non-snglar ransformaon 5.5. Law of nera of qarac forms. If a ceran ransformaon 5.5 reces he form 5.4 o he form 5.6 hen he ransformaon 5. wh mar eqal o he ranspose nverse of a k reces 5. o he form where n j A* j K n L j n ± A* K f j m j 5.7 j n A* K f j or j> m. Here we have wren o onl erms conanng ervaves of he hghes orer of he fncon. The form 5.7 of eqaon 5. s calle n s canoncal form a he pon K. n I s hs possble o ehb for each pon K of he regon
G a non-snglar ransformaon of he nepenen varables whch reces eqaon 5. a ha pon o canoncal form. In general he ransformaon 5. whch reces eqaon 5. o n canoncal form a a gven pon K vares wh hs pon.e. ma no rece eqaon 5. o canoncal form a a pon fferen n from K. I can be shown b means of eamples ha as soon as he nmber of nepenen varables ecees wo s n general mpossble o ehb a lnear ransformaon wh consan coeffcens or for ha maer an oher ransformaon whch wol rece a gven lnear eqaon of he secon orer o canoncal form n an arbrarl small regon. In he case of wo varables sch a ransformaon ess ner ver general assmpons on he coeffcens of eqaon 5.. Ths fac wll be emonsrae n he followng secon. The classfcaon of eqaons of he secon orer s base on he possbl of recng eqaon 5. o canoncal form a a pon.. n Eqaon 5. s sa o be ellpc a a pon K f all n A* K are fferen from zero an have he same sgn. n Eqaon 5. s sa o be hperbolc a a pon K f all n b one A* K n 5.7 have he same sgn he eceponal A* s of oppose sgn an m n. n Eqaon 5. s sa o be lrahperbolc a a pon K n f n 5.7 he nmber of posve A* K ecees one he
n nmber of negave A* K ecees one an m n. v Eqaon 5. s sa o be parabolc n he broa sense a a pon n n K f some A* K are zero.e. f m < n. v Eqaon 5. s sa o be parabolc n he resrce sense or n smpl parabolc f onl one of he coeffcens A* K A * sa s zero an all oher A* have he same sgn an he coeffcen of / s fferen from zero. Eqaon 5. s sa o be ellpc hperbolc lrahperbolc ec. n he whole regon G f s ellpc hperbolc lrahperbolc ec. a each pon of he regon G.. The non-lnear eqaon of he secon orer Φ K K n K K n j n one nknown fncon s sa o be ellpc hperbolc or parabolc n he broa sense for a gven solon K a a pon n K or n he regon G f he eqaon N j Aj K n j * n where Φ Aj K n 5.8 j s ellpc hperbolc or parabolc n he broa sense a he pon 4
n K or n G. On he rgh se of 5.8 he fncon an s ervaves are replace b he fncon K an s * n approprae ervaves. Laer we shall s onl lnear eqaons of he secon orer n one nknown fncon whch are eher ellpc or hperbolc or parabolc n he whole regon ner conseraon. We shall no be nerese n lrahperbolc eqaons for he o no appear n phscs or n engneerng. We shall lkewse ake no neres n eqaons whch are parabolc n he broa sense for he rn p selom n apple work. When speakng of parabolc eqaons n Chaper IV we shall accorngl have n mn eqaons parabolc n he narrow sense. 6. Recon o canoncal form n a regon of a paral fferenal eqaon of he secon orer n wo nepenen varables. Conser he eqaon A B C F. 6. Here he coeffcens A B an C are fncons of an whch are wce connosl fferenable. We shall assme ha he fncon s also wce connosl fferenable. We now go from he nepenen varables o he nepenen varables. Le he fncons 6. be wce connosl fferenable an le he Jacoban 5
6 be fferen from zero n he regon ner conseraon. I s hen possble o solve he ssem 6. nqel for an n some regon of he plane. The reslng fncons an wll also be wce connosl fferenable fncons of an. In he new nepenen varables eqaon 6. akes he form F C B A C B B A C B A 6. We now conser he fferenal eqaon C B A 6.4 n he nknown fncon. We shall have o eal separael wh he cases AC B > AC B < an AC B n all of he regon ner conseraon. We shall no conser he case when he epresson
B AC changes sgn n he regon consere or when vanshes a some pon of he regon who vanshng encall n all of.. Hperbolc We frs nvesgae he case B > AC n he regon ner conseraon.e. he case when eqaon 6. s hperbolc cf. he efnon of hperbolc gven n he preceng paragraph. In hs case eqaon 6.4 s eqvalen o he wo eqaons A A [ B B AC ] / [ ] B B AC / 6.5 If he coeffcen A vanshes hen one of he eqaons 6.5 becomes neermnae f for nsance B > hen he secon eqaon n 6.5 becomes neermnae. Is neermnac however s easl remove b mlplng boh ses b he encall eqal epressons / / [ B B AC ]/ A C /[ B B AC ]. 6.6 We now eermne he fncons as solons of he eqaons 6.5 b prescrbng her vales on ceran crves l whch are nowhere angen o he characerscs of he corresponng eqaon. If he crves l an he vales of he fncons prescrbe on hem are sffcenl smooh we oban solon fncons havng connos ervaves wh respec o an of orer p o an nclng he secon. If n aon we assme ha 7
he nal vales on l have been chosen so ha he ervave of n he recon of l oes no vansh anwhere hen a no pon wll he ervaves of he fncons wh respec o an vansh smlaneosl. Snce he vanshng a some pon of hese wo ervaves wol mpl he vanshng a ha pon of he ervave of n an arbrar recon or saemen wll have been prove f we can show ha a no pon A oes he ervave of vansh n all recons. Le s move n a non-characersc recon from he pon A o some pon A'. Le s enoe b B he pon of nersecon of l an he characersc hrogh A an b B' he pon of nersecon of hrogh A' Fg. 4. l an he characersc Then n vew of he fac ha s consan along a characersc we have A B A' B' A' A B' B The wo sances l beween he pon A an A' an l' beween he pons B an B' approach zero smlaneosl. Also n vew of he assme smoohness of he coeffcens of eqaon 6. hese wo sances are of he same orer of smallness. Snce b assmpon B' B lm l l' 8
we ms also have A' A lm. l l I follows ha a no pon o he ervaves wh respec o an of eher one of he fncons an vansh smlaneosl. Ths means ha he Jacoban of hese fncons canno vansh a an pon of he regon ner conseraon; for snce neher row of he Jacoban vanshes a necessar conon for s vanshng s he proporonal of s colmns b / / B B AC : A B B AC A : becase B AC. An neermnac n an one of or relaons can be remove b sng he en 6.6. Snce he Jacoban of an wh respec o an oes no vansh we ma p n 6. If we o hs he erms n 6. whch conan 6.7 / an / ms vansh. A he same me he coeffcen of 9
/ wll be fferen from zero n he whole regon consere. Oherwse a change from he coornaes o he coornaes wol ecrease he orer of he eqaon an conseqenl he oppose change from he coornaes o he coornaes wol ncrease he orer of he eqaon whch s an obvos mpossbl. Dvng eqaon 6. b he coeffcen of o he canoncal form F / we rece 6.8 n all of he regon of efnon of he fncons an. If we p β an β eqaon 6.8 becomes β Φ β. 6.9 β The laer form of he eqaon s also calle canoncal. If a hperbolc eqaon has been rece o he canoncal form 6.8 s somemes possble o negrae n close form.e. o fn a formla gvng all solons of hs eqaon. Eample. B means of he sbson v he eqaon
becomes v v. 6. The laer eqaon can be easl negrae b he meho of separaon of varables for eners n v onl as a parameer. The consan of negraon wll be a fncon of hs parameer. We have or Whence Here ln v ln ln C v C. C C. C C s an arbrar n vew of he arbrarness of C fferenable fncon of an C s an arbrar fncon of. Eample. The ransformaon reces he eqaon
o he form 6. Hence ψ ψ 6. 6. where an ψ are arbrar wce connosl fferenable fncons.. Parabolc If B AC n all of he regon G hen eqaon 6. s parabolc n G cf. he efnon of parabolc n he preceng secon. In hs case boh eqaons 6.5 are he same an can be replace b he sngle eqaon or eqvalenl b he eqaon Le A B 6.4 B C. 6.5 C be he general negral of hs eqaon n some regon G of he plane. We assme ha he fncon s wce connosl
fferenable an ha s frs paral ervaves o no vansh smlaneosl cf. Para.. Le ψ C be a faml of crves n he regon G sch ha he fncon ψ s sffcenl smooh an he Jacoban ψ ψ 6.6 oes no vansh anwhere n he regon G. If for eample > a all pons of he regon G we ma assme ha Le s p n 6.. Then he coeffcen of coeffcen of ψ. an ψ / becomes / n 6. vanshes. The ψ A B B C. Accorng o 6.4 an 6.5 hs coeffcen wll also vansh. We assme ha a no pon of he regon G o all hree coeffcens A B an C of eqaon 6. vansh smlaneosl. Snce B AC hs assmpon mples ha a each pon of G one of he coeffcens A an
C s fferen from zero. Le A a some efne pon. Then he coeffcen of A / n eqaon 6. akes he form ψ ψ ψ ψ ψ ψ B C A B. A Ths epresson canno be zero: for oherwse n vew of 6.4 he Jacoban 6.6 wol vansh a he pon ner conseraon. We can show n he same wa ha he coeffcen of / oes no vansh a pons a whch C. Ths means ha hs coeffcen canno vansh anwhere n he regon G. Conseqenl we are allowe o ve eqaon 6. b hs coeffcen. We hen oban F. 6.7 Accorng o he efnon of canoncal form gven n 5 eqaon 6.7 s n canoncal form n he regon G. If eqaon 6. s lnear hen so s eqaon 6.7. Sppose ha 6.7 s of he form A B C D. 6.8 I s possble o smplf hs eqaon somewha b nrocng a new nknown fncon z n place of. We p zv where v s a fncon of an sll o be efne. Then eqaon 6.8 becomes 4
z v z z z v A v Bv C z D. 6.9 We have wren o n eal onl erms conanng ervaves of z. All erms conanng he fncon z self have been combne n z C. We choose he fncon v so ha he coeffcen of he ervave z / n eqaon 6.9 shol vansh. Eqang he coeffcen of z / o zero we ge where z z A Cz D. 6. C C / v D D / v an v s v ep B. 4. Ellpc Fnall we nvesgae he case when n all of he regon consere AC > B. Then eqaon 6. wll be ellpc n ha regon cf. he efnon of ellpc gven n 5. In hs case we assme ha he coeffcens A B an C are analc fncons of an. Then he coeffcens of eqaons 6.5 are also analc fncons of an. Le * ** be an analc solon of one of he eqaons n 6.5 n a neghborhoo of a pon. We p * an ** 6. 5
6 n 6.. Snce he Jacoban J 6. oes no vansh anwhere we can solve eqaons 6. for an. In fac separang real an magnar pars n he secon eqaon 6.5 we ge B AC B A B AC B A / / 6. Sbsng he epressons for / an / obane from 6. n he Jacoban 6. we oban / A B AC J. I follows ha or eermnan can be eqal o zero onl a hose pons a whch or n vew of eqaons 6. onl a hose pons a whch an.
7 B or regon conans no sch pons for a sch pons. Separang real an magnar pars n he en C A B we ge C A C A B B 6.4 C A B 6.5 Snce he form β β C B A AC < B s posve-efne boh ses of 6.4 can vansh onl f. 6.6 We chose he fncon n sch a manner however ha he eqaons 6.6 are no sasfe a he same me. Hence we ma ve eqaon 6. b one se of he eqaon 6.4. We ge F. 6.7 Ths form of he ellpc eqaon s calle s canoncal form.
We have rece or eqaon o canoncal form n a neghborhoo of n whch here ess an analc solon of eqaon 6.5 wh non-zero ervaves. I can be shown b means of more nrcae conseraons ha sch recon s possble who assmng ha A B an C are analc. I sffces o assme ha hese coeffcens are wce connosl fferenable. 8
CHAPTER II HYPERBOLIC EQUATIONS PART I THE CAUCHY PROBLEM FOR NON-ANALYTIC FUNCTIONS 8. The reasonableness of he Cach problem. Formlas gvng he solon of he Cach problem for he wave eqaon. Three-Dmensonal Le G n he space be he oman of efnon of a hree mes connosl fferenae fncon an a wce connosl fferenae fncon. We wsh o eermne n G he regon escrbe n remark n he solon of he eqaon. sasfng for he conons.a '.b We frs seek a solon of eqaon. when he nal conons have he specal form.a ' I s hen easl checke ha he fncon..b 9
v sasfes for he conons v ' v. Therefore he solon of eqaon. sasfng boh conons. s gven b he formla..4 Ths he general Cach problem for eqaon. s rece o he problem of fnng. We clam ha 4 S σ π.5krchhof s formla Here S enoes he sphere wh ras an cener n he hperplane where he fncon s prescrbe an σ enoes an elemen of he srface of hs sphere. We assme ha he fncon s connos an bone ogeher wh s ervaves of orer p o an nclng k k. Then as wll become apparen from formla.6 he fncon wll have connos ervaves of orer p o an nclng k. We frs show ha gven b he formla.5 sasfes he
nal conons.. Tha sasfes he frs of hese conons follows from he fac ha of bon pper leas S 4 π σ whch means ha as. As for he secon conon we noe ha f we p k k k β k he negral.5 becomes 4 S σ β β β π.6 where he negraon s carre o over he srface of he sphere S : β β β σ σ fe for all vales of. Hence 4 4 S k k k S σ β β β β π σ β β β π..7 Here k enoes he ervave of wh respec o k. I s eas o see ha as he frs erm on he rgh approaches an he secon erm approaches zero; he laer s re becase he
negral n ha erm sas bone. I remans o show ha as efne b Krchhof's formla sasfes eqaon.. From eqaon.6 we fn ha 4 4 S S σ π σ π..8 To compe / we rewre he eqal.7 as follows: I V S π π π 4 4 4.9 where I V an V s he srface of he sphere of ras wh cener a n he hperplane. From.9 we ge
I I I I 4 4 4 π π π.. Now s easl seen ha S I σ. Comparng he eqales.8. an. we see ha he fncon efne b Krchhof's formla acall sasfes he wave eqaon.. Remark. If he fncon s onl known o be connos ogeher wh s frs ervaves hen he fncon efne b he eqales.4 an.5 s onl a generalze solon of he Cach problem.. Two mensonal We now conser he specal case when oes no epen on. I s eas o see ha n ha case he fncon gven b Krchhof's formla lkewse oes no epen on an so sasfes he eqaon.
4 In hs case we can replace he negral over he sphere S b he oble negral aken over he nersecon K of hs sphere wh he plane. Projecng a srface elemen σ of hs sphere on he plane we ge [ ] / σ an Krchhof's formla can hen be rewren as follows: [ ] K S σ π σ π / 4 Therefore he solon of eqaon. sasfng he conons ' s gven b he formla [ ] [ ] K K ga ga / / π π. Ths s he so-calle Posson formla.. One mensonal If he fncon oes no epen on eher or hen he fncon
5 gven b Krchhof's formla s also nepenen of an an so sasfes he eqaon.4 In ha case s possble o rewre Krchhof's formla as follows: S 4 σ π Here we have mae se of he fac ha he area of he poron of he sphere S conane beween wo planes. cons an cons. whch nersec S s eqal o π an he fncon has almos consan vale a all pons of hs poron of he sphere. Therefore he solon of eqaon.4 sasfng he conons ' s gven b he formla.5 Ths formla s known as 'Alember's formla. We recall ha accorng o he nqeness heorem prove n here es no solons of he Cach problem oher han hose gven b.4 for eqaon. b. for eqaon. an b
.5 for eqaon.4. The meho se n obanng solons of he Cach problem for eqaons. an.4 from he solon of he Cach problem for eqaon. s calle he meho of escen. We have fon he solon of he Cach problem for >. The case < reces o he case prevosl consere f we replace b ; hs ransformaon oes no change eqaons.. an.4.. Eamnaon of he formlas whch gve he solon of he Cach problem. Connos epenence of he solon on he nal conons.. Dffson of waves. Formlas.4 an.5 show ha he vale a he pon n K of he solon of he wave eqaon. for n epens onl on he nal conons on he bonar of he base of he characersc cone wh vere a K. On he oher han for n n or n K epens on he conons as gven on he n whole base of ha cone. Ths s clear from formlas. an.5. Le s assme ha he nal vales of an ' for ffer from zero onl nse a small regon Ge abo some pon K. n Begnnng wh we conser he vales of a he pons K for fe K n an for ncreasng. In he case n n he magne of K can ffer from zero onl on a small n 6
poron of he consere sragh lne n he space K ; n namel on ha poron of he lne on whch are locae he verces of hose characersc cones he bonares of whose bases nersec he regon. Ge. On he oher han f n or n an he pon or oes no belong o Ge hen or s eqal o zero for sffcenl small an s n general fferen from zero begnnng wh he vales of for whch he segmen or he crcle nersecs he regon Ge. Conseqenl a srbance se p a n some small n neghborhoo of he pon K wll for n an > affec he vales of he fncon onl n hose pons of he space K n whch le close o he sphere of ras wh cener K. Ths a n srbance se p a a he pon gves rse o a sphercal wave wh cener a ha pon an hs wave has a well-efne fron an back. On he oher han f n or n hen a n srbance se p a n a neghborhoo of he pon K affecs n general all pons n he neror of he sphere wh ras n an cener K. The reslng wave has a well-efne fron b he back of he wave s blrre. We sa ha n hs case he wave s ffse. For n no ffson akes place. I can be shown ha no ffson akes place for he solons of eqaon. for an arbrar o n. Dsrbances se p n a small regon Ge of a hree-mensonal rg elasc bo or of a gas gve rse o waves whch leave no race afer 7
hem prove he vbraons sasf eqaon.. In he case of a gas enoes for nsance he evaon from normal ar pressre a he pon a he momen. On he oher han srbances se p n a small regon of a wo-mensonal connm as for eample a gh membrane or a waer srface gve rse o waves whch n heor leave a permanen race afer hem prove hese vbraons sasf eqaon.. In pracce he e own ver qckl becase of he esence of frcon whch s no aken no conseraon n ervng eqaon.. Lkewse a race s lef afer he passng of a wave n a one-mensonal connm cf. Para. of hs secon.. Eamnaon of 'Alember's formla. 4. The Lorenz ransformaon. In we menone he fac ha ecep for a consan mlpler he epresson s he onl lnear combnaon of secon ervaves whch oes no change s form ner a roaon of he space.e. ner an arbrar orhogonal ransformaon of he coornaes. We now conser n some eal a ceran class of lnear ransformaons of he varables wh consan real coeffcens whch are closel connece wh he wave eqaon 8
9. 4. B a Lorenz ransformaon of he varables we mean an lnear homogeneos ransformaon of hese varables j j j a 4. wh real coeffcens j a whch leaves nvaran he qarac form. 4. Ths means ha n he new varables hs qarac form becomes I s eas o check ha he oal of Lorenz ransformaons forms a grop. In parclar s eas o see ha he proc of wo Lorenz ransformaons sbsons s agan a Lorenz ransformaon. We now wre own a formla for a specal class of Lorenz ransformaons whch have he proper of leavng nvaran wo of he las hree space coornaes. Sch ransformaon has he form δ γ β 4.4 For sch ransformaons he en ms hol. Sbsng n hs en he epressons for an n 4.4 we ge
Whence β γ δ. γ β δ β γδ In parclar hese eqaons are sasfe f we p δ coshψ β γ snhψ where ψ s an arbrar nmber. Then 4.5 Le s p anh ψ coshψ snhψ snhψ coshψ β. We hen oban he sal formlas for he class of Lorenz ransformaons ner conseraon: β β /β / β / / Here β s an arbrar nmber smaller han one n absole vale becase anh ψ < for an ψ. 4.6 The formlas 4.6 are of fnamenal mporance snce as we are gong o show an Lorenz ransformaon s a combnaon of an orhogonal ransformaon of he varables whch leaves 4
fe a ransformaon of he form 4.6 an a possble change of sgn of one of he varables a refleon. If we ranspose he mar of each of he nermeae ransformaons we wll agan oban a mar of a ransformaon of he same pe. I follows from hs ha he ranspose of a mar of a Lorenz ransformaon s agan a mar of a Lorenz ransformaon. From he efnon of a Lorenz ransformaon follows ha he nverse of a Lorenz ransformaon s also a Lorenz ransformaon.. We now prove a fnamenal fac whch clarfes he close connecon beween Lorenz ransformaons an he wave eqaon. THEOREM. Ever non-snglar lnear ransformaon of he varables wh real consan coeffcens whch oes no change he form of eqaon 4. s a combnaon of a Lorenz ransformaon a ranslaon of he orgn n he space an a smlar ransformaon n ha space. 5. The mahemacal fonaons of he specal prncple of relav The specal prncple of relav assers ha all laws of nare have he 4
same form for all observers who move wh respec o one anoher wh nform sragh-lne moon. More precsel for each of hese observers here ess a 'local' space-me coornae ssem s he me coornae an are he space coornaes n whch an gven naral law s epressble b means of he same eqaons. In parclar for each of hese observers he spee of lgh n hs local coornae ssem s he same n all recons. For he sake of smplc we assme ha hs spee s eqal o. We wsh o fn a conneon beween he local space-me coornaes of wo observers A' an A" of whom A" s assme o be movng nforml an along a sragh lne wh respec o A'. The veloc of A" s eqal o β β <. In vew of he assmpon ha space an me are homogeneos an soropc we conser he reqre conneon o be lnear an s coeffcens o be fncons of β onl. We enoe he local space-me coornaes of A' b ' ' ' ' an he local space-me coornaes of A" b '' '' '' ''. For smplc n noaon we shall somemes wre ' nsea of ' an '' nsea of ". Ths le ' ' j a β ' j. 5. We shall fn he conneon beween he coornaes ' ' ' ' an he coornaes '' '' '' '' sng onl he assmpon ha he veloc of lgh s consan for he observers A' an A". 4
We escrbe he reclnear propagaon of a plane lgh wave b means of some non-consan fncon f a ' a ' a ' a ' 5. whose vales move as a resl of he changng me ' n a recon perpenclar o he plane a ' a ' a' cons. wh he spee a a a a whch s eqal o b assmpon. Here a a a a are consans. I follows ha a a a a. 5. Snce he veloc of lgh for he observer A" n hs local coornaes " " " " s also eqal o we fn on gong from he coornaes ' o he coornaes " ha he epresson a goes over no he epresson an ' a' a ' a' a ' '' a' '' a '' a ' b ' ' a' a' a' a '. 5.4 We shall show ha he coornaes '' ' ' '' ' ' are obane from he coornaes ' ' ' ' b means of a Lorenz ransformaon an a ranslaon of he orgn. B means of a ranslaon of he orgn we can replace he coornaes '' ' ' '' ' ' b 4
coornaes whose conneon wh ' ' ' ' s gven b he homogeneos lnear eqaons Now le he fncon go over no he fncon a β '. 5.5 j j f a ' a' a ' a' f a' a' a' a' where he nmbers a ' a' a' a' sasf he relaon a ' a' a' a' as soon as he nmbers a a a a sasf he relaon a a a a. Here a a a a s an arbrar ssem of nmbers sasfng eqaon 5. an a ' a' a' a' s he corresponng ssem of nmbers reslng from he ransformaon 5.5. We shall show ha now follows ha 5.5 els he Lorenz ransformaon for he coeffcens a ha s ha a a a a a' a' a' a'. In fac wha we o know from he form of he sbson 5.5 s ha he relaon beween he be of he form We frs show ha j a an her ransforms ner 5.5 ms a a a a kj β a' a' j 5.6 j 44
In fac j j k β a' a' k β a' a' a' a'. j 5.7 k β a' a' 5.8 j j mples a ' a' a' a' 5.9 an conversel. Ths means ha he srfaces gven n he for-mensonal space a ' a' a' a' b 5.8 an 5.9 conce. I s eas o see ha hs mples he correcness of 5.7. Conseqenl a a a a a k β a' a' a' '. If we conser he moon of he frs ssem wh respec o he secon we have analogosl whence a' a' a' a' k β a a a a k β k β. On he oher han snce neher ssem s n an wa sngshe we ms have an conseqenl k β ±. When he varables he varables k β k β ' are sbjece o he ransformaon 5.5 a are also sbjece o a lnear ransformaon. I follows ha he nmber of pls an mns sgns n he qarac form n he a 45
canno change. Hence k β an he form a a a a ms reman nvaran ner he ransformaon 5.5. Ths he ransformaon of he varables lnear ransformaon o whch he a s a Lorenz ransformaon. The a are sbjece when he are ransforme b means of 5.5 s gven b a mar whch s he ranspose nverse of he mar 5.5. Conseqenl 5.5 s self a Lorenz ransformaon cf. Para. 4 whch s wha we se o o prove. o be conne 46