Über die Fortpanzung ebener Luftwellen von endlicher Schwingungsweite The Propagation of Planar Air Waves of Finite Amplitude

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1 Übe ie Fopanzung ebene Lufwellen von enliche Schwingungweie The Popagaion of Plana Ai Wave of Finie Ampliue Benha Riemann Abhanlungen e Geellchaf e Wienchafen zu Göingen 8 pp Even hough he iffeenial equaion fo eemining he moion of gae have long been known hei inegaion ha been caie ou almo ecluively fo he cae when he peue iffeence can be viewe a infinieimally mall facion of he enie peue an unil ecenly one ha o be aifie wih aking accoun of only he fi powe of hee facion Unil quie ecenly Helmholz wa able o bing he econ oe em ino he calculaion an in hi way he claifie he objecive geneaion of combinaion one Bu ill he pecie iffeenial equaion can be compleely inegae fo he cae whee he iniial moion ake place in he ame iecion an whee he velociy an peue ae conan in evey plane pepenicula o hi iecion An even hough he fome eamen i eniely ufficien even fo eplaining he epeimenal phenomena foun o ae ill in all i i poible -- given he emenou poge mae mo ecenly by Helmholz in he epeimenal eamen of acouic poblem-ha he eul of hi moe pecie calculaion will povie aiional inigh ino epeimenal eeach in he no oo ian fuue Regale of i puely heoeical inee peaining o he eamen of non-linea paial iffeenial equaion hi migh conibue owa a fuhe eluciaion of he poblem Boyle law coul be ue a he bai fo he epenence of peue on he eniy when he peue change caue by empeaue iffeence ae compenae o quickly ha he empeaue of he ga migh be viewe a conan Bu pobably he hemal echange houl be neglece eniely an hu fo hi epenence we mu ue he law ha ae ha he peue of he ga change wih i eniy when i i no abobing o eleaing any hea Accoing o Boyle law an he Gay-Luac law when v i he volume of he uni weigh p he peue an T he empeaue calculae fom -73 C we have he epeion: p v T con Le u look a T a a funcion of p an v an call he pecific hea a conan peue c an ha a conan volume c boh elaive o he uni weigh an hen if p an v change by p an v he amoun of hea abobe by hi uni weigh will be

2 T T c v c p v p o ince T T v p he amoun of hea abobe by hi uni weigh will be T c v c p Now if no hea abopion ake place hen c p v c an hu if we make he Poion aumpion ha he aio of he wo pecific hea c/c k i inepenen of empeaue an peue hen p k v con Accoing o moe ecen e by Regnaul Joule an W Thomon hee law ae quie pobably vey cloely applicable fo oygen niogen an hyogen an hei miue une all feaible peue an empeaue Now Regnaul aceaine a vey cloe elaionhip o Boyle law an o he Gay-Luac law fo hee gae an eemine ha he pecific hea c i inepenen of empeaue an peue Fo amopheic ai Regnaul foun: beween - 30 C an 0 C c 0377 beween 0 C an 00 C c 0379 beween 00 C an 5 C c 0376 Thu fo peue fom o 0 amophee no noable iffeence in pecific hea wa foun Accoing o epeimen by Regnaul an Joule fuhemoe fo hee gae Maye aumpion a aope by Clauiu eem o be vey nealy coec ha i ha a ga epaning a conan empeaue will abob only enough hea a neee o pouce he eenal wok If he volume of he ga change by v while he empeaue ay conan hen p v he abobe quaniy of hea T c c v he pefome wok i p v Thi Diffeeniaing T p v con wih epec o v we have T T T v v T v T v v T v So we have T T v v v v v v v v T v v T v v Similaly iffeeniaing wih epec o p we have T T p p p p p p p p T p p p T p Theefoe T T c v c p ct v c T p v p hypohei will yiel he following epeion when A i he mechanical equivalen of hea: o AT c c v pv pv c c AT

3 hu inepenen of peue an empeaue Accoingly k c/c i inepenen of peue an empeaue An accoing o Joule A 4455 kg m an auming c an fo he empeaue of 0 C o T 00 C/ an accoing o Regnaul pv kg m he faco k become equal o 40 The pee of oun in y ai of 0 C amoun o k an wih hi value of k i wa foun o be equal o m/ while he wo mo complee e eie by Moll an van Beck calculae epaaely yiel 3358 m/ an m/ fo an aveage of 337 m/ an he epeimen by Main an A Bavai accoing o hei own calculaion yiel 3337 m/ pv c c AT c k c Iniially i i no neceay o make any paicula aumpion abou he elaion of peue on eniy; we hu aume ha a a eniy he peue will be an we will iniially leave he funcion abiay Now le u inouce ecangula cooinae y z wih he -ai in he iecion of moion an le be he eniy p he peue an u he velociy fo cooinae a ime an le ω be an elemen in he plane whoe cooinae i The conen of he aigh cyline aning on he elemen ω wih heigh i hen ω an he ma hel by i i ω The change in hi ma uing he ime elemen o he quaniy ma flowing ino i which i foun o be ω i eemine by he u ω I acceleaion i u u u an he foce iving i in he iecion of he poiive -ai i p ω ω whee enoe he eivaive of Thu fo an u we have he wo iffeenial equaion: u u u u O in a iffeen epeion u u u u u u u u u u 3

4 u u u u u Du D D u D If we muliply he econ equaion wih ± an a i o he fi one an ue he abbeviaion: f f f f f f u f u hen hee equaion will be in a imple fom 3 u u whee u an ae paicula funcion of an eemine by equaion Fom hi i follow 4 { u } 5 { u } Une he aumpion geneally applicable in eal iuaion ha i poiive hee equaion ae ha emain conan when change wih o Du D Df D f u D u f D D u f D D D D D u f u f ha u an emain conan when change wih o ha u One paicula value of o f u move owa lage value of wih he velociy u a paicula value of o f u move owa malle value of wih he velociy u One paicula value of will hu coincie gaually wih each occuing value of an he velociy of i popagaion will epen a evey momen on he value of wih which i coincie Now he analyi offe he mean of anweing he queion of whee an 4

5 5 when a value of encoune a value of ha i o eemine an a funcion of an In fac if we inouce an a inepenen vaiable in equaion 3 hen hee equaion go ove ino linea iffeenial equaion fo an an hu hey can be inegae accoing o known meho In oe o anfom he iffeenial equaion o a linea one i i be o e equaion 4 an 5 of he peviou ype ino he following fom: { } u { } u If we view an a inepenen vaiable hen fo an we obain he wo linea iffeenial equaion: [ ] [ ] u u Conequenly 3 { } { } u u i a complee iffeenial whoe inegal w aifie he equaion w w m w whee m i a funcion of If we e σ f hen we have σ an conequenly we alo have

6 m σ σ Une he Poion aumpion a k f k aa an wih k / k con an if we elec he value zeo fo he abiay conan we hen have a k k k 3 u k 3 k u k 3 m k σ k Une he aumpion of Boyle law f a k aa we obain u a u a m a ha i value ha obain fom he peviou cae if we omi fom f he conan a k k an fom an he conan a k an hen e k k The inoucion of an a inepenen vaiable quaniie i only poible when he eeminan of hee funcion of an ha ae equal o o no iappea ha i only when an ae boh iffeen fom zeo When 0 hen fom we have 0 an fom [ u ] i a funcion of Conequenly even he epeion 3 i a complee iffeenial an w will be a mee funcion of Fo imila eaon when 0 hen i alo conan in ega o [ u ] an w ae funcion of If finally an ae boh equal o zeo hen a a eul of he iffeenial equaion an w ae conan 6

7 3 In oe o olve he poblem fi we mu eemine w a a funcion of an o ha i will aify he iffeenial equaion: w w w m 0 an he iniial coniion which eemine i wihin an abiay conan Whee an when ome value of encoune ome value of eul fom he equaion [ { u ] } { [ u ] } w an hen we fin u an ecluively a funcion of an by binging in he equaion: 3 f u f u In fac when o ae no zeo along ome finie ineval an conequenly o ae vaiable hen fom we obain he equaion: 4 w [ u ] w 5 [ u ] Though linkage wih 3 we obain u an epee in an Bu when iniially ha he ame value in a finie ineval hen hi pah gaually move off owa geae value of Wihin hi egion whee fom equaion we hen canno eive he value of u ince 0; an in fac he queion of whee an when hi value of encoune a paicula value of canno be efiniively anwee Equaion 4 hen will apply only a he bounaie of hi egion an inicae beween which value of he conan value of will appea a a paicula ime an uing which ime ineval hi value of will ei a a paicula locaion Beween hee limi u an ae eemine a funcion of an fom equaion 3 an 5 By imila mean we fin hee funcion when ha he value in a finie egion while i invaian an alo when an ae boh conan In he lae cae hey aume conan value aiing fom 3 beween ceain limi eemine by 4 an 5 4 Befoe we pocee wih he inegaion of equaion fom he peviou 7

8 ecion i eem epeien o eamine how o cay ou hi inegaion By mean of he funcion he only aumpion ha i neee i ha i eivaive oe no eceae wih inceaing which in ealiy i ceainly alway he cae; an le u commen igh hee -- a will be ue in he following ecion ha [ α α ] α 0 eihe emain conan o ie an fall when only one of he quaniie an change fom which i follow ha he value of hi epeion alway e beween an Fi le u eamine he cae whee he iniial iubance of equilibium i limie o a finie egion boune by he inequaliie a < < b o ha ouie of hi egion u an an hu alo an ae conan The value of hee quaniie fo < a can be enoe by applicaion of he ine ; fo > b by aachmen of he ine The egion whee i invaian move gaually fowa accoing o ecion an of coue i ea boun ecee wih he velociy u while he fon bounay of he egion whee i invaian ecee wih he velociy u Afe paage of ime b a u u boh egion move apa an a gap fom beween hem whee an an conequenly he ga paicle ae again a equilibium Fom he iniially iube locaion wo wave will emanae going in oppoie iecion In he fowa iecion ; hu wih a paicula value of eniy he velociy u f will alway be aache an boh value move fowa wih a conan velociy u f Bu in he backwa-moving iecion he velociy f i connece wih he eniy an hee wo value move backwa a a velociy f The ae of popagaion i geae fo geae eniie ince boh an f inceae imulaneouly wih Now if we imagine a he oinae of a cuve fo he abcia hen 8

9 each poin of hi cuve move paallel wih he abcia ai a a conan velociy an of coue he velociy will be all he geae he lage i oinae I i eay o ee ha une hi law poin wih lage oinae will evenually oveake peceing poin wih malle oinae o ha moe han one value of woul belong o one value of Now ince in ealiy hi canno occu hen a cicumance woul have o occu whee hi law will be invali Bu in fac he eivaion of he iffeenial equaion i bae on he fac ha u an ae conan funcion of an have finie eivaive Bu hi peequiie op being me a oon a he eniy cuve become pepenicula o he abcia ai a any poin an fom hi momen on a iconinuiy occu a hi cuve o ha a lage value of will iecly follow a malle one; hi i a cae ha will be icue in he following ecion The compeion wave ha i he poion of he wave whee he eniy eceae in he iecion of popagaion will accoingly become inceaingly moe naow a i pogee an finally goe ove ino compeion hock; bu he wih of he epanion o eleae wave gow popoional wih ime I can be eaily hown -- a lea une he peequiie of he Poion o Boyle law -- ha even when he iniial iubance of equilibium i no limie o a finie egion ha compeion hock will alway fom in he coue of he movemen ecep fo quie pecial cae The velociy a which a value of move fowa une hi aumpion i k k 3 ; hu lage value will on aveage move a geae velociy an a geae value woul finally have o oveake a peceing malle value " unle he value of coincien wih " i on aveage malle by k 3 k han he one ha coincie imulaneouly wih In hi cae woul go negaive infiniy fo a poiive infinie an hu fo he velociy u o inea of hi he eniy will be infiniely mall accoing o Boyle law Ecep fo pecial cae he cae woul alway have o occu ha a value of geae by a finie amoun will iecly follow a malle one; conequenly a goe o infiniy he iffeenial equaion will loe hei valiiy an fowa-moving compeion hock woul have o appea Likewie almo alway backwa moving compeion hock will fom ince goe o 9

10 infiniy To eemine he ime an locaion fo which o go o infiniy an uen compeion begin we obain he following epeion fom equaion an fom ecion by ineing he funcion w heein: w w 5 Now ince uen compeion almo alway e in even when he iniial eniy an velociy vay coninuouly eveywhee we mu eek he law fo popagaion of compeion hock We aume ha a ime fo ξ a uen change in u an will occu an le u call hei value an he quaniie epenen on hem fo ξ 0 by appening he ine an fo ξ 0 by he ine The elaive velociie of he ga moving owa he uneay ae ξ u ξ u will be enoe by v an v The ma paing hough an elemen w of he plane whee v ξ in he poiive iecion in ime elemen i hen w v w ; he acing foce w an he eulan inceae in velociy v v ; hu we have an fom which follow heefoe w v v vw v v v m ξ u ± u ± Fo a compeion hock mu have he ame ign a v an v an of coue fo he fowa-moving hock i will be negaive fo a backwa-moving hock i will be poiive In he fi cae he uppe ign will Ma: v w Foce: w v v Acceleaion: Eq of moion: v v v w w 0

11 apply an i geae han Theefoe une he aumpion mae a he beginning of he peceing aicle egaing he funcion we have ξ u > > u an conequenly he iconinuiy poin will move moe lowly way han he following one an will move fae han he peceing value of Thu an ae eemine a any momen by mean of he iffeenial equaion applicable on boh ie of he iconinuiy poin The ame will apply alo fo an conequenly fo an u ince he value of go backwa wih he velociy u bu no fo The value of an ξ ae eemine fom an u ae eemine uniquely hough he equaion In fac he equaion: 3 f f [ ] will only aify a value of becaue when gow fom o infiniy he igh ie will ake on evey poiive value ince boh f an alo he wo faco an l in which he la em can be boken own gow eaily o only he lae faco ay conan Bu when i eemine by mean of equaion we obain evienly compleely eemine value fo u an ξ A backwa-moving compeion hock will behave quie imilaly 6 Now we have foun ha in a popagaing compeion hock beween he value of u an on boh ie of i he equaion will alway appea u u [ ] Now he queion aie of wha will happen if a a given ime an a a given poiion anom iconinuiie ae peen Then fom hi ie epening on he value of u u eihe wo compeion hock will emanae in oppoing iecion o a ingle fowa-moving o a backwa-moving hock o finally no compeion hock a all will occu In he la cae movemen will

12 hu ake place accoing o he iffeenial equaion If we call he value ha u an ake on afe o beween he compeion hock in he fi momen of popagaion by aing in a pime hen in he fi cae > an > an we have: [ ] u u [ ] u u [ ] [ ] u u Thu ince boh em on he igh ie of inceae wih poiive an [ ] u u > ; an u u i an conveely when hee coniion ae me alway one an only one pai of value of u an ae obaine ha will aify equaion In oe fo he lae cae o occu an hu he moion be eemine accoing o he iffeenial equaion i i neceay an ufficien ha an ha i u u i negaive an u [ f f ] u The value an an will move apa ince he peceing value avance wih a geae pee o ha he iconinuiy will iappea If neihe he fi no he la coniion i me hen he iniial value will aify one compeion hock an of coue eihe a fowa o backwa moving hock will occu epening on whehe i geae o malle han In fac when > o f f u u i poiive becaue an a he ame ime becaue Thu fo he eniy u < [ f f ] u [ ] u u f f [ ] u u afe he compeion hock a value ha aifie he coniion 3 of he peviou ecion will be foun an i i malle han Conequenly ince f f alo o ha he moion afe he compeion hock can occu accoing o he iffeenial equaion

13 The ohe cae when < i evienly quie iffeen fom hi 7 In oe o eplain he foegoing wih a imple eample whee he movemen can be eemine wih he eive equaion le u aume ha he peue an eniy will epen on each ohe accoing o Boyle law an he iniial eniy an velociy change uenly a 0 bu ae conan on boh ie of hi locaion Now we make a iincion beween he fou cae above I When u > u ha i when he wo ga mae move owa each ohe an u u a > hen wo oppoiely moving compeion hock will fom Accoing o Secion 6 when 4 / i enoe by a an he poiive oo of he equaion u a u θ a / a θ i enoe by θ he eniy beween he compeion hock θθ an accoing o Secion 5 fo he fowa moving compeion hock we have: ξ a u aαθ u αθ an fo he backwa-moving compeion hock: ξ θ a u a u α θ Thu he value fo velociy an eniy wih epec o ime when ae u an u an u θ a < < u aαθ α Fo a malle hey ae u an an fo a lage hey ae II When u u 0 conequenly he ga mae move apa an a he ame ime < u u a hen wo gaually boaening epanion o eleae wave move ou in oppoie 3

14 iecion fom he bounay Accoing o Secion 4 beween hem we have u In he fowa-moving wave an u a i a funcion of whoe value i equal o zeo fom he iniial value 0 0 Bu fo he backwa unning wave we have an u a 0 The one equaion o eemine u an when < a < u a i u a / fo malle value of an fo lage value ; he ohe equaion when u a < < a i u a / fo a malle an fo a lage III If neihe of hee wo cae occu an > hen a backwa-unning iluion wave an a fowa-moving compeion hock occu Fo he lae fom Secion 5 3 when θ i he oo of he equaion θ θ a θ hen θθ an fom Secion 5 we have ξ a u aθ u θ Afe paage of ime befoe he compeion hock ha i when > u aθ hen u u Bu behin he compeion hock we have an in aiion when u a < < u a we have u a / ; fo a malle we have u u an fo a lage we have u u IV Now finally if he fi wo cae o no occu an < hen he pofile i eacly a in III only he iecion i change 8 Now o olve ou poblem in geneal accoing o ecion 3 he funcion w mu be eemine o ha i aifie he iffeenial equaion w w w m 0 an he iniial coniion If we eliminae he cae whee iconinuiie occu hen evienly accoing o Secion he value of an fo which a paicula value of of 4

15 encoune wih a paicula value of ae fully eemine when he iniial value of an beween an ae given ove he -ineval an when he iffeenial equaion 3 of Secion hol eveywhee in he whole egion S which coni of all inemeiae value of beween he one whee an he one whee fo any value of Theefoe he value of w fo an i alo compleely eemine when w aifie he iffeenial equaion eveywhee in S an when he value of w an w ae given fo he iniial value of an -- which eemine he iniial value of w ecep fo an abiay aiive conan Acually hee coniion ae equivalen o hoe above Fom Secion 3 i alo follow ha w - wll ake on iffeen value on boh ie of a value " of when hi value occu in a finie iance bu change eaily wih ; likewie w change wih bu he funcion w ielf change eaily wih an alo wih Afe hee pepaaion we can now olve ou poblem o eemine he value of w fo wo anom value an of an Fo illuaion le u imagine an a he abcia an oinae of a poin in a plane an aw he cuve in hi plane whee an whee have conan value Of hee cuve he fi one i eignae by he lae one by an in hem he iecion in which inceae i viewe a poiive The lage egion S will hen be epeene by a piece of he plane ha i limie by he cuve he cuve an he piece of he abcia ai locae beween he wo; we ae ineee in eemining he value of w in he poin of ineecion of he wo fome value given in he lae line Le u genealize he poblem omewha an aume ha he lage egion S i boune by an abiay cuve c hough hi lae line an none of he cuve an ineec moe han once an ha he pai of value of an belonging o hi cuve ae given by he value of w an w A will be een fom he oluion of he poblem hee value of w an w ae ubjec only o he coniion of changing eaily wih he locaion in he cuve an can be aken abiaily Thee value woul no be inepenen of each ohe when he cuve c ineec one of cuve o moe han once In oe o eemine funcion ha ae o aify linea paial iffeenial equaion an linea bounay coniion we can ue an eniely imila meho 5

16 Thi i one by muliplying all equaion by abiay faco aing hem up an hen eemining hee faco o ha fom he um all known quaniie epec one will op ou Le u image he piece S of he plane a being cu by he cuve an ino an infinie numbe of mall paalleam an ue δ an δ o enoe change ha he quaniie an unego when he cuve elemen ha fom he ie of hee paalleam ae anie in a poiive iecion Fuhemoe we call v an abiay funcion of an ha i coninuou an ha coninuou eivaive A a eul of equaion we hen have w w w 0 v m δδ epane aco he enie lage egion S Now he igh ie of hi equaion mu be aange accoing o he unknown; ha i he inegal will be change by paial inegaion o ha beie he known quaniie only he eie funcion an no i eivaive ae conaine Duing eecuion of hi opeaion he inegal fi goe ove ino he inegal epane aco S: w mv mv w δδ an a imple inegal ha een only aco he bounay of S becaue w change wih w change wih an w change eaily wih boh quaniie If an mean he change in an in a bounay elemen when he bounay i anie in he iecion ha i oppoie he iecion inwa like he poiive iecion in he cuve i oppoie he poiive iecion in he cuve hen hi bounay inegal i equal o w w v mv w mv The inegal hough he enie bounay of S i equal o he um of he inegal hough he cuve c ha fom hi bounay ha i when hei poin of ineecion ae eignae by c c c c c c Of hee hee em he fi one conain only known quaniie beie he funcion v; he econ conain only he unknown funcion w ielf ince in i 0 an no i eivaive; he hi em can be convee hough paial inegaion ino 6

17 c v vw vw c w mv o ha i likewie will conain only he eie funcion w ielf Afe hee anfomaion equaion evienly povie he value of he funcion w a poin epee by known quaniie when he funcion v i eemine accoing o he following coniion: v mv mv Eveywhee in S: 0 v Fo : mv 0 v 3 Fo : mv 0 4 Fo : v l We hen have w v 4 w vw c v mw w mv 9 By mean of he meho ju ecibe he poblem become one of eemining a funcion w of a linea iffeenial equaion an linea bounay coniion an applying i o he oluion of a imila bu much imple poblem fo anohe funcion v Deeminaion of hi funcion i uually aaine mo eaily by eamen of one pecial cae uing he Fouie meho We will have o be aifie hee meely wih poining ou he calculaion bu fining he eul by ohe mean If we ubiue σ an u ino equaion of he peviou ecion fo an a inepenen vaiable quaniie an fo he cuve c we elec a cuve in which σ i conan hen he poblem can be eae accoing o Fouie ule an by compaion of he eul wih equaion 4 of he peviou ecion when v π 0 σ u we obain: coµ u u σ [ ψ σ ψ σ ψ σ ψ σ ] wheein ψ an ψ ae wo uch paicula oluion of he σ iffeenial equaion ha σ ψ m ψ µµψ 0 µ 7

18 8 σ ψ ψ ψ ψ If we aume he Poion law accoing o which σ k m hen we can epe ψ an ψ by ceain inegal o ha fo v we obain a hee-fol inegal ha can be euce o: 3 k k F v k Now we can eaily emonae he accuacy of hi epeion by howing ha i acually oe aify he coniion 3 of he peviou ecion If we e y e v m σ σ hen fo y hey go ove ino 0 y mm m y σ an y boh fo an alo fo Une he Poion aumpion hee coniion can be aifie when we aume ha y i a funcion of z Becaue if we enoe k by λ hen σ λ m an hu σ λ λ σ mm m an z y z z y y σ Conequenly we have y v λ σ σ an y i a oluion of he iffeenial equaion 0 zy z y z z y z λ λ o accoing o he eignaion ue in my eamen of he Gauian eie we have a funcion

19 0 P 0 λ λ 0 z 0 an of coue ha paicula oluion i equal o when z 0 Accoing o he anfomaion pinciple evelope in ha pape y canno be epee meely by he funcion P 0λ 0 bu alo by he funcion P 0 λ P 0 λ λ Thu we obain fo y a lage quaniy of epeenaion hough hype-geomeic eie an ceain inegal of which we will commen only on he following one: y F λ λ z z z λ z Fλ λ z λ z F λ λ z ha will be ufficien in all cae In oe o eive he eul applicable fo Boyle law fom hee eul foun fo he Poion law accoing o ecion he quaniie mu be euce by an a k k an hen we can le k o ha we obain m a a n 0 v e n n! n!a n n 0 If we ubiue he epeion fo v foun in he peviou ecion ino equaion 4 of ecion 8 hen we obain he value of w fo hough he value of w w - an w epee in cuve c; bu ince in ou poblem in w w hi cuve - an ae alway iecly given an w ha o be foun fom a quae oo hen i i epeien o anfom he epeion fo une he inegal ign only he eivaive of w will appea Le u call he inegal of he epeion w o ha 9

20 an v mv mv v mv mv ha ae complee iffeenial a a eul of he equaion v mv mv 0 a P an Σ ; likewie he inegal of P Σ i a complee iffeenial ue o P mv Σ If we eemine he inegaion conan in hee inegal o ha ω ω an ω iappea fo hen w aifie he equaion ω ω v ω mv Boh fo an alo fo he equaion w 0 an i i fully eemine by hi bounay coniion an by he iffeenial equaion ω ω ω m 0 Now if we ine he funcion w in he epeion of can conve i by paial inegaion ino w w c ω c c w fo v hen we w ω w In oe o eemine he moion of he ga fom he iniial ae we have o ake he cuve c in which 0; in hi cuve we hen have w w an by mean of epeae paial inegaion we have c w wc ω c conequenly fom Secion 3 4 an 5 we have 0

21 u u ω ω Bu hee equaion only epe he moion a long a w an w ae no equal o zeo A oon a one of hee quaniie iappea a compeion hock will occu an equaion will only apply wihin ha egion ha e on one an he ame ie of hi compeion hock The pinciple evelope hee in geneal will no be ufficien o eemine he movemen fom he iniial ae; bu of coue by mean of equaion an he equaion obaine fo he compeion hock in Secion 5 we can eemine he movemen when he ie of he compeion hock i known a ime ha i ξ i a funcion of We will no puue hi fuhe an likewie will ignoe a eamen of he cae when he ai i boune by a fie wall ince he calculaion peen no ifficulie an a compaion of he eul wih epeimen i peenly impoible