Lecture 10: Wave equation, solution by spherical means

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1 Lecure : Wave equaion, soluion by spherical means Physical modeling eample: Elasodynamics u (; ) displacemen vecor in elasic body occupying a domain U R n, U, The posiion of he maerial poin siing a U in he undeformed sae is + u (; ) a ime. " (; ) srain ensor " ij (; ) ui + u j j i u he symmeric par of he velociy gradien Du i j, an n n mari We consider linear elasiciy so displacemens are supposed o be "in niesimal". (; ) sress ensor F (; ; ) racion, i.e., force per area on an area elemen vds a a ime ( uni normal vecor for he area elemen) Cauchy s sress principle: F (; ; ) (; ) Consiuive relaion (Hooke s law) for isoropic incompressile elasic medium (e.g., biological issue on shor imescales) ij (; ) " ij where is called a Lamé consan. Densiy Toal momenum vecor of a subdomain of he elasic body: m () u (; ) d

2 Force on he subdomain : F i () [Gauss heorem] X " ij j ds j " ij X ds j j X j X j u i B j + u j ds i {z j C } A i u j j u i ds since X j j u j j so F i () u i ds Newon s law dm () d F () gives u i (; ) u i d valid for all smooh subvolumes, i.e., u i (; ) u i i.e., each componen u i of he displacemen vecor sais es he wave equaion r u i c u i wih he speed of sound c Noe: speed of sound larger for si er maerial (larger ) or ligher maerial (smaller ). Boundary condiions: Clamped boundaries: u Free boundaries: v " F, given racion force (force per area) on he boundary wih uni normal vecor. Eample: A uni square clamped a he boom and wih a ime dependen uniform racion force applied on he lef verical side.

3 Domain U [; ] [; ] PDE Boundary condiion on boom side Srain ensor " u c u u c u u ( ; ; ), [; ], > " 4 u u + u (; ; ) u (; ; ) + u (; ; ) u + u u Boundary condiion on lef side, uni normal ( racion force F () " u # 3 5 ; ), given uniform F (), F () [; ], > Boundary condiion on righ side, uni normal (; ), racion force " u # " (; ; ) u (; ; ) + u, (; ; ) [; ], > Boundary condiion on op, uni normal (; ), racion force " # u " ( ; ; ) + u ( ; ; ), u ( ; ; ) [; ], > Iniial condiions: iniial displacemen and velociy are zero u ( ; ; ), ; [; ] u ( ; ; ), ; [; ] We ge a sysem of wo scalar wave equaions, coupled hrough he boundary condiions. 3

4 Soluion by spherical means D wave equaion on R for given g, h. Noe ha Inroducing coordinaes u u in R (; ) and u (; ) v (y; z) such ha which gives u g, u h on R f g y z + {z } {z } y z y + z + v yz and hence inegraing in wo seps v y (y; z) ~ (y) v (y; z) (z) + (y) so Iniial condiions give u (; ) ( ) + ( + ) () + () g () () + () h () Di ereniaing he rs equaion, and adding and subracing he equaion, respecively, gives () g () + h () () g () h () 4

5 which gives for > g ( + ) g () ( + ) () + + g () g ( ) () ( ) + + h () d h () d which gives d Alembers formula (using () + () g () o cancel g ()) u (; ) ( ) + ( + ) g ( + ) + g ( ) + + h () d Re ecion mehod The D wave equaion on he posiive hal ine wih homogeneous boundary condiions can be solved by d Alembers formula and odd re ecion. For <, we have he original d Alembers formula, using only daa on he posiive hal ine u (; ) (g ( + ) + g ( )) + + h () d, < < bu for >, he lefgoing wave has reached he lef boundary and re eced here o a righgoing wave, so he soluion is hen u (; ) (g ( + ) g ( + )) + + Soluion wih spherical means Consider he wave equaion in R n, n, m u C m (R n [; )) u u in R n (; ) + u g, u h on R n f g h () d, < < Laplacian is roaionally symmeric. Consider PDE for spherical mean (symmerizaion) wih respec o, of soluion U (; r; ) u (y; ) ds (y) and symmerized iniial condiions G (; r) H (; r) B(;r) B(;r) B(;r) 5 g (y) ds (y) h (y) ds (y)

6 Lemma The symmerized soluion of he wave equaion in R n sais es he Euler Poisson Darbou equaion U C m R+ [; ) n U U rr U r r U G, U H on R + f g Soluion mehod Transform he Euler Poisson Darbou equaion o he D wave equaion and use d Alembers formula! Soluion for n 3, Kirchho s formula The ransformaion ~U ru, ~ G rg, ~ H rh does he rick, ~U Urr ~ in R + (; ) ~U G, ~ U ~ H ~ on R + f g ~U on fr g (; ) d Alembers formula gives an eplici formula for ~ U (; r; ), and nally we ge u (; ) by leing r &, which gives u (; ) u (; ) lim r& ~ U (; r; ) r gds + hds B(;) B(;) Evaluaing he ime derivaive we ge Kirchho s formula u (; ) h (y) + Dg (y) (y ) ds (y), R 3, > B(;) Noe ha he values of u a he spaial poin a ime depends only on he iniial daa on B (; ), i.e., daa on he "ligh cone". Soluion for n, he mehod of descen, Poisson s formula No corresponding ransformaion eiss for n. Insead consider u as a funcion in R 3 [; ) independen of he hird spaial coordinae 3 and 6

7 use Kirchho s formula. Paramerize he sphere means B (; ) R 3 over he projeced disc B (; ) R, viz., f (y) ds (y) f y; ; (y) + jd ; (y) j dy B(;) B(;) + f y; ; (y) + jd ; (y) j dy where and B(;) ; (y) jy j + jd ; (y) j ; (y) Subiuing in he soluion formula for R 3 we ge u (; )! g (y) dy + B(;) ; (y) B(;) h (y) ; (y) dy and subsiuing in Kirchho s formula, we ge Poisson s formula for he soluion of he wave equaion in R, u (; ) g (y) + h (y) + Dg (y) (y ) dy, R, > B(;) ( jy j ) Noe ha in his case, he values of u a he spaial poin a he ime depends on he iniial daa on he full disc B (; ), no only he values a he boundary as in he hree dimensional case. Talking o each oher in a planar world is di cul! 7