HUYGENS S ENVELOPING-WAVE PRINCIPLE

Transcription

1 CONTRIBUTION TO THE THEORY OF HUYGENS S ENVELOPING-WAVE PRINCIPLE BY J. VAN MIEGHEM TRANSLATED BY D. H. DELPHENICH BRUSSELS PALAIS DES ACADEMIES RUE DUCAL, 96

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3 TABLE OF CONTENTS INTRODUCTION FIRST CHAPTER Itegratg the wave equato. The wave equato.. Itegratg the wave equato Specal cases.. 4. Bcharacterstcs. CHAPTER II Prcple of evelopg waves 5. Geometrc costructo of a tegral surface for the wave equato Prcple of evelopg waves Rays of the wave (.. 7 CHAPTER III Waves ad rays 8. Retur to the homogeeous wave equato Homogeeous dfferetal system for the bcharacterstcs (or space-tme rays.. Equato for the elemetary Huyges wave.. CHAPTER IV Wave trasport. Metrc o geometrc space 4. Dfferetal equatos of the rays for a wave 5. Wave trasport geometrc space... 5 CHAPTER V Jacoba form of the wave equato ad Hamltoa form of the ray equatos 4. Jacoba form of the wave equato Hamltoa form of the ray equato Case of a homogeeous medum.. 7. Trasport velocty. 8. Hyper-refrgece Elemetary Huyges wave a homogeeous medum whe the metrc s Euclda 4 Page

4 Table of Cotets CHAPTER VI Reflecto ad refracto of waves Page. Icdet, reflected, ad refracted waves. 7. Applyg the prcple of evelopg waves. 8. Geometrc laws of reflecto ad refracto.. 8 CHAPTER VII Applcato of Huyges s prcple to the propagato of electromagetc waves. Revew of the geeral equatos Case of a homogeeous, asotropc, o-absorbet medum Defto of a electromagetc wave Compatblty codtos Cosequeces of the compatblty codtos Partal dfferetal equatos of electromagetc waves 5 9. Electromagetc rays. 54. Elemetary Huyges wave 6. Case of a homogeeous medum. Fresel wave surface.. 6 CHAPTER VIII Applcato to secod-order lear equatos A. Waves ad rays.. Secod-order equato Wave equato Dfferetal system of the rays 67 B. Geodescs of a quadratc dfferetal form. 5. Prelmares Deftos ad Lagraga equatos of the geodescs 7 7. Hamltoa equatos of the geodescs 7 8. Jacob s theorem 7 9. Geeral tegral of the dfferetal equatos of geodescs Parametrc equatos for geodescs Dfferetal parameters.. 77 C. Elemetary Huyges wave that s assocated wth a secod-order lear equato. 4. Lagraga form of the bcharacterstc equato Characterstc Hadamard cood ad elemetary Huyges wave... 8

5 INTRODUCTION Chapter I begs wth a revew of the partal dfferetal equato for waves. We the study the tegrato of that equato by utlzg a method that s dfferet from the classcal oe. Chapter II s dedcated to the evelopg-wave prcple of Chr. Huyges. We show that ths prcple results from the rule for tegratg the partal dfferetal equato for waves. We wll the defe the ray to be the locus of the pot of cotact of a elemetary Huyges wave wth ts evelope. I Chapter III, we smultaeously study waves ad rays. They are othg but the bcharacterstcs of J. Hadamard, whch are evsoed as oe-dmesoal mafolds, ether geometrc space or space-tme. I Chapter IV, we wll defe the trasport of a wave geometrc space wth the ad of the dfferetal system of the bcharacterstcs. The troducto of the oto of trasport velocty of a wave oblges us to defe the metrc o geometrc space. That velocty wll the be gve by a varat; cosequetly, t wll have a trsc meag. I Chapter V, we dcate how oe ca wrte the partal dfferetal equato for waves the form of a Jacob equato. The Hamltoa form of the ray equatos wll result from them mmedately. Whe those equatos are appled to the case of homogeeous physcal meda, that wll permt us to establsh some remarkable results. I partcular, we wll show that the case of a Euclda metrc, the equato for the elemetary Huyges wave wll be the pot-lke equato that correlates wth the partal dfferetal equato for waves, whe t s cosdered to be a tagetal equato. Chapter VI s dedcated to the reflecto ad refracto of waves. We show that wth the ad of Huyges s prcple, oe ca costruct reflected ad refracted waves that are produced by a cdet wave, whch s assumed to be gve. We the establsh the geeral geometrc laws of reflecto ad refracto the case where the separato surface s movg. Those laws are othg but the compatblty codtos that relate to wave fuctos, whch are codtos that must be verfed at the separato surface space-tme. We coclude ths moograph wth two applcatos of Huyges s evelopg-wave prcple. Frst, Chapter VII, we apply t to the equatos of electromagetsm. After recallg some geeral equatos, amog whch, oe fds Maxwell s equatos ad the dfferetal equatos of trasport for electromagetc eergy, we look for the partal dfferetal equato for electromagetc waves the case of a homogeeous medum. That equato wll lead to a geeralzato of Fresel s equato. We the show that the rays that are assocated wth the electromagetc wave frot are the trajectores of the electromagetc eergy. The partal dfferetal equato of electromagetc waves ad the dfferetal system of electromagetc rays the represet the wave-lke ad corpuscular aspects of the equatos of electromagetsm. More geerally, we have see (Chap. I how oe ca assocate a dfferetal system of mathematcal physcs wth a partal dfferetal equato for waves that s compatble wth the equatos of that system. O the other had (Chaps. II ad III, we have see how the problem of tegratg that wave equato wll ecessarly troduce rays alog whch the trasport of the waves takes place. Cosequetly, the aalyss that leads us to state Huyges s

6 Itroducto evelopg-wave prcple wll show both the wave-lke ad corpuscular aspect of the equatos of physcs (wave-corpuscle dualty. We coclude that chapter wth the search for the elemetary electromagetc wave equato the case of a homogeeous medum. That equato wll clude the equato of the Fresel wave surface as a specal case. Fally, Chapter VIII s dedcated to the study of the waves ad rays that are compatble wth a secod-order, lear, partal dfferetal equato, whe t s cosdered to be the fudametal equato of mathematcal physcs. That wll lead us to meto some geeral results that relate to the geodescs of a quadratc dfferetal form that s assocated wth a secod-order, lear, partal dfferetal equato. We have obtaed those results by systematcally utlzg Jacob s drect ad verse theorems, whch wll result whe a varatoal prcple s appled to a quadratc dfferetal form.

7 FIRST CHAPTER INTEGRATING THE WAVE EQUATION. The wave equato. Cosder a physcal pheomeo whose mathematcal law s expressed by the dfferetal system of order c: H ( x, z, z, z,, z ( s r r r r C = r, s =,,, m,,, r =,,, that s composed of m partal dfferetal equatos m ukow fuctos z,, z m of depedet varables x,, x, ad whch we have set: z r = a z a r a,, a =,,, r =,,, m. ( The th varable x wll represet tme t; we ca the set: t x. ( Suppose that the pheomea evsoed space-tme (x,, x ; x bouded by the hypersurface whose equato s: t are ( (x,, x ; t =. (4 That amouts to assumg that at the stat t, all pots that are foud o oe sde of the -dmesoal surface of equato (4 the -dmesoal geometrc space (x,, x are uder the fluece of the pheomea questo; all of the pots that are foud o the other sde of that surface ca have o fluece o that stat. Let I ad II represet the regos thus-defed at the stat t the geometrc space (x,, x ; let I deote the rego that s swept out (Fg. The, let: r r zi I z (x,, x ; t (5 be the soluto to equato ( that represets the physcal state of the medum I at the stat t. The physcal state of the medum II at the same stat s defed by the soluto: r r zii II z (x,, x ; t. (6

8 4 Chapter I Itegratg the wave equato. x ( -dmesoal geometrc space (x, x,, x x ( -dmesoal wave frot at the stat t r z I I II Fgure. r z II Assume that the rego I expads at the expese of rego II; the ( -dmesoal varety of equato (4 that bouds the rego I that s swept out geometrc space (x,, x at the stat t s what oe calls the wave frot ( physcs. The ( - dmesoal surface ( space-tme that s defed by equato (4 costtutes a sythess of the progress of the wave frot ( the geometrc space (x,, x whe tme t x vares. That beg the case, suppose that the fuctos (5 ad (6 ad ther spato-temporal dervatves up to order c clusve vary cotuously whe oe crosses the wave surface (. I that case, oe wll have: r r z II = I z, z r II x z = r I x, z c r II c = z c r I c [o ( ]; (7.e., by vrtue of (x, t =. As a result, the two solutos m z I,, I z ad m z II,, II z to equatos ( have cotact of order c alog the wave surface (. It the results that the wave ( s a characterstc Cauchy varety of system ( of partal dfferetal equato mathematcal physcs. Cosequetly, the partal dfferetal equato of waves that s compatble wth the dfferetal system ( of physcs s othg but the equato of the characterstc varetes: H s O( x, r = (8 c = = c = z c (r, s =,, m whose left-had sde s a determat of order m. The otato:

9 . The wave equato. 5 H z s r c represets what oe wll obta after substtutg the: z r, z r,, r z c H s r c z wth ther values (5 ad (6 whe oe takes (7 to accout. The expressos: H z s r c are fuctos of oly the depedet varables x,, x t the. We have set: Whe the dervatves: ( =,,,. (9 H s r c z are depedet fuctos z r of ther dervatves, the wave equato (8 wll rema the same for ay soluto of the dfferetal system (. The same thg wll the be true for the equatos of mathematcal physcs that are lear wth respect to the hgher-order dervatves whose coeffcets deped upo oly the x. By defto, ay fucto (x of the spato-temporal varables such that (x, = s a relatve soluto of the wave equato (8. Oe says that the fucto (x s a absolute soluto of the wave equato (8 f oe obtas a detty x,, x whe oe substtutes that fucto the equato. Ay relatve or absolute soluto of the wave equato (8 wll defe a wave surface ( space-tme whe t s equated to zero, as well as a wave surface ( geometrc space. We remark that the left-had sde of the wave equato (8 s a homogeeous polyomal wth respect to the partal dervatves,,, ; let µ be ts degree of homogeety. That beg the case, solve equato (4 wth respect to t; hece: ( t = t (x,, x ( ad t t x ( =,,,. (

10 6 Chapter I Itegratg the wave equato. It wll the result from the homogeety of (8 ad the aforemetoed defto of a relatve soluto of the wave equato that the partal dfferetal equato: t O x, t, = ( that oe obtas by dvdg the left-had sde of (8 by ( µ wll admt the fucto t (x that s defed by ( as a (absolute soluto. I equato (, x represets the spatal varables x,, x. Ay relatve soluto to the wave equato (8 wll defe a absolute soluto of the wave equato ( whe t s equated to zero.. Itegratg the wave equato. That amouts to fdg a fucto: t (x,, x ( of the ( geometrc varables (x,, x such that whe oe substtutes t to the wave equato (, oe wll obta a detty x,, x (vz., a absolute soluto. As (, we set: t t ( =,,, ( here. I the ( -dmesoal space of elemets (x,, x, t ; t,, t, we let: x,, x, t, t,, t (4 be the coordates of a pot-elemet the ( -dmesoal varety E ( of the equato (. Now, let: δx,, δx, δ t, δ t,, δ t (5 be the compoets of ay elemetary dsplacemet that s taget to that varety E at the pot whose coordates are (4; oe wll the have: δo * O O O δ x + δ t + t δ =. (6 = t = t By vrtue of (, oe wll have the relato that s called cotact or uted elemets : δt t δ x =. (7 =

11 . Itegratg the wave equato. 7 We the say that (4 represets a elemet of a tegral of the proposed equato (. A elemet of the tegral surface or wave: t (x,, x (8 of the equato s the defed by a pot o that ( -dmesoal surface spacetme (x,, x, t x whose coordates are x,, x, t x ad the taget plae at that pot that s defed by t,, t. That beg the case, we subject the elemets of a arbtrary tegral of ( to a trasport alog the les that are determed the geometrc space by the dfferetal equatos (x,, x : dx = du ( =,,,, (9 X whch the deomators X wll be specfed later, ad u refers to a parameter. Now, totally dfferetate the fucto (, coformg to equatos (9; hece: dt du = X t. ( Fally, prolog the dfferetal equatos (9; hece, from (: = dx X = k = dt X t k k = dt T j j = du (, j =,,,. ( The fuctos X ad T : X X x x t t t (,,,,,, (,,,,,, T T x x t t t ( =,,, ( are defed by the followg codto: The X ad T must be such that f the elemet (4 of a tegral surface (8 vares accordg to ( the the relatos (6 ad (7 wll persst. The elemet evsoed (4 wll the cotue to verfy the proposed equato ad, at the same, rema a cotact elemet whe t dsplaces alog a le (9 of the geometrc space (x,, x. I order for that to be true, s t ecessary ad suffcet that oe should have the trasport relato: d t t x du δ δ A δo * + B δt t δ x, ( = =

12 8 Chapter I Itegratg the wave equato. whch A ad B are udetermed. That detty relato δx,, δx, δ t, δ t,, δ t shows that f the cotact codto (7 for a elemet of a tegral of ( s verfed at a pot of a le (9 the that wll persst all alog that le. Replace δo * ( wth ts value ad vert the order of the operators d / du ad δ ; after some easy calculatos, oe wll get the detty: O O O T Bt + A δ x + A + B δ t + A X δt = t = t (4 δx,, δx, δ t, δ t,, δ t. Hece: O B A, t O X A, t O O do T A + t A x t dx ( =,,,. (5 Fally, replace the dfferetal equatos: X ad T ( wth ther values (5; we wll the get the dx dt dt = = = dv O O do tk t k= tk dx ( =,,,, (6 whch we have set: dv A du. (7 The dfferetal system (6 defes the characterstc Cauchy varetes ( of the wave equato (. Let: x = x ( v; x,, x, t, t,, t, t = t( v; x,, x, t, t,, t, ( =,,, (8 t = t( v; x,, x, t, t,, t, represet the geeral soluto of the dfferetal equato (6. The equatos (8 are the oes of the smple ftude of characterstc elemets that cotas the tal elemet x, x,, x, t, t,, t. The frst equatos (8 are those of the ( E. GOURSAT, Cours d Aalyse mathématque, t. II (Pars, Gauther-Vllars, 95; see pp. 69 ad 64, ad especally equato (.

13 . Itegratg the wave equato. 9 characterstc Cauchy le that ssues from the pot x, x,, x, t ( space-tme to whch oe assocates the cotact elemet x, x,, x, t, t,, t. Now, cosder a ( -dmesoal varety V ( the -dmesoal space (x, x,, x, t, t,, t that s defed by the parametrc equatos: (V ( whch oe supposes s such that: ad x = x ( u,, u, t = t( u,, u, (9 t = t ( u,, u, O x t t (,, t ( t δ uk k = uk = u ( k are two dettes, the frst of whch s u,, u, ad the secod of whch s δu,, δu. Hece: t = t (k =,,,. ( u u k = k The trasport of the varety V ( whose parametrc equatos are (9 coformty wth equatos (6 geerates a -dmesoal tegral surface or wave ( the space-tme (x, x,, x, t. I order to obta the parametrc equatos: x = x ( u,, u, t = t ( u,, u, ( =,,, ( for the wave (, t wll suffce to replace the x, t, ad t the frst equatos (8 wth ther values (9. We have set u v (. Fally, upo solvg ( the frst equatos ( for u,, u ad substtutg the values for the u thus-obtaed as fuctos of x the last equato of (, oe wll obta the requred tegral the form (. The correspodg tegral surface (or wave wll the be represeted by equato (8 (. Theorem. If two tegral surfaces ( ad ( of the equatos: t = t (x, x,, x ad t = t (x, x,, x (4 ( That soluto wll be possble f the Jacobas of the x wth respect to the u k ( =,,, ; k =,,, are ot all zero. ( The tegrato method that was developed above s applcable to ay frst-order partal dfferetal equato.

14 Chapter I Itegratg the wave equato. have the elemet x,, x, t, t,, t commo space-tme (x, x,, x, t the they wll agree all alog the characterstc Cauchy le that ssues from that elemet. That le s defed by the frst equatos (8. Ideed, the dfferetal equatos (6 are depedet of ay tegral of the proposed equato (.. Specal cases.. Let S ( represet the ( -dmesoal varety space-tme (x, x,, x, t that s defed by the frst parametrc equatos (9. The last ( equatos (9 are the deduced from ( ad (. It wll the result from the cosderatos above that the tegral surface of the wave equato ( that cotas the varety S ( s the locus of characterstc Cauchy les that ssue from the pots of that varety; hece: Theorem II. The hypersurface space-tme that s geerated by the characterstc Cauchy les that ssue from the pots of ay gve ( -dmesoal varety wll be a wave surface (.. We ow propose to determe the tegral surface ( of ( that passes through a ( -dmesoal varety S that s take at the tal stat t = the geometrc space (x, x,, x. Let: ( t = S( x = ψ ( x,, x (5 be the equatos of that varety, whch the fucto ψ s assumed to have bee gve explctly. Adopt x,, x as depedet varables ad set: Hece, t wll result from ( that: k u k x (k =,,,. (6 ψ ( x,, x k + t k t = (k =,,,. (7 Upo replacg the fucto of x,, x t,, t ( wth ther values (7, oe wll fd ; hece, from (7, oe wll fally have: t as a t = t ( x,, x ( =,,,. (8

15 . Specal cases. Now, replace the ad the replace x t that fgure the frst relatos (8 wth ther values (8, wth ψ ( x,, x ad replace t wth zero; hece: x = x ( v; x,, x, t = t ( v; x,, x. (9 Now, solve the frst ( equatos (9 for v, substtute the values thus-obtaed for v, from (8, oe wll get the equato: x, x,, x, x,,, ad the x x to the last relato (9; ( ( t t x x = (,, (4 of the tegral surface ( of (.e., of the wave surface ( that s compatble wth ( that passes through the tal varety S that was gve advace (. Theorem III. The wave surface ( of equato (4 s determed completely by the followg codtos:. It satsfes the wave equato (.. It passes through the tal varety ( ( S equatos (5. Theorem IV. The wave surface ( of equato (4 s the locus of characterstc Cauchy les that ssue from the pots of the tal varety S of equatos (5.. Cosder a fxed pot P space-tme (x, x,, x, t wth coordates ( x, x,, x, t ad a drecto at that pot that s defed by t,, t. Now, express the dea that the coordates of the pot P ad the drecto coeffcets t,, t verfy the proposed equato (; hece, oe wll have the detty relato (. Oe ca the suppose that: t = t (u,, u ( =,,,, (4 whch the fuctos t are such that whe oe substtutes them (, oe wll get a detty u,, u. Next, replace the t the frst equatos (8 wth ther aforemetoed values. Hece, upo settg u v *, oe wll get the parametrc equatos: ( ( The varety S s the tal posto of the wave surface ( geometrc space. Recall ( ( that, space-tme, the wave ( wll costtute a sythess of the advace of the wave ( geometrc space.

16 Chapter I Itegratg the wave equato. x = x ( u,, u, x,, x, t, t = t ( u,, u, x,, x, t ( =,,, (4 of the wave surface ( that s the tegral of ( that s geerated by the characterstc Cauchy les that ssue from the pot P ( x,, x, t. Upo solvg the frst ( equatos (4 for u,, u ad replacg the u the last of equatos (4 wth ther values thus-calculated, oe wll get the equato of that tegral surface the explct form: t = t( t ; x,, x ; x, x,, x. (4 That wave surface ( admts the pot P as a multple pot. 4. Bcharacterstcs. J. Hadamard s bcharacterstcs of the physcal dfferetal system ( are, by defto, the Cauchy characterstcs of the correspodg wave equato (. Those bcharacterstcs are the determed by the dfferetal equatos (6 or by ther geeral tegral (8. Hece, oe has the theorems: a The surface space-tme that s geerated by the bcharacterstcs of the dfferetal system ( that ssue of the pots of ay ( -dmesoal varety that s gve advace s a wave ( that s compatble wth that system. b Two wave surfaces ( ad ( space-tme that have a commo cotact elemet wll cocde all alog the bcharacterstc that ssues from that elemet. The locus of bcharacterstc les that ssue from the pot P ( x,, x t s, by defto, J. Hadamard s characterstc cood whose summt s P. That surface s gve by equato (4. The followg proposto wll the result from Theorem I of : c The wave ( of equato (4 s the evelope of the characterstc coods whose summts are foud o the tal varety S of equatos (5. d More geerally, the evelope space-tme of the characterstc coods whose summts are the pots of ay ( -dmesoal varety that s gve advace s a wave (. Remark. Istead of cosderg the explct form ( for the equato of a wave surface, we retur to the mplct form (4. The wave equato ( wll the take the homogeeous form (8, amely: O (x, =. (44 (

17 4. Bcharacterstcs. Now, thaks to ( ad (, the dfferetal equato (6 of the bcharacterstcs wll lkewse take a homogeeous form, amely: dx β O( x, β = d ( x β, β O = γ γ = γ d β O( x, β (, β =,,,. (45 Note that, thaks to Euler s theorem o homogeeous fuctos, oe wll have: hece, (45 wll become: ( x β, β O γ µ O (x β, β ; (46 γ = γ dx O d d = = O ( =,,,, (47 whe oe takes (44 to accout. As a result, ay bcharacterstc that ssues from a pot P that s take o the wave ( whose equato s (4 wll be cotaed etrely o that wave (see Fg.. t x x P ( Fgure.

18 CHAPTER II PRINCIPLE OF ENVELOPING WAVES 5. Geometrc costructo of a tegral surface for the wave equato. Cosder the ( -dmesoal varety space-tme (x, x,, x, t whose equatos are: t = t ( ( (47 ( x,, x =. We propose to costruct the tegral surface to the wave equato ( that passes through the varety (, whch s assumed to be kow. We just saw that oly oe ( wave surface space-tme passes through that varety; let: be ts equato. The varety ( ( (x, x,, x, t = (48 ( s the tal posto (t = t of the wave ( the geometrc ( space (x, x,, x. At the stat t, the wave ( wll occupy the posto geometrc space of the ( -dmesoal varety that s defed by the equato: ( (x, x,, x, t =. (49 ( Equato (48 represets a ( -dmesoal varety space-tme (x, x,, x, t that costtutes the sythess of the advace of the wave ( the geometrc space whe t vares (see Fg.. The tersecto of the wave surface ( wth the plae ϖ whose equato s: ( ( ( ( ϖ t = t (5 ( determes the posto order to obta whose equato s: ( geometrc space of the wave at the stat t. Ideed, (, t wll suffce to project the varety oto the plae ( ϖ ( ( ϖ t = t (5 (

19 5. Geometrc costructo of a tegral surface to the wave equato. 5 ad s parallel to the tme axs (see Fg. (. Let P be a pot o coordates are x,, x, t, ad let whose summt s P wth the plae ( whose ( Γ deote the tersecto of the cood Γ ( ϖ (. The cood Γ ( s taget to the wave surface ( alog the bcharacterstc c that ssues from P. As a result, taget to the varety descrbes ( ( at the pot Γ wll be ( P where c perces the plae ϖ. Whe P (, the bcharacterstc c wll geerate the desred tegral surface (or wave ( ; furthermore, at the same tme, the cood whose summt s P wll evelope the surface (, ad the pot P wll descrbe the varety ( that evelopes the varety Γ. ( t t = t Γ ( P ( ϖ ( x Γ ( c ( t = t P Γ ( ϖ ( x c ϖ ( t = t ( P Γ ( Γ ( ( P P ( ( ( r Fgure. ( I that fgure, geometrc space (x,, x s represeted by the plae ϖ space-tme. The tme t s reckoed alog a axs perpedcular to that plae. The fgure s therefore oly schematc. (

20 6 Chapter II Prcple of evelopg waves Smlarly, project the varety wll the obta a varety projecto of P oto the plae ( Γ oto the plae ϖ parallel to the t-axs. Oe ( Γ that s taget to ϖ ( t s othg but the evelope of the varetes ( at the pot P that s the (. As a result, the posto of the wave at the stat Γ (. We have thus costructed the posto of the wave at ay later stat whe we start from ts tal posto (whch s assumed to be kow; that wll succeed tegratg the wave equato. 6. Prcple of evelopg waves. Cosder the surface geometrc space (x,.., x to be the tal posto of the frot ( of a dsturbace that s govered by partal dfferetal equatos ( of order c. I, we recalled how that dsturbace propagates geometrc space such a fasho that the frot of the dsturbace occupes the posto of at the curret stat t. We lkewse ote that the space-tme ( (x,.., x, t, the surface ( wll gve a sythess of the advace of the frot of the dsturbace geometrc space. I partcular, whe reduces to the pot P, the frot of the dsturbace ( geometrc space at the stat t wll be the surface Γ ( (. The advace of the frot s exhbted space-tme by the characterstc cood whose summt s at P. Cosequetly, oe wll see that the varety Γ s othg but the posto at the stat ( t of the elemetary Huyges wave that ssues from the geometrc pot P ( x,, x at the tal stat t. The propagato of the elemetary Huyges wave that ssues from the geometrc pot P s the represeted space-tme by the characterstc cood whose summt s P ( x,, x, t. We say that the Hadamard cood s the elemetary Huyges wave space-tme. Now, assume, wth Huyges (, that each pot P wth coordates, x of the posto of the wave ( at the tal stat t geometrc space ( commucates ts dsturbace to the ambet medum. We just saw that the propagato of the dsturbace s represeted space-tme by the characterstc cood whose summt s the pot x, x, t, ad that at ay stat t > t, the frot of that dsturbace, wll occupy the posto of the surface evelopg waves s expressed by: (, Γ geometrc space. The prcple of ( C. HUYGENS, Traté de la Lumère (coll. Les Maîtres de la Pesèe scetfque, Pars, Gauther- Vllars, 9; see pp., secod le: such a way that there s a wave aroud each partcle whose ceter s at that partcle See also, pp., Fg. 6.

21 6. Prcple of evelopg waves. 7 The wave ( the elemetary Huyges waves posto ( geometrc space, whe take at the stat t, s the evelope of Γ that ssue from the varous pots of the tal ( of the wave evsoed at the stat t. Now, we have see that ths s true ( 5; that prcple s therefore oly a theorem. More geerally, cosder a arbtrary ( -dmesoal varety S ( space-tme (x,.., x, t. We kow that such a varety wll defe oe tegral surface of the wave equato (;.e., a wave surface ( space-tme. Thaks to the last two propostos 4, oe ca determe the posto of the wave geometrc space (x,, x at ay stat wth the ad of elemetary Huyges waves. Ideed, let P be a pot o the varety S ( whose spato-temporal coordates are ξ,.., ξ, τ. Let Γ ( represet the pot the geometrc space at the stat t of the elemetary Huyges wave that ssues from the geometrc pot (ξ,.., ξ at the stat τ. Hece, t wll result mmedately from the propostos that were metoed 4 that the posto of the elemetary Huyges wave geometrc space at the stat t s the evelope of the elemetary Huyges waves Γ ( that are take at the same stat t. We remark that ths prcple s othg but the rule for tegratg frst-order partal dfferetal equatos whe stated from the vewpot of wave theory. 7. Rays of the wave (. We retur to the tal posto wave ( (Fg.. At the stat t, the elemetary Huyges wave from the pot P wave ( ( x,, x of ( ( of the ( Γ that ssues at the tal stat t < t wll be taget to the at the pot P. By defto, the ray r of the wave that ssues from the pot P s the locus of the pot P whe t vares. Oe the sees that the projectos oto the plae ϖ of the Hadamard characterstcs that ssue from the pots ( x,, x, t of ( ( are othg but the thgs that physcsts call the rays of the wave (. They are the the Hadamard bcharacterstcs, whe cosdered to be oedmesoal varetes geometrc space. Those cosderatos show that the ray r that ssues from the geometrc pot P s the trajectory of the frot of the dsturbace that ssues from that pot at the tal stat t. Theorem. Two wave surfaces geometrc space ( ad ( that are taget at a geometrc pot P at a certa stat t wll rema taget at that pot whe t traverses a ray. ( That beg the case, let be the posto of the wave ( at the stat t, let ( Γ be the posto at the same stat of the elemetary Huyges wave that ssues from the geometrc pot P at the stat t, ad let Γ be the posto at the stat t (

22 8 Chapter II Prcple of evelopg waves of the elemetary Huyges wave that ssues from the geometrc pot P at the stat t (see Fg. 4. The surfaces Γ ad Γ are taget to the surface at the pot ( ( ( P of the ray r that ssues from P ad passes through P. The wave surface the evelope of both the elemetary waves Γ ad ( wll the see that order to deduce the posto Γ ( ( s (. I a geeral fasho, oe of a wave frot ( that s kow at the stat t whe t s at a later stat t > t, oe ca frst seek the posto at a termedate stat t (t < t < t, ad the costruct the fal posto ( at the fal stat t wth the ad of that termedary posto ( (. That proposto (whch s a obvous cosequece of the prcple of evelopg waves makes the mathematcal oto of group appear ts most elemetary form. ϖ ( Γ ( ( P Γ ( Γ ( P Γ ( c ( ϖ ( c ( P P ( ( Γ ( Γ ( Γ ( P ( ϖ ( r Fgure 4.

23 CHAPTER III WAVES AND RAYS 8. Retur to the homogeeous wave equato. We saw Chapter I that ay relatve soluto to the wave equato (8 wll correspod to a wave surface (. Theorem. Ay relatve soluto to the wave equato (8 ca be assocated wth a absolute soluto to that equato. Proof. Ideed, let (x be a relatve soluto; oe wll the have: O (x, =, (5 by vrtue of (x =. Solve the equato of the wave surface ( : (x,, x = (5 for oe of the varables x for example, the varable x t ; that wll gve the equato of the wave surface ts explct form: That beg the case, set: hece, oe wll get the equato: O * (x, t, t =. (54 * t t (x,, x ; (55 * (x, t = (5 for the wave surface ( evsoed. Oe deduces from (55 that: t = t ad However, t wll the result from (54 that: = ( =,,,. (56 hece: O (x, O * x, t, = ; (57 = ( =,,,. (58 The fucto * that s defed (55 s a absolute soluto to the wave equato (8 ts homogeeous form. Q. E. D.

24 Chapter III Waves ad rays. Cocluso. Oe ca the put the equato of a wave surface to a form such that the left-had sde of the equato s a absolute soluto to the wave equato (8 ts homogeeous form. We gve the ame of wave fucto to the left-had sde of the equato for the wave surface that s defed that way. 9. Homogeeous dfferetal system for the bcharacterstcs (or space-tme rays. Let be a absolute soluto of the wave equato (8; hece, thaks to (46, oe wll have the detty x,, x t: β ( x, β O γ γ. (59 γ = Hece, the homogeeous dfferetal system of the bcharacterstcs (or rays spacetme s deduced from (45, whe oe takes (59 to accout; t has the form: dx O d d = = = dθ O ( =,,,, (6 whch θ represets a arbtrary parameter. Oe ca ow repeat the cosderatos that were developed Chapter I whe we started wth the dfferetal system (6. That s what we shall ow do brefly (. The geeral tegral of (6 has the form: x = ξ ( θ; x,, x,,,, = varat, = W ( θ; x,, x,,, ( =,,,. (6 As before, set: x t ad x t. (6 Oe ca the fer θ as a fucto of t, x,, x, t,,,, t from the th equato (6, ad upo replacg θ wth ts value thus-obtaed the other equatos (6, oe wll get: x = ξ ( t; x,, x, t,,,,, (6 = Ivarat, (64 t ( Note that here the space of elemets (x, x,, x, t, t,, t s replaced wth the space of elemets (x,, x,,,,.

25 9. Homogeeous dfferetal system of the bcharacterstcs. = W ( t; x,, x, t,,,,, t t = W ( t; x,, x, t,,,, t ( =,,,. (65 That beg the case, we propose to look for a soluto to the wave equato (8: O(x, = or O x, t,, =, (66 t whch wll reduce to a fucto (x,, x of the spatal varables for: We shall utlze the varables at the pot ( x,, that s gve by: Cosequetly: Now, replace the t = t. (67 x, x,, x for the depedet varables. The soluto x geometrc space at the stat t wll the have a value : ( x,, x. (68 ( x x ( =,,,. (69 O( x,, x, t,,, wth ther values (69 that are deduced from (68; hece: t = t (, (7 t x,, x, t. (7 Now retur to formulas (6 ad (64, whch oe must replace ad ther values that were calculated above (69 ad (7, resp.; oe wll fd that: t wth x = x (t; t, x,, x, (7 = ( x,, x. (7 The relato (7 s a mmedate cosequece of the varace of wth respect to the dfferetal system (6. As for equatos (7, they represet the ray geometrc space that ssues from the tal pot ( x,, x at the tal stat t. Upo solvg the relatos (7 for x, oe wll fd the equatos: x = x (t, t, x,, x, (74

26 Chapter III Waves ad rays. whch the fuctos x are detcal to the oes that fgured (7. Fally, replace the tal coordates x (7 wth ther values (74; that wll mply the requred tegral: = (x,, x, t. (75 It results from the varat character of wth respect to the dfferetal system (6 that the surface whose equato s: (x,, x, t = (76 wll be the posto at the stat t geometrc space (x,, x of the wave ( that cocdes wth the surface whose equato s: (x,, x, t = (77 at the tal stat t. I order to geometrcally obta the wave surface ( at a arbtrary stat t whe oe starts from ts tal posto, t wll suffce to cosder the pot P(x,, ( x o the ray (7 that ssues from the pot P ( x,, x o ( at the stat t < t that correspods to the stat t. The wave surface ( wll the be the locus of the pot P at the stat t whe P descrbes the surface. Oe ca recall Theorems I, II, III, ad IV of ad here; t wll suffce to replace the Cauchy characterstc le wth bcharacterstc (or space-tme ray. We cofe ourselves to recallg the followg two propostos here: (. Two wave surfaces ( ad ( whose equatos are: (x,, x = ad (x,, x =, resp., (78 ad have a elemet (x,, x,,, commo space-tme (x,, x wll agree alog the bcharacterstc (or space-tme ray that ssues from that elemet.. Two wave surfaces ( ad ( whose equatos are: (x,, x, t = ad (x,, x, t =, resp., (78 ad are taget at a pot geometrc space (x,, x at a certa stat wll rema taget at that pot f they traverse a ray geometrc space (. ( See 7.

27 Equato for the elemetary Huyges wave.. Equato for the elemetary Huyges wave. Cosder a fxed pot P geometrc space (x,, x whose coordates are ( x,, x ad a arbtrary drecto at that pot that s defed by the umbers p, p,, p. Oe wll the have: = (p,, p ; (8 hece, from (7 ad (7: t = t (p,, p. (8 Fally, replace the ad t (6 wth ther values (8 ad (8, resp.; that wll gve the parametrc equatos: x = χ (t ; x,, x, t ; p,, p ( =,,, (8 for the posto that s occuped at the stat t by the elemetary Huyges wave that ssues from the pot P whose coordates are ( x,, x at the stat t. I order to obta the elemetary wave (8 geometrcally, t wll suffce to cosder the pot P (x,, x o oe ray (6 that ssues from the pot P ( x,, x at the stat t that correspods to the stat t > t. The elemetary wave wll be the locus of the pot P whe the taget to the ray at P vares. The elmato of the parameters p, p,, p from the ( equatos (8 wll lead to the fte equato of the elemetary Huyges wave. Recall that a elemetary Huyges wave wll geerate a characterstc Hadamard cood space-tme whe tme t vares. Fally, t wll result from the propostos that were metoed at the ed of 9 that order to obta the wave ( equato (76 geometrcally whe startg from ts tal posto equato (77, oe ca always take the evelope at the stat t of the elemetary Huyges waves that ssue from the pots of at the tal stat t < t (recall the prcple of evelopg waves. ( (

28 CHAPTER IV WAVE TRANSPORT. Metrc o geometrc space. By defto, the metrc o geometrc space (x,, x at the stat t.e., whe: δt, (8 s determed by the dfferetal quadratc form: (δσ j B δ j x δ x (, j =,,,, (8 = j= whch s assumed to be varat uder a arbtrary chage of spatal varables x,, x. By hypothess, oe wll have: B j (x t B j (x t, (84 whch x represets the geometrc or spatal varables x,, x. Let B j the mor of the ( -order determat B kl that correspods to the elemet B j, dvded by B j : B j mor of Bj. (85 B Hece, the B j (x, t ad the B j (x, t are the covarat ad cotravarat compoets, resp. uder a arbtrary chage of varables x,, x of secod-order symmetrc tesor. Set: kl j = B j. (8 = j= The covarat ad cotravarat compoets of the ut vector N that s ormal to the wave ( are, tur, gve by: N, N j B N j ( =,,,, (8 j= respectvely, whe the equato for the wave ( s gve by (4. Recall that the Lamé dfferetal parameter s varat uder a arbtrary chage of varables x,, x, whch the wave fucto (x, t s cosdered to be a varat the broader sese.

29 . - Dfferetal equatos of the rays of a wave. 5. Dfferetal equatos of the rays of a wave. Let: (x, t (88 be a wave fucto (see 8. Oe wll the have the detty x ad t: O x, t,, t. (89 It wll the result from the dfferetal system (6 that the rays of the wave equato: are gve by the dfferetal equatos ( : (x, t = (9 wth: dx dt = = dθ O O t ( =,,, (9 d. (9 dθ O O The otatos ad sgfy that oe has replaced the ad t t O ad wth the frst partal dervatves of the wave fucto (88. Recall that: t O, t t. (9 The detty relato (9 expresses the dea that ay wave fucto (x, t s a varat of the dfferetal system of the rays.. Wave trasport geometrc space (. Now, set: w w (x, t O O : t ( =,,,. (94 ( Upo replacg / wth the wave fucto (88, the last equatos of the dfferetal system (6 wll be satsfed detcally. ( J. VAN MIEGHEM, Étude sur la théore des odes (Pars, Gauther-Vllars, 94; see pp. 99, et seq.

30 6 Chapter IV Wave trasport. The trasport of the wave ( of equato (9 geometrc space (x,, x s defed by the dfferetal system: dx w = dt ( =,,,. (95 The detty (9 becomes: d = + w, (96 dt t = whch s a detty x,, x ad t. Theorem. Ay wave fucto s a varat of the dfferetal system of wave trasport that s defed by that fucto. Defto. The velocty of trasport T of the wave ( s the compoet w N ormal to the wave ( of the velocty vector w whose compoets are w,, w. Oe wll the have: T w N. (97 It wll the result from the detty (96 ad formulas (87 that: w N = w N =, (98 whch w,, w are the cotravarat compoets of the velocty vector w whose covarat compoets are represeted by w,, w ; hece: w j Bj w, = w j B wj. (99 = However, oe obvously has: w N w N = w N. ( = It the results from (98 that the compoet w N of the velocty w that s ormal to the wave s a varat. Fally, by vrtue of the defto (97, that varat s othg but the trasport velocty T of the wave ( ; hece:

31 . Wave trasport geometrc space. 7 ( x, t / t T ( x, t. ( x, t (

32 CHAPTER V JACOBIAN FORM OF THE WAVE EQUATION AND HAMILTONIAN FORM OF THE RAY EQUATIONS 4. Jacoba form of the wave equato. The homogeety of the wave equato (8 permts oe to wrte t the form: O(x, P ( µ + P ( µ + + P k ( µ k + + P µ =, ( whch P, P,, P k,, P µ, are homogeeous polyomals,, that have degree,,, k,, µ, respectvely; µ s the degree of the homogeety of O(x,. We shall always deote the temporal varable by x or t. Let: H x, t,,, H r x, t, t ( be the r ( µ dstct roots of equato ( or t (x, t / t. The fuctos ( are homogeeous ad of degree oe,,. Recall that / represets the partal dervatves of wth respect to the spatal varables x,, x. Each of the r Jacoba equatos: + H p x, t, = t (p =,,, r (4 s the equato of a famly of wave surfaces the. We let (p deote a wave of that famly. 5. Hamltoa form of the ray equatos. The Cauchy characterstcs of the wave equato (4 are defed by the Hamltoa equatos: dx = dt H p, ( =,,,. H p = d dt x (5 (6 Equatos (5 admt ay soluto: (p (x, t

33 5. Hamltoa form of the ray equatos. 9 of the wave equato (4 as a varat, because f oe takes the homogeety of H µ to accout the oe wll have, tur: d dt H m ( p ( p p + ( p t = t + H p x, t, ( p ( p. (7 The dfferetal equatos (5 deed defe the trasport of a wave (p the; the les alog whch that trasport takes place are (as we saw above the rays of the wave whose equato geometrc space (x,, x s: Theorem I. I say that the Pfaff form: (p (x, t =. (8 ( p δ x (9 = s a absolute dfferetal varat of the Hamltoa system (5 ad (6. The symbol δ represets a trucated varato;.e., δt. Proof. Ideed, thaks to (5 ad (6, oe wll have: d dt H H ( p δ x p ( p p x ( p = x δ + δ = =. ( However, due to the homogeety of H p, t wll result that: hece, from (: = ( p H p ( p H p x, t, ( p ; ( d dt H H H ( p δ x p ( p p p ( p x ( p ( p = x δ + δ δ = = =, ( or rather: d dt ( p δ x δh p + = δ H ( p p ( p. ( = Hece, upo takg ( to accout, oe wll fally have:

34 Chapter V Jacoba ad Hamltoa forms of the equatos d dt ( p δ x =. (4 = Q. E. D. Theorem II. Whe the Hamltoa H p s depedet of tme t, that fucto wll be a varat of the Hamltoa equatos (5 ad (6. Proof. Ideed, t s easy to verfy that the detty: dh p dt H p t (5 s a cosequece of the dfferetal equatos (5 ad (6. As a result, the relato: wll mply that: H p t dh p dt (6. (7 Q. E. D. Remark. Whe H p does ot deped explctly upo tme t, the wave fucto (p (x, t wll have the form: (p ht + ϕ (p (x, (8 whch h s a costat, ad ϕ (p s a fucto of the spatal varables that s a soluto to the partal dfferetal equato: ϕ H p x, = h. (9 6. Case of a homogeeous medum. Whe the coeffcets of the wave equato (4 are costat, oe wll have: H p ad H p t. ( I that case, the Hamltoa dfferetal system (5 ad (6 s tegrated mmedately; oe wll have:

35 6. Case of a homogeeous medum. =, H p x = x + ( t t, ( ( whch ad oe wll get after replacg wth x are tegrato costats, ad the otato H p. H p represets what We the see that the rays are straght les here. That stuato wll be produced wheever the coeffcets of the hghest-order dervatves that appear the physcal equatos ( reduce to costats. I that case, oe says ( that the physcal medum that s govered by those equatos s homogeeous. Theorem I. I a homogeeous medum, the space-tme rays (, as well as the rays geometrc space, wll be straght les. Corollares: The Hadamard cood a homogeeous medum has rectlear geerators. The elemetary Huyges waves that ssue from a pot of a homogeeous medum are mutually homothetc; ther locus space-tme s J. Hadamard s characterstc coe. H We remark that p results that the coeffcets s homogeeous of degree zero H p,,. It wll the of equatos ( deped upo oly ( parameters, amely, the ( relatoshps betwee the ( costats,, ad ay oe of them. The elmato of those ( parameters from equatos ( leads to the equato of the surface that occupes the posto of the elemetary Huyges wave that ssues from the pot (x,, x at the tal stat t < t whe the curret stat s t. I order to obta the preset posto of a wave whe oe starts from ts tal posto, draw the le through the pot P (x,, x of that s defed by equatos ( ad cosder the pot P (x,, x o t that s take at the curret stat t. wll the be the locus of the pot P whe P descrbes. Recall that s ( T. LEVI-CIVITA, Caratterstche de sstem dfferezal e propagazoe odose (Bologa, N. Zachell, 9; see pps. 54 ad 55. ( Or J. Hadamard s bcharacterstcs.

36 Chapter V Jacoba ad Hamltoa forms of the equatos also the evelope at the curret stat t of the elemetary Huyges waves that ssue from the pots of at the tal stat t < t. We just saw that the pots of are deduced from the pots of by traslato; we remark that ths traslato s ot the same for every pot of, except whe s a plae geometrc space. Theorem II. A plae wave a homogeeous medum wll rema a plae wave as tme vares. Corollary. No matter what pheomeo s evsoed a homogeeous medum, the propagato of plae waves wll always be possble. Theorem III. The trasport of a pot o a wave a homogeeous medum s a uform, rectlear moto. That trasport wll be the same for all pots of a wave oly whe t s a plae. Corollary. I a homogeeous medum, the trasport velocty of a wave depeds upo oly the drecto of propagato. 7. Trasport velocty. The metrc o geometrc space (x,, x s defed by ( the varat dfferetal quadratc form (8. The covarat compoets N p ad cotravarat oes N ( p of the ut vector N (p that s ormal to the wave (p whose equato s: (p (x, t = ( are gve by formulas (87, ad the trasport velocty T p of the wave (p s gve by formula (. Havg recalled that, dvde equato (4 by ( p, whch the Lamé parameter (p s defed by (86; hece, the trasport velocty T p of the wave (p wll be: T H x t N (4 p ( p p (,,. Remark. Suppose that the Hamltoa fuctos H p ad H q are depedet of tme t ;.e., that: H p H, q, (5 t t ad that oe has the detty: H p x, + H q x, (6 x ad, moreover. Hece, the two Jacoba equatos:

37 7. Trasport velocty. ad t + H p x, = (7 t + H q x, = (8 wll represet the same famly of waves that propagate the cotrary sese. Ideed, ay soluto: (p ht + ϕ (x (9 to the wave equato (7 ca be assocated wth the soluto: (p + ht + ϕ (x ( to the wave equato (8. Recall that h represets a arbtrary costat. Now, thaks to (5 ad (6, the dfferetal equatos for the trasport of the waves (p ad (q ca be wrtte: dx dt = ( =,,,, ( ϕ ± H x, ϕ whch: H H p H q. ( Equatos ( exhbt the fact that the trasport of the waves (p ad (q takes place the cotrary sese alog the same rays, amely, alog the les: dx = = H ϕ dx H ϕ ( geometrc space (x,, x. Now, ote that by vrtue of (4 ad (6, the trasport veloctes T p ad T q of that wave are coupled by the relato: T p + T q =. (4 The famles of waves (p ad (q are detcal, but they propagate the opposte sese alog the same les or rays wth the same velocty. We shall assume what follows that ths stuato presets tself ρ tmes;.e., that amog the r famles of waves (4, there are ρ pars of wave famles such that the two famles of the same par wll propagate opposte seses alog the same les or spacetme rays.

38 4 Chapter V Jacoba ad Hamltoa forms of the equatos 8. Hyper-refrgece. Let: t + H x, t, =,, t + H r ρ x, t, = (5 be the Jacoba equatos of the r ρ dstct famles of waves that propagate alog rays that are all dfferet ad gve by the dfferetal equatos: dx dt d dt = H H = dx dt d dt = H H = r ρ r ρ ( =,,,, (6 (7 respectvely. It wll the result from equatos (5 ad (6 that at ay pot (x,, x of geometrc space there wll be (r ρ propagato veloctes for the wave ad (r ρ dstct rays that correspod to each drecto (N,, N that s gve advace ad ssues from the pot (x,, x. We wll the say that the physcal medum that occupes the geometrc space (x,, x s (r ρ-tmes refrget. 9. Elemetary Huyges wave a homogeeous medum whe the metrc s Euclda. I that case, the wave equato has the form: + H =, (8 t whch H s a homogeeous fucto of degree oe wth respect to the partal dervatves /,, /, ad the otato / represets all of those dervatves. The drecto coses N,, N of the ormal to the wave are gve by (87: N N, (9 ad the trasport velocty T of the wave s gve by (: wth T, (4

39 9. Elemetary Huyges wave ad Euclda metrc. 5 (. (4 = It the results from equato (8 ad the homogeety of H, from (4: T H ( N, (4 whch N takes the place of the drecto coses N,, N. The rays that ssue from the org O of the coordates at the tal stat t are defed by equatos (: H x = t, (4 or rather, thaks to the homogeety of H, by the equatos: x H ( N = t, N (44 whch, by vrtue of (, the ad the N are costats, respectvely. Note that the costats N are ormalzed by: ( N. (45 = That beg the case, let P be a pot that s take at the stat t > o the elemetary Huyges wave that ssues from the org O at the stat t. The dstace from the pot O to the taget plae at P to the elemetary wave that s evsoed s defed by the well-kow formula: so, thaks to (44: = N x, (46 = H ( N = N t, (47 N = or rather, f oe takes to accout the homogeety of H ad the relato (4: T t. (48 As a result, the elemetary Huyges wave that ssues from the org O at the stat t = s, at the stat t >, the evelope of the plae waves of the equato:

40 6 Chapter V Jacoba ad Hamltoa forms of the equatos N x T t =, (49 = whose coeffcets N, N,, N, T verfy the relato (4. The elemetary Huyges wave s the the evelope geometrc space (x,, x of a ( -parameter famly of plaes. We fally remark that the equato of the elemetary Huyges wave s the pot-lke equato x,, x that correlates wth the taget equato (4 N, N,, N.

41 CHAPTER VI REFLECTION AND REFRACTION OF WAVES. Icdet, reflected, ad refracted waves. Cosder two physcal meda ad geometrc space (x,, x that are separated at each stat t by a movg surface S whose equato s: S (x, t =. (5 Let: O x, t,, =, t (5 O x, t,, = t be the partal dfferetal equatos of waves that are compatble wth the equatos of physcs that gover the meda ad, resp. Let µ ad µ be the degrees of homogeety of O ad O, resp., wth respect to the partal dervatves of. From (, let: H s x, t, H s x, t, s =,,, ν µ s =,,, ν µ (5 (5 deote the v (v, resp. roots of equato (5 [(5, resp.] (whch have degrees µ ad µ / t, respectvely, whch correspod to the famles of waves that propagate alog the varous rays ( 7. The Jacoba equatos: + H s x, t,, = t + H s x, t, =, t (s =,,, v, s =,,, v (5 (5 are the those of the v ad v famles of dstct waves that are compatble wth the physcal equatos that relate to the meda ad, resp. Oe the sees that medum s v -tmes refrget, ad medum s v -tmes refrget. s Let ( x, t deote a soluto to oe of the Jacoba equatos (5, so: s ( x, t = (54

42 8 Chapter VI Reflecto ad refracto of waves wll be the equato of a wave medum ; by defto, t wll be a cdet wave. The s surfaces S ad ( x, t tersect space-tme (x,, x, x t alog a ( - dmesoal varety V (. It results from Theorem III of that the cdet wave wll gve rse to v reflected waves ad v refracted waves that pass through V (. Let: r ( x, t s = ad ( x, t = (55 r be the equatos of a reflected wave, whch s a soluto of oe of equatos (5, s ad a refracted wave, whch s a soluto of oe of equatos (5, respectvely. r s The waves ad are determed completely space-tme by the demad that they must be cotaed the varety V (.. Applyg the prcple of evelopg waves. Let ξ,, ξ, τ be the coordates of a pot o the varety V ( space-tme. I geometrc space (x,, x, let: ( r ( s Γ ad Γ ( represet the posto at the stat t meda ad, respectvely, of the elemetary Huyges waves that ssue from the pot whose coordates are (ξ,, ξ at the ( r ( s stat τ. The elemetary waves Γ ad Γ are tegral surfaces of the equatos: ( ( ( + H r x, t, t = ad + H s x, t, =, (56 t respectvely. Hece, t results from the prcple of evelopg waves that ( the reflected wave ( r ( r geometrc space at the stat t s the evelope of the elemetary wave Γ whe the pot (ξ,, ξ, τ descrbes the varety V ( space-tme. Smlarly, the ( s ( s refracted wave at the same stat t s the evelope of the elemetary wave Γ. ( (. Geometrc laws of reflecto ad refracto. (. By hypothess, the wave ( s ( r ( s fuctos (x, t, (x, t, ad (x, t are dfferetable o the surface S spacetme; hece, from (54 ad (55: ( J. HADAMARD, Leços sur la Propagato des odes et les équatos de l Hydrodyamque (Pars, Herma, 9; see, pp. 95. ( J. VAN MIEGHEM, Ws- e Naturkudg Tjdschrft (Get; deel VII, 94; see pp. 5 ad 8.

43 . Geometrc laws of reflecto ad refracto. 9 = = = ( s ( s δ x + δ t =, t ( r ( r δ x + δ t =, t ( s ( s δ x + δ t =, t (57 whch the δx ad δt verfy the sgle codto: S S δ x + δt = t =. (58 The dfferetal relatos (57 ad (58 say that space-tme the wave surfaces ( s ( r ( s,, ad, ad the surface S have a ( -dmesoal varety commo, amely, the varety V (. Oe deduces mmedately from the relatos (57 ad (58 that: ad ( r ( s ( r ( s S = λ, = λ S t t t ( s ( s ( s ( s S = λ = λ S,, t t t (59 (59 whch λ ad λ are two arbtrary fuctos: λ λ (x, t ad λ λ (x, t (6 of the spato-temporal varables x,, x, ad t. ( s ( r The compatblty codtos (59 ad (59 that relate to the fuctos,, ( s ad o the separato surface S express the geometrc laws of reflecto ad refracto. We propose to put the relatos (59 ad (59 to a form that s close to the classcal form of the laws of reflecto ad refracto (vz., Descartes s laws. To fx deas, suppose that 4, ad set: N S x, ( s S N ( s ( s, ( r N ( r ( r, ( s N ( s ( s, (6 ( =,,,

44 4 Chapter VI Reflecto ad refracto of waves ( whch N, N s (, N r (, N s are the covarat compoets of the ut vectors that are ( s ( r ( s s r ormal to the surfaces S,,,, resp. Ther trasport veloctes T, T, T, s T geometrc space (x, x, x are gve by: T S t S, s T t ( s T ( s, r ( r ( r, s T t ( s ( s, (6 respectvely. By coveto, the postve ormal to S s drected from the medum towards ( s ( r ( s medum. As for the postve ormals to the wave surfaces,,, they pot the drecto of ther propagato. By vrtue of (6 ad (4, oe wll have: T H ( x, t, N, s ( s s T H ( x, t, N, r ( r r s ( s s T H ( x, t, N. (6 Thaks to the deftos (6 ad (6, t s easy to trasform formulas (59 ad (59 ; oe wll fd that: ad ( r ( r ( s ( s N N = λn S, ( =,,, (64 T T = λ T S, ( r ( r ( s ( s ( s ( s ( s ( s N N = λn S, (64 T T = λ T S. ( s ( s ( s ( s Elmate λ ad λ from relatos (64 ad (64, respectvely; that wll gve: N N = ( r ( r ( s ( s T T r ( r s ( s T N (65 ad N N = ( s ( s ( s ( s T T s ( s s ( s T N ( =,,. (65 Now multply formulas (65 ad (65 by the cotravarat compoets N of the ut vector N that s ormal to S; hece: