G. Herglotz s treatment of acceleration waves in his lectures on Mechanik der Kontinua applied to the jump waves of Christoffel

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1 From: E. B. Chrstoffel: the Influence of hs Wor on Mthemtcs nd the Physcl Scences, ed. P. L. Butzer nd F. Feher, Brhäuser, Boston, G. Herglotz s tretment of ccelerton wves n hs lectures on Mechn der Kontnu ppled to the jump wves of Chrstoffel Ernst Hölder 1 ) Mthemtcs Insttute, Johnnes-Gutenberg-Unverstät, Mnz (FRG) Trnslted by D. H. Delphench Followng lecture delvered by Herglotz n 195/6, we brefly tret ccelerton wves n hyperelstc mterls. Our mn result s dvergence equton for the squred Eucldn norm of the soclled wve vector. We then pply Herglotz s method (devsed for ccelerton wves) to the propgton of such frst order dscontnutes n elstc bodes s were treted by Chrstoffel n [1]. Introducton In the prevous rtcle we brefly referred to the contents of the mddle prt of n unpublshed three-prt lecture Mechn der Kontnu tht Herglotz gve t Göttngen n 195/6. It ws prepred for the Göttnger Mthemtsche Insttut by Fru Ph. Slé. I m grteful to her for the nowledge of ths lecture. Now, we re rememberng the onehundredth brthdy of Herglotz (b , d ). As fr s I cn see, fter Chrstoffel, Herglotz ws the frst to crry out the tretment of the spredng of dscontnutes to such n extent tht the dvergence relton for the norm of the wve vectors emerges s the orthogonlty condton for the determnton of the hgher (.e., thrd) order jumps. Ths gves the menng of the rys s the shdow boundry of geometrc optcs. At the concluson of ths rtcle, we show how the computtons of Herglotz for the second order dscontnutes cn be derved from Chrstoffel s frst order dscontnutes wthout hs symbolsm. 1. Herglotz s tretment of ccelerton wves In the followng, let D R 4 be n open set tht decomposes nto two open subsets D 1, D by mens of porton of suffcently flt, orented hypersurfce n R 4 : Σ = {(x, t) x R 3, t = τ(x) R}; n ths, we let τ: G R, G R 3 be suffcently mny tmes contnuously prtl dfferentble functon. 1 ) Ths wor ws commssoned by the Deutschen Forschungsgemenschft. My collbortor s Herr Dpl.-Mth. Hns-Jürgen Böttger.

2 Herglotz nd Chrstoffel on dscontnuty wves We now consder moton of hyperelstc contnuum n Σ whose frst prtl dervtves of the Eulern poston coordntes nd whose second prtl dervtves of these coordntes re dscontnuous on Σ. The dscontnuty surfce Σ wll lso be clled the wve surfce n wht follows, nd the jump cross Σ becomes functon g tht s contnuous n the complement of Σ, whch, snce the tme of Chrstoffel, hs been denoted [g]. The Eulern coordntes of prtcle t tme t wll be denoted by x = x (t, x), = 1,, 3, n whch x = (x 1, x, x 3 ) men the (Lgrngn) poston coordntes of the prtcle t n rbtrry, but fxed, tme t = t 0 ( number). Let the functons x, = 1,, 3 belong to C 1 (D) C (D 1 D ), nd ts second prtl dervtves wth respect to t, x, = 1,, 3 shll possess regulr, fnte lmtng vlues on D 1 (D, resp.) under one-sded pproxmton; the second prtl dervtves of the Eulern poston coordntes wll therefore hve jump dscontnuty on Σ, s requred. On both sdes of Σ the x, = 1,, 3 stsfy the equtons of moton: ɺɺ x = X ρ dw, = 1,, 3, wth ɺɺ x t x ; ρ s the densty n fxed reltve confgurton, X s the th component of the externl force, nd W s one component of the (frst) Pol-Krchhoff tensor. We set: p x x,, = 1,, 3, nd demnd tht W = W p wth n nternl energy functon W = W ( p ( t, x), x) (.e., hyperelstcty) tht shll be suffcently mny tmes contnuously dfferentble. The equtons of motons on both sdes of the wve surfce then red: (B) ɺɺ = X ρ x W W p x p p x, = 1,, 3; n ths nd everywhere n the sequel we sum over ll ndces tht pper twce. One sets: τ p nd P,,, = 1,, 3. x x Wth the help of well-nown lemm of Hdmrd (cf. [], 7 or [3], 174), one derves the followng jump relton for the functons p on Σ: [ p ] = [ ɺɺ x ] P P.

3 Herglotz nd Chrstoffel on dscontnuty wves 3 If one tes ths nto ccount, long wth x C 1 (D), then t follows, wth the ssumpton tht [X ] = 0, = 1,, 3, nd when one further ntroduces the notton η [ ɺɺ x ], = 1,, 3, tht: ρη W = P P η, = 1,, 3. We wrte: Ths gves: η = X j η j, wth X j 1 W ρ 0 = (X j δ j )η j, = 1,, 3, n whch δ j refers to the Kronecer symbol. The vector η = (η 1, η, η 3 ) wll be referred to s the wve vector. Due to the dscontnuty of ɺɺ x on Σ, one hs η 0, nd t follows tht: 0 = det{x j (t, x, P) δ j } = : F(t, x, P) wth P (P 1, P, P 3 ). We cll the surfce n R 3 tht s defned by F(t, x, P) = 0 (wth the runnng vrble P) the norml surfce. F s polynoml n P of the form F = F 6 F 4 F 1, n whch the F, =, 4, 6 re homogeneous polynomls of degree n P. An nterestng nterpretton for F derves from the fct tht the wve moton cn be descrbed by one-prmeter group of contct trnsformtons of the surfce elements (t, x, P), whose genertng functon s F(t, x, P) precsely. Wth the cnoncl prmeter Θ, one obtns the followng dfferentl equtons for the ndvdul pthlnes of the surfce elements of the wve surfce, whch re dsplced nto ech other by the trnsformtons of the group n such wy tht F = 0: P P. dθ = F = : Π, P dp dθ = F F x t P, = 1,, 3, dt dθ = P Π = : Φ. These re obvously the equtons for the chrcterstcs of the frst order prtl dfferentl equton: τ 0 = F(t, x, p ), t = τ(x), p = ( p 1, p, p 3), p = P =, = 1,, 3. x Hdmrd clled these dfferentl equtons for the rys the bchrcterstcs of the equtons of moton for the chrcterstc wve t = τ(x). Along wth the wve vector η, one lso ntroduces the ry vector p = (p 1, p, p 3 ) through: p 1 Ω Ω P, = 1,, 3, wth Ω X j η η j

4 Herglotz nd Chrstoffel on dscontnuty wves 4 (snce η 0, one obvously hs Ω 0). On the grounds of the homogenety of Ω n the P, = 1,, 3, one next remrs tht: 1 = P p. In the cse where the mtrx (X j δ j ) hs rn two, one further hs: F P = p, R {0}, = 1,, 3. One sees ths perhps n the followng wy: Let F h, h, = 1,, 3 be the djont tht s ssocted wth X j δ j ; snce rn(x j δ j ) =, one hs: F h = η h Λ, h, = 1,, 3, wth Λ R. Snce for t lest one ordered pr (h, j ), h, j {1,, 3} one must hve F h 0, one deduces, becuse F h = F h : Λ = η λ, λ R {0}, = 1,, 3,.e., Fh = 1 λ ηh η. It follows tht: F P = F h X P h = 1 λ ηh η X h P = 1 Ω λ P = λω p. Hence Ω/λ = = Φ: p = Π, = 1,, 3. Φ Π The vectors p = (p 1, p, p 3 ), p = tht emnte from the null pont defne the so-clled Φ ry surfce. If one tes nto ccount the equtons P p = 1 tht we just proved nd the fct tht F = p, 0, = 1,, 3 then one esly sees tht the sme reltonshp P exsts between the rys nd the norml surfce F(t, x, p ) = 0, wth p = ( p 1, p, p 3), p P, = 1,, 3 through the recprocl trnsformton nmely, the polr reltonshp n the rel unt sphere whch, ccordng to Mnows, defnes n ordnry vrtonl problem between the ndctrx nd the fgurtrx. If rn(x j δ j ) < then the foregong conclusons re not vld; n the sequel, ths cse shll be omtted.. Herglotz s proof of dvergence equtons for the wve nd ry vectors In the followng, let D R 4 gn be n open set tht decomposes through the wve surfce Σ = {(x, t) x R 3, t = τ(x) R} nto two open subsets D 1, D : D = D 1 Σ D.

5 Herglotz nd Chrstoffel on dscontnuty wves 5 Let the functons x, = 1,, 3 belong to C 3 (D 1 D ) C 1 (D). For mult-ndex µ = (µ 0, µ 1, µ, µ 3 ), µ 0, µ 1, µ, µ 3 N {0} wth µ 0 µ 1 µ µ 3, one sets: D µ µ 0 µ 1 µ µ 3 1 µ 1 µ 3 ( t) ( x ) ( x ) ( x ) µ µ 0 3 ; one hs for ll such mult-ndces nd ll ponts (x, t(x)) Σ: 0 = lm sup{ D µ x n (y) D µ x n (z) ; y, z D, y (x, τ(x)) 4 ε 0 E z (x, τ(x)) E 4 < e} = 1,, 3, = 1,. We ntroduce the followng nottons: nd: pɺ x, t x pɺ γ 3 x t x x γ, ɺɺ p t d η [ ɺɺ x ]. 3 x x, ζ [ ɺɺɺ x ], ɺɺɺ x ɺɺ x t, A smple clculton gves the jump reltons on Σ: (1) [ ɺ ] = η P, p () [ p γ ] = [ ɺ ] P = η P P γ, p (3) [ ɺɺ p ] = η ζ P, (4) [ ɺ ] = η P γ η P p γ γ η P γ ζ P P γ ; one proves ths, sy, by repeted pplcton of the forementoned lemm of Hdmrd. We now consder the equtons of moton: (B) ɺɺ x = X ρ W W p x, = 1,, 3, nd dfferentte wth respect to t: ɺɺɺ x = ρ W W W X ɺ p ɺ p ɺ p p ɺ, = 1,, 3. j j γ x γ The equtons re true on both sdes of the wve surfce. It follows tht:

6 Herglotz nd Chrstoffel on dscontnuty wves 6 (5) ρζ W W d W j = η P ( P ) P P P [ p p ] η η ζ ɺ j γ, x γ = 1,, 3, n whch we hve ssumed tht [ X ɺ ] = 0. One sets: j pɺ γ j x lm ( y) t x γ y ( x, τ ( x)) y D1 nd j p γ j x lm ( y), (x, τ(x)) Σ. γ x x y ( x, τ ( x)) y D1 Snce x C 1 (D) (s one sees from the Hdmrd lemm): Ths yelds: d p γ = j j pγ pγ P ɺ,, γ, j = 1,, 3 on Σ. nd d W (6) ρζ d = Now, on ccount of (1), ():.e.: (7) ρζ d = Wth: W W d W j j η P = η P ( η P ) ( p pɺ P ) η P j x γ γ γ W W W η P ( η P P P ) ζ ([ j p pɺ ] j ) j γ pɺ γ P η P. γ W j [ p pɺ ] = ( p η P P )( pɺ η P ) p pɺ one hs, due to (1), the equton: γ j j j γ γ γ j j j = pɺ η P P η η P P P p η P γ γ γ W W j 1 j η P ( η P P P ) ζ ( pɺ ) j γ η Pγ η P P. γ j pɺ γ j x lm ( y) t x γ y ( x, τ ( x)) y D j pɺ γ 1 ηj P γ = 1 ( j pɺ γ j pɺ γ ). We ntroduce the followng bbrevtons:

7 Herglotz nd Chrstoffel on dscontnuty wves 7 nd: Ω(P ζ) 1 W ρ j P P ζ ζ j, 3 1 W Q ρ j γ P η j P ( Ω 1 W ρ j j pɺ γ pɺ γ ). η P, When one observes tht: d W d η P = ( ρω ), nd one rrves t: or (8) W W Ω dη η P = η P = ρ, η W P P ρ Ω ζ = ( P ζ ), ζ ρζ = ( ) d ρω Ω dη ρ Ω ρ ( P ζ ) ρq η ζ 1 Ω ρ ( P ζ ) ζ ζ = ( ) d ρω Ω dη ρ ρq. η The vector ζ = (ζ 1, ζ, ζ 3 ) s therefore soluton of symmetrc nhomogeneous lner system of equtons; the determnnt of the coeffcent mtrx s once gn precsely F(t, x, P). The soluton of the homogeneous lner system of equtons tht s ssocted to (8) wll be generted by η = (η 1, η, η 3 ). Due to the symmetry of the coeffcent mtrx, (8) hs precsely one soluton when η s orthogonl to the rght-hnd of the system of equtons, hence, when one hs: d Ω dη (9) η ( ρω ) ρη ρη Q η = 0 or, due to the homogenety of Ω n η: d dη η ( ρω ) ρω ρη Q = 0, hence: d (10) ( ρη Ω ) = ρη Q.

8 Herglotz nd Chrstoffel on dscontnuty wves 8 The term ρη Ω cn, wth the help of the ry vector p = (p 1, p, p 3 ), be expressed s: Ω η W = ρ 1 η η P = 1 Ω ( P η) = Ω(P η) p. P Wth E (1/)ρΩ (P η) = (ρ/)η, one obtns Ep = ρω(p η) p, nd the solvblty condton (10) becomes: d ( Ep ) = Q wth: ρ (11) Q η 3 W 1 Q = ( j j P P p p ) j η η ɺγ ɺ γ, δ whch ultmtely delvers the dvergence equton: de de de (1) grd E,, nd dv p dp. = 1 In order to better understnd the menng of the dvergence relton, we consder the ry lnes or rys of the wve moton. They re the solutons of the followng system of ordnry dfferentl equtons: ds = p v, = 1,, 3 wth v s p p. s These dfferentl equtons gve the chnge n E wth the rc length s long the ry. From (1), t follows tht: de (13) vs E dv p = Q. ds We now me the ssumpton tht one hs F = 0 long the pece of ry n queston, whle F h = 0 does not hve to be true for ll determnnts F h, h, = 1,, 3 (cf., supr) of the two-rowed submtrces of the coeffcent mtrx of the system (8); lewse, let v s 0. There then exsts normlzed soluton η 0 = ( η 1 3 0, η0, η0 ) to the homogeneous lner system of equtons: 1 Ω ( P η) η = 0, = 1,, 3, η whch wll be true t no plce on the ry where ll of the components re null. The ctul wve vector s then gven by: η(s) = λ(s) η 0.

9 Herglotz nd Chrstoffel on dscontnuty wves 9 From ths, t follows tht: E = λ E 0, E 0 ρ (η0 ) > 0 nd Q = Rλ Sλ 3 wth nown coeffcents R nd S. Equton (13) then gves: (14) d A ds λ Bλ Cλ = 0, A = v s E 0 0, n whch the expressons B nd C re nown functons. One then determnes λ, nd therefore η, long the ry by the Rctt dfferentl equton (14). In prtculr, f λ = 0 t ny locton s = s 0 then t follows from the unqueness theorem tht λ 0 long the entre ry tht goes through ths locton. However, wth λ 0, one hs η 0. Thus, f the length of the wve vector η tht mesures the ntensty of the wve s equl to zero t ny locton then t hs the vlue zero ll long the entre ry tht runs through ths locton. In optcs, the rys re chrcterzed precsely by property tht s nlogous to the fcts presented here. One thns of, sy, lght source tht shnes upon lmpshde. Of ll of the rys tht fll upon ths shde the ones tht fll upon the boundry of the shde wll defne shdow. From our results, the rys re the lnes long whch the ntensty of the lght remns zero when t s once zero. Ths expresses the fct tht the shdow boundry s ndeed constructed from rys. Ths fct s true n wve optcs only for wvelength tht converges to zero. If the wvelength s not zero, so one hs dffrcton phenomen, then there wll be no shrp shdow boundry. By consderng the ccelerton wves, one obtns these chrcterstc propertes of rys n ther full strength. Geometrc menng One tes closed surfce n R 3. Let N be the externl norml to t nd let do be the hypersurfce element wth the projectons: do = do cos(n, x ). It then follows from (11) tht the hypersurfce ntegrl s the volume ntegrl: E p do = Q dv. As closed surfce, we now choose ry tube wth norml cross-secton ω. If we pply the formul bove to ths ry tube, whch s bounded by the ponts 1 nd of the cross-secton, then we obtn: ω Ep cos(n, x ) = (Ev 1 sω) (Ev s ω) 1 = Q dv = s s1 Qw ds.

10 Herglotz nd Chrstoffel on dscontnuty wves 10 In the event tht the dfferentl equtons re lner, so Q 0, one hs: Ev s ω = const. long the ry tube. E cn be sutbly regrded s the energy of the wve. 3. Applcton of the Hdmrd-Herglotz tretment of the ccelerton wves to the dscontnuty wves of Chrstoffel Chrstoffel treted dscontnuty wves of frst order. Therefore, let: Σ = {(x, t) x R 3, t = τ(x) R} be frst order dscontnuty surfce,.e., the solutons x, = 1,, 3 of the Euler equtons of moton (B) re contnuous on Σ, whle ther frst prtl dervtves possess (fnte) jump dscontnutes. All dervtves nvolved shll hve equl two-sded lmts on Σ. Furthermore, let ll of the nottons nd defntons gree wth the ones bove, nd one hs x C 0 (D) C (D 1 D ), = 1,, 3, wth open sets D, D 1, D, s bove. For the frst order dscontnutes tht were exmned by Chrstoffel we requre jump relton on the wve surfce Σ tht one obtns wth the help of Hmlton s prncple of sttonry cton: From Zemplén [4], the vnshng of the frst vrton (the Hmlton ntegrl) on the edge of the n of n extreml x (t, x), 1 3, wth dscontnutes of frst order on Σ gves the Weerstrss-Erdmnn corner condtons on Σ: 0 = δxx (r [ x ɺ ] () [ W ] do ), = 1,, 3, wth do 1 = dt 3 = P 1 (), do = dt 1 3 = P (), do 3 = dt 1 = P 3 (). Ths gves the jump reltons on Σ: ρξ [ W ] P = 0, = 1,, 3 wth ξ [ x ɺ ]. Chrstoffel treted clsscl elstcty; there, one hs the followng expresson for the nternl energy W: W = 1 c γ j j p p ; γ j c γ = c γ j = const., j, γ, = 1,, 3. Anlogous to the second order dscontnutes, one ntroduces: Ω(P ξ) 1 c γ ξ ξ j P ρ j P.

11 Herglotz nd Chrstoffel on dscontnuty wves 11 Ths s bqudrtc form n the ξ, P wth constnt coeffcents tht re symmetrc n, j. Wth: [ p γ ] = ξ P γ nd [ W γ ] = c [ p j ] c γ ξ P γ, ths gves: j γ = j P [ W ] = c γ j ξ j ρ Ω P P = ( P ξ) ξ Thus, there exst lner equtons for ξ = (ξ 1, ξ, ξ 3 ): (*) 0 = ξ 1 Ω ( P ξ), = 1,, 3. ξ Snce ξ 0, the determnnt of the coeffcent mtrx of the homogeneous lner system (*) hs the vlue F(t, x, P) = 0, just le the second order dscontnuty wves. We now obtn dvergence relton for ξ n the followng wy: From the Euler equtons of moton: ρ ɺɺ x = X dw, = 1,, 3, t follows, under the ssumpton tht [ρ] = 0, [X ] = 0, nd snce:. tht W = 1 c j j p p ; ρη j = cj [ p ], = 1,, 3. An esy clculton (wth the help of the Hdmrd lemm) delvers: nd.e.: [ p ] = [ pɺ ] P [ pɺ ] P ξ P η P P,,, = 1,, 3 ξ d ξ =[ ] pɺ P η ; [ p ] = η P P ξ P ξ P ξ P. From ths, one gets: Hence, wth: one hs: η 1 Ω ( P η) = 1 j η ρ c Q 1 j ρ c ( ξ P ( ξ P ξ P ξ P ). ξ P ξ P ),

12 Herglotz nd Chrstoffel on dscontnuty wves 1 (**) η 1 Ω ( P η) η = Q, = 1,, 3. The coeffcent mtrx of the nhomogeneous lner system of equtons (**) grees wth the coeffcent mtrx of the homogeneous lner systems of equtons (*), nd s symmetrc. η = (η 1, η, η 3 ) s therefore ndeed soluton of (**), s long s one hs: ξ Q = 0 (orthogonlty condton). One sets: Ω Ω(P ξ) nd p 1 Ω, = 1,, 3. Ω P One thus obtns: ρωp ρ Ω = = c j ξ ξ j P, P nd furthermore, snce c j = c j : d ( ρω p ) = c ( j j j ) j ξξ P ξ ξ P ξ ξ P = c ( ξ P ξ j P ξ j P ) ξ = ρ ξ. j Together wth the orthogonlty condtons, ths gves: d d ( ρω p ) = Q ρ ξ p = 0. ( ) Ths s the desred dvergence relton. Chrstoffel [1] hs lredy dscussed the geometrc corollres of t concernng the ry tube (cf., supr). BIBLIOGRAPHY [1] Chrstoffel, E. B.: Über de Fortpflnzung von Stössen durch elstsche feste Körper, Ann. Mt. Pur Appl. () 8 (1877), [] Hdmrd, J.: Leçons sur l propgton des ondes et des equtons de l hydrodynmque. Hermnn, Prs [3] Truesdell, C., nd Toupn, R.: The clsscl feld theores. Flügge s Hndbuch der Phys, Sprnger, Berln-Göttngen-Hedelberg [4] Zemplén, G.: Krteren für de physlsche Bedeutung der unstetgen Lösungen der hydrodynmschen Bewegungsglechungen. Mth. Ann. 61 (3) (1905),