APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS. Classification of second order quasi linear partial. Solutions of one dimensional wave equation

Transcription

1 APPICATIONS OF PARTIA DIFFERENTIA EQUATIONS Cassifiaio of sod ordr quasi iar paria diffria quaios Souios of o dimsioa wav quaio O dimsioa ha quaio Sad sa souio of wo-dimsioa ha quaio Isuad dgs udd Fourir sris souios i Carsia oordias. I mahmais paria diffria quaios PDE ar a p of diffria quaio i.. a raio ivovig a ukow fuio or fuios of svra idpd variabs ad is or hir paria drivaivs wih rsp o hos variabs. Paria diffria quaios ar usd o formua ad hus aid h souio of probms ivovig fuios of svra variabs; suh as h propagaio of soud or ha rosais rodamis fuid fow ad asii. Smig disi phsia phoma ma hav idia mahmaia formuaios ad hus b govrd b h sam udrig dami.

2 I mahmais i h fid of diffria quaios a boudar vau probm is a diffria quaio oghr wih a s of addiioa rsrais ad h boudar odiios. A souio o a boudar vau probm is a souio o h diffria quaio whih aso saisfis h boudar odiios. Boudar vau probms aris i svra brahs of phsis as a phsia diffria quaio wi hav hm. Probms ivovig h wav quaio suh as h drmiaio of orma mods ar of sad as boudar vau probms. A arg ass of impora boudar vau probms ar h Surm-iouvi probms. Th aasis of hs probms ivovs h igfuios of a diffria opraor. To b usfu i appiaios a boudar vau probm shoud b w posd. This mas ha giv h ipu o h probm hr iss a uiqu souio whih dpds oiuous o h ipu. Muh horia work i h fid of paria diffria quaios is dvod o provig ha boudar vau probms arisig from siifi ad girig appiaios ar i fa w-posd. Amog h aris boudar vau probms o b sudid is h Dirih probm of fidig h harmoi fuios souios o apa's quaio; h souio was giv b h Dirih's priip.

3 Iiia vau probm A mor mahmaia wa o piur h diffr bw a iiia vau probm ad a boudar vau probm is ha a iiia vau probm has a of h odiios spifid a h sam vau of h idpd variab i h quaio ad ha vau is a h owr boudar of h domai hus h rm "iiia" vau. O h ohr had a boudar vau probm has odiios spifid a h rms of h idpd variab. For amp if h idpd variab is im ovr h domai [] a iiia vau probm woud spif a vau of ad ' a im whi a boudar vau probm woud spif vaus for a boh ad. If h probm is dpd o boh spa ad im h isad of spifig h vau of h probm a a giv poi for a im h daa oud b giv a a giv im for a spa. For amp h mpraur of a iro bar wih o d kp a absou zro ad h ohr d a h frzig poi of war woud b a boudar vau probm. Tps of boudar vau probms Th boudar vau probm for a idaisd D rod If h boudar givs a vau o h orma drivaiv of h probm h i is a Numa boudar odiio. For amp if hr is a har a o d of a iro rod h rg woud b addd a a osa ra bu h aua mpraur woud o b kow. If h boudar givs a vau o h probm h i is a Dirih boudar odiio. For amp if o d of a iro rod is hd a absou zro h h vau of h probm woud b kow a ha poi i spa. If h boudar has h form of a urv or surfa ha givs a vau o h orma drivaiv ad h probm isf h i is a Cauh boudar odiio. Asid from h boudar odiio boudar vau probms ar aso assifid aordig o h p of diffria opraor ivovd. For a ipi opraor o disusss ipi boudar vau probms. For a hprboi opraor o disusss hprboi boudar

4 vau probms. Ths agoris ar furhr subdividd io iar ad various oiar ps. Rad mahmais: Phsia appiaios: Numria agorihms: iiia vau probm diffria quaios Gr's fuios Sohasi prosss ad boudar vau probms Eamps of boudar vau probms Surm-iouvi hor Dirih boudar odiio Numa boudar odiio Sommrfd radiaio odiio Cauh boudar odiio wavs orma mods rosais apa's quaio poia hor shooig mhod dir muip shooig mhod CASSIFICATION OF PARTIA DIFFERENTIA EQUATIONS OF THE SECOND ORDER a sod ordr paria diffria quaio i h fuio u of h wo idpd variab b of h form. u u u A B C u u f u

5 Eq is assifid as ipi paraboi or hprboi a h pois of a giv rgio R dpdig o whhr B -AC < B -AC < B -AC < [ ipi quaio] [ Paraboi quaio] [ hprboi quaio] CASSIFY THE FOOWING EQUATIONS :- f f > >. So : - Hr A ; C B - A - - v wh > > Th quaio is ipi. f - f > > So :- Hr A; B - B A - v Th quaio is hprboi u -u u Hr A; B - B - AC - Th quaio is paraboi f f f > > Hr AB C 5

6 B -AC v Th quaio is ipi 5 f - f for - << << Hr A ; B ; C- B -AC is awas v i -<< - is gaiv i -<< B -AC - v Th quaio is ipi If B -AC Th quaio is paraboi. ONE DIMENSIONA WAVE EQUATION i Probms o Vibraig srig wih zro voi ii Probms o vibraig srig wih o zro voi O dimsioa wav quaio is 6

7 a Th various souios of h wav quaio is [ C p C -p ] [C pa C -pa ] [ C 5 osp C 6 sip] [ C 7 ospa C 8 sipa] [ C 9 C ] [ C C ] No: I probms of vibraio of srigs w awas ak h foowig as h orr souio; [ C osp C sip] [C ospa C sipa ] i Probms o Vibraig srig wih zro iiia voi A srig is srhd ad fasd o wo pois ad aspr. Moio is sard b dispaig h srig io h form k- from whih i is rasd a im o. Fid h dispam of a poi o h srig a a disa of from o d a im. So:- Hr w hav o sov h wav quaio a wih giv boudar odiios i Ths wo odiios ar fid odiios i wav Eq ii iii iiia voi is o giv i probm. So w ak i as. iv K iiia dispam is giv i probm i sf. Th orr souio of is 7

8 [C osp C sip] [C ospa C sipa] App odiio i i w g C [C ospa C sipa] >C ospa C sipa C Sub C i w g h souio C sip [ C ospa C sipa] App odiio ii i w g C sip [C ospa C sipa] > C ad C ospa C sipa sip i. p si - i. p i. p Sub p i w g h souio a C si [ C os C a si 8

9 9 `P.Diff w.r.o w g App odiio iii i I w g Sub C i w g h souio Bu suprposiio priip w g h souio App odiio iv i 5 w g This show ha his is h haf rag F.S.sris of K-. Usig Fourir offii formua a a C a a C C os si si si & & si si C aso a C a C C a C C os si ] os si ] C C whrc a C a C C a C os si 5 Si b K Si C I

10 b V si u V -os u V - si u V os C K os - -si os Sub C i 5 w g whih is h rquird so d k si C K d K si os K [ ] K K a K os si

11 A igh srhd srig wih fid d pois ad is iiia i a posiio giv b si p/.if i is rasd from rs from his posiio. Fid h dispam a a disa from d a a im. So : - Hr w hav o sov h wav quaio a Wih boudar odiios i ii iii iv Th orr souio of is si si si osp sip ospa sipa App odiio i i w g [ ospa sipa] > ospa sipa sub i w g sip ospa sipa App odiio ii i w g sip ospa sipa > o ad ospa sipa sip i. p

12 sub p i w g Paia diff w.r. w g a a a a os si si App odiio iii i w g B suprposiio priip w g App odiio iv i 5 w g p si a a si os os si os si si si si whr a a wg i sub ad a a a a > a os si o si si si 5

13 o si si si si o si si... si si Equaig h offi w g ; ; ; 5... Subsiu hs vaus i 5 w g o a o a si os si os whih is h rquird so A srig is igh srhd ad is ds ar fasd a wo pois ad. Th mid poi of h srig is dispad rasvrs hrough a sma disa b ad h srig is rasd from rs i ha posiio. Fid a prssio for h rasvrs dispam of h srig a a im durig h subsqu moio. So Hr firs w hav o fid iiia dispam i[o ] of h srig i giv probm. D /b b / / A C B

14 Now w fid Equaio of h srig i is iiia posiio ADB Th quaio of h srigor i AD is. Th quaio of h srig or i DB is / / / b b b < < [ ] b b b b b b b b b b b b b b b b b b b b b < <

15 H iiia h dispam of h srig is i h form. b X b / o < < < < / Now w hav o sov h wav quaio a wih boudar odiios i ii Th orr souio of is iii b iv < < / b / < < os p si p os pa si pa App odiio i i w g [ ospa sipa] > ospa sipa Sub i w g si p[ ospa si pa] 5

16 6 App odiio ii i w g Y sip[ ospa ospa] > ad ospa sipa sip i p Sub Paria diff w.r.o w g si App odiio iii i w g si si si > ad a a a p wg i p si a a si os a a a a os si

17 7 Sub i w g C C si a os C si os whr a B supr posiio priip w g App odiio iv i 5 w g whr b a os si < < < < b b b si / si / / si si d d b si si d b d b 5

18 8 / / si os si os X X X b u vsi u- u v -os u - v - si Sub i 5 w g whih is h rquird so A au srig of gh has is ds fid. Th poi whr / is draw asid a sma disa h h dispam saisfis a.drmi a a im. si os si os b si b si 8 b os si si 8 a b

19 So Hr firs w hav o fid B/h h iiia dispam of h srig i h i. o fid q of OBA / Eq. of h srig or i OB is A - - -/ -h i. - - /h i. h i <</ Eq of h srig or i BA is -/ -h / h- -h - h - -h h- - h- h - h- h h- h - i / << Th iiia dispam of h srig is h <</ h- / << 9

20 Now w hav o sov h wav q a wih boudar odiios i ii iii iv h <</ h- /<< h prod his probm sov as prvious probm. 5 Th pois of risio of a srig ar pad asid hrough a disa b o opposi sids of h posiio of quiibrium ad h srig is rasd from rs. Fid h dispam of h srig a a subsqu im. So : - Hr firs w hav o fid a iiia dispam of h srig.

21 B /a a B A -a C /-a Hr B & C b h pois of h risio of h srig OA. Iiia h srig is hd i h form OB C A whr BB CC a Eq of OB is - - -/ -a > a i / Eq of B is -/ /-/ -a aa > -/ -/ -a a > a- - -a > a- a > a-6aa

22 a-6a a- i // Eq of C A is -/ a / -a - a -a -a- - a a- - a a-a-a a-a a- i / H hr iiia dispam of h srig is a </ a- / </ a- / </ Now w hav o sov h wav q a

23 wih boudar odiios i ii iii iv - a </ a- / </ a- / </ h prod his probm sam as prvious probms Probms o vibraig srig wih o-zro voi A igh srhd srig wih fid d pois ad is iiia a rs i is quiibrium posiio ad ah of is pois is giv h voi. Drmi h dispam fuio V si << So : - Hr w hav o sov h wav quaio a wih boudar odiios i ii iii iv V si V / si -/ si Th orr so of is [C osp C sip ] [C ospa C sipa]

24 App odiio i i w g [ ospa _ sipa] > ospa C sipa Sub i w g [ sip [ Cospa sipa] App odiio ii i w g sip [ ospa sipa] sip i. psi - p p sub p i w g si os a si a App odiio i w g si si & si

25 5 Sub i w g si si a si si a si si a B suprposiio priip w g Paria w.r.o ; w g App odiio iv w g Equa h offiis w g C si si a 5 a a os si V a si si si > V V a a a si si... si si si V a a V o V a

26 Vo a & 5.. Sub hs vaus of s i 5 w g V a Vo a si si si si a a A srig is srhd bw wo fid pois a a disa apar & h pois of h srig ar giv iiia voiis V whr V is << / - i < < big h disa from a d poi.. fid h dispam of h srig as a im Souio:- Hr w hav o sov h wav quaio a Whr for ovi Wih boudar odiios i ii iii iv <</ 6

27 - / < < Th orr So of is osp sip ospa sipa App odiio i is w g ospa sipa > ospa sipa sub i w g sip ospa sipa App odiio ii i w g sip ospa sipa > ospa sipa & sip psi - p p Sub p i w g 7

28 si os a si a App odiio iii i w g si > si > & si sub i w g si si a si si a B suprposiio priip w g si si a 5 Paria w. r. o w g a si si a App odiio iv w g 8

29 9 <</ a si b si - /<< / whr a b si d - si d / u v si u- u v -os u - v -si / si si d d / si os si os

30 > Sub i 5 w g Rpa b w g si os. si os si si 8 si 8 si 8 a a C 8 si / a si si si 8 a a si si si 6 a a

31 whih is h rquird so Fid h dispam of a igh srhd srig of gh 7 ms vibraig bw fid d pois if iiia dispam is si So ad iiia voi is 5 si Hr w hav o sov h wav quaio u u Hr 7 m wih boudar odiios w g i ii hr boh iiia voi iii si & iiia dispam ar giv. iv 5 si Th orr so of is [ osp sip] [ ospa sipa] App odiio i i w g [ ospa sipa]

32 > ospa sipa Sub i w g App odiio ii i w g sip [ ospa sipa] > ospa sipa & sip p si - p Sub p i w g sip ospa sipa p a a si os si > > a d a a a si si os si si si os si

33 B supr posiio priip w g si osa d si si a sia d si a si App odiio iii i. w g si si N si si si. si Equa ik offiis w g &. A From w g a -sia d aos a si App odiio iv w g d a si 5 si 9 d 9 9a 5 [Equa ik offiis] d 9 5 & d d.d 8 d B 9a sub A & Bi w g

34 a 5 9a 9 si si si si 9a whr 7 ONE DIMENSIONA HEAT FOW EQUATION O dimsioa ha quaio is u α u Th various souios of h ha quaios is u [ ] p p [ ] α p u 5 [ os p si p] α p u No Th orr so of o dimsioa ha fow q as u α p [ Aos p B si p] Probms. Sov h quaio u α u subj o h boudar odiios u u u. So:- Hr w hav o sov h ha fow quaio u α u

35 wih giv boudar odiios i u ii u iii u Th orr so of is u A osp B sip α p App odiio i i w g u A α p > α p A Sub A i w g u B sip α p App odiio ii is w g u Bsip α p > B α p > sip p p 5

36 Sub p i w g u Bsi I gra u α α B si B suprposiio priip w g u B si α App odiio iii is w g u B si b si - whr B b / si u v si u v - os v si B - os si / / / -- / 6

37 B - Sub B i w g α u - si - STEADY STATE CONDITIONS AND ZERO BOUNDARY CONDITIONS Probms: A rod of gh has is ds A ad B kp a of ui sad sa odiio prvai. if h mpraur a B is rdud sudd o & kp so whi ha of A is maiaid fid h mpraur u a a disa from A ad im. SOUTION: Sad sa A B I giv probm Wh sad sa prvais h d A is a ad h d B is a Now w hav o fid h mpraur wh sad sa odiio prvais i. o fid u bfor h d B is rdud o H wh sad sa odiios prvais h ha fow quaio boms u > d u d 7

38 > uab wih boudar odiios i u ii u App i i w g u b > b Sub b is w g u a App ii i w g u a >a/ Sub a is w g u I sad sa h mpraur disribuio is u Now h mpraur disribuio rahd a h sad sa boms h iiia disribuio for h usad sa i u is << 8

39 Usad sa A B h d A is a C & B is a C H wh usad sa odiio prvais u u wih boudar odiios i u ii u iii u i << Th orr so of 5 is u A osp Bsip α p 6 App odiio i is 6 w g u A α p > α p A Sub A is 6 w g u Bsip α p 7 9

40 App odiio ii is 7 w g u B sip α p > B & sip p p Sub p i u Bsi I gra 7 α p w g α u B si α B suprposiio priip w g u α B si 8 App odiio iii i 8 w g u B si b si Whr B b / si d u u v si v -os /

41 v si B os si B - Sub B i 8 w g u α si STEDY STATE CONDITIONS AND NON-ZERO BOUNDARY CONDITIONS Probms : A bar m og wih isuad sids has is ds A&B kp a C ad C rspiv ui sad sa odiios prvai. Th mpraur a A is h sudd raisd o 5 C & a h somim ha a B is rdud a C.Fid h mpraur a a poi of h bar a a im So: - I sad sa odiio h ha fow quaio boms Sad sa d u d m uab A B C C wih boudar odiios i u ii u App i i w g

42 ub > b Sub b i w g ua App ii i w g ua >a >a sub a i w g u I sad sa h mpraur disribuio is u Now h mpraur a A & B ar hagd A his sag h sad sa is hagd io usad sa. From his sag h iiia mpraur disribuio is u i << H wh usad sa odiio prvais h ha fow quaio is u α u 5 wih boudar odiios i u5 m ii u5 A B iiiu 5 Th orr souios of 5 is uaosp Bsip α p 6

43 App i i 6 w g ua α p 5 7 App ii i 6 w g uaosp Bsip α p 8 From & i is o possib o fid h osas A & B. 7 8 I his as w spi h souio u io wo pars. i. uu s u 9 whr u s is a souio of h u α u ad is a fuio of ao ad saisfig h odiios u s 5 & u s ad u is a rasi souio saisfig To fid u s u s a b 9 whih drass as irass. wih boudar odiios i u s 5 ii u s App i is u s b 5 w g > b 5 Sub b i w g u s a 5

44 App ii i w g u s a 5 a 5 a -5 a - sub a i w g u s - 5 To fid u > u u-u s 9 is a rasi souio of u α u Now w hav o fid h boudar odiios for u i u u-u s 5-5 ii u u-u s - iii u u-u s Th orr souio of is u Aosp Bsip α p 5 App odiio i w g u A α p > α P A Sub A i 5 w g u B p α p si 6

45 App ii i 6 w g u Bsi p α P > B & si p p p α P Sub p i 6 wg u B si I gra u B si α α B suprposiio priips w g u B si α 7 App odiio iii i 7 w g u B si 6 si d 5 b si WhrBu- b 6 si v si d u v os 5

46 v si B 5 os si 5 5 X B -6 - Sub B i 7 w g u 6 si α Bu u u s u > u 5 6 si α whih is h rquird so :- Thrma isuad ds: - If hr wi b o ha fow passs hrough h ds of h bar h ha wo ds ar said o b hrma isuad. No : If h d sa is hrma isuad h w hav 6

47 Probms : ad if h d sa is hrma isuad h w hav Th mpraur a o d of a bar 5 m og wih isuad sids is kp a C ad ha a h ohr d is kp a C ui sad sa odiios prvai. Th wo ds ar h sudd isuad so ha h mpraur gradi is zro a ah d hrafr. Fid h mpraur disribuio. So :- A 5m B Wh sad sa odiio prvais C C Th ha fow quaio boms d u d > u ab wih boudar odiios i u ii u5 App i i w g u b > b sub b i i w g u a App ii i w g u5 a[5] a 7

48 > Sub a i w g u B I sad sa h mpraur disribuio u Now h mpraur a A & B ar hagd. A his sag h sad sa is hagd io usad sa. From his sag h iiia mpraur disribuio is u Whr usad sa odiios prvais h ha fow quaio is u α u 5 wih boudar odiios i u h wo ds ar hrma isuad ii u5 iii u <<5 Th orr souio of 5 is u A os p B si p α p 6 Diff. Paraia w. r.o. w g u α p [ Ap si p Bp os p] App odiio i w g 8

49 > α p & p u α p Bp & p Sub B i 6 w g u A os p α p 7 Paria diff w.r.o w g u α p A si p App odiio ii w g u5 Asi 5 p α p si 5P α p > A & > 5 p p 5 Sub p 5 i 7 w g u Aos 5 α 5 B suprposiio priip w g 9

50 u A os 5 α 5 8 App odiio iii w g u A os 5 A A os 5 a a os 5 Now Whr A a & A a a d 5 5 a A 5 a os os 5 5 d d u v os 5 si 5 5 5

51 u v v os si 5 5 os X 5 [ ] a A Sub A & A i 8 w g u 5 α 5 os 5 whih is h rquird so TRY YOURSEF: 9Th mpraur a o d of a bar m og & wih isuad sids is kp a C & ha a h ohr d is kp a 6 C ui sad sa odiios prvai. Th wo ds ar h sudd isuad so ha h mpraur is C a ah d hr afr. Fid h mpraur disribuio of h bar. 5

52 A isuad ma rod of gh m has o d A kp a C & h ohr d B a C ui sad sa odiios prvai. A im h d B is sudd isuad whi h mpraur a A is maiaid a C. Fid h mpraur a poi of h rod a a subsqu im. So Wh sad sa odiios prvai i h rod h mp disribuio is giv b d d h orrspodig boudar odiios ar u u soig w g u usig & i w g & u 5 O d B is isuad hough h mpraur a A is o ard h ha fow udr usad sa odiios & h subsqu mpraur disribuio is giv b u u α 6 wih boudar odiios i u for a o ii u for a iii u for o<< Th orr souio is u Aos p Bsi p α p 5

53 App odiio i i 7 w g > p u A α α p A sub A i 7 w g u Bsi p α p 8 Paria diff 8 w.r.o w g App odiio u u pb os p pb os p α p α p ii 8 w g α p > p ; ; B osp > p os - a odd muip of / or - / p whr... sub p i 8 w g u B si α 5

54 h mos gra souio is u B si α 9 App odiio iii i 9 w g u B si This is of h Fourir fi sris form whrb b si d si os si 8 B 8 Sub i 9 w g 5

55 8 u si α Whih is h rquird souio PROBEMS WITH NON ZERO BOUNDARY VAUES & STEADY STATE CONDITIONS. A bar m og wih isuad sids has is ds A & B kp a C & C rsp. ui sad sa. Codiios prvai. Th mpraur a A is h sudd raisd o 5 C & a h sam im ha a B is rdud a C.Fid h mpraur a a poi of h bar a a im. C C So I sad sa odiios h ha fow quaio boms d d A B 5 C C > u a b wih boudar odiios i uo ii u App i i w g u b > b sub b i w g 55

56 u a App ii i w g u a >a > a Sub a i w g u I sad sa h mpraur disribuio is u Now h mpraur a A & B ar hagd. A his sag h sad sa is hagd io usad sa. From his sag h iiia mpraur disribuio is u i < < H wh usad sa odiio prvais h ha fow quaio is u u α 5 m wih boudar odiios A B i u 5 5 C C ii u iii u Th orr so of 5 is u α p Aos p Bsi p 6 56

57 App i i 6 w g u o A App ii i 6 w g α p 7 5 u Aos p Bsi p α p 8 From 7 & 8 i is o possib o fid h osas A & B. I his as w spi h so u io wo pars. i. u u s u 9 Whr u s is a so of h q u u α & is a fuio of ao u [ i d u i d ] & saisfig h odiios u s 5 & u s ad u is a rasi souio saisfig 9 whih drass as irass To fid u s u s a b wih boudar odiios i u s 5 ii u s App i i w g 57

58 u s b 5 > b 5 sub b i w g u s a 5 App ii i w g u s a 5 a 5 a -5 a - sub a i w g u s - 5 To fid u 9 > u u u s u u is a rasi souio of α Now w hav o fid h boudar odiios for u i u u u s 5-5 ii u u u s - iii u u u s Th orr so of is u α p Aos p B si p 5 App odiios i w g 58

59 u o A α p > α p A Sub A i 5 w g u Bsi p α p 6 App ii i 6 w g u B si p α p > B & α p si p p Sub p p i 6 wg u Bsi u B si α α I gra B suprposiio priip w g u α B si 6 59

60 App odiios iii i 6 w g u B si 6 b si whrb b 6 si d si d u - v si u v v os si B os si B 6 Sub B i 6 w g u 6 si α 6

61 Bu u u s u > u 5 6 si α whih is h rquird so TWO DIMENSIONA HEAT FOW EQUATION Th wo dimsioa ha fow quaio or apa quaio is u u α Th various souios of quaio is i u p p os p si p ii u 5 os p 6 si p 7 p 8 p iii u 9 Probms A squar pa is boudd b h is o &. Is fas isuad. Th mpraur aog h uppr horizoa dg is giv b u -wh << whi h ohr hr dgs ar kp a C. Fid h sad sa mpraur i h pa. So 6

62 us ak h sids of h pa b for ovi Th mpraur disribuio is giv b u u Y wih boudar odiios i u u- ii u iii u C C iv u - << Th orr souio of q is X u osp sip p -p App odiios i i w g u p -p > p -p Sub i w g u sip p -p App odiio ii i w g u sip p -p > & p -p sip p si - p p 6

63 6 sub p i w g App odiio iii i w g > - Sub - i w g Th mos gra souio is App odiio iv i 6 w g u ρ si si u & si > u si si sih si sih si u sih si sih si u 6

64 6 whih is of h form of F.S. sris whr i. u - u - u - sih si f u si b f d b si sih d si sih os si os si v v v v [ ] os si os sih sih

65 os h Sub i 6 w g u osh sih si Rpa b w g u osh sih si u 6 osh sih si Fid h sad sa mpraur a a poi of a squar pa whos wo adja dgs ar kp a C & h ohr wo dgs ar kp a h osa mp C. So h sid of h squar pa b Th mpraur u is giv b u u wih boudar odiios Y C C C i u for << ii u for << iii u for << iv u for << u u u whr u & u ar saisfig h foowig boudar odiios. 65

66 a u a u b u b u u u d u d u To fid u Th orr souio is u osp sip p -p App odiio a i w g u p -p > p -p Sub i w g u sip p -p App odiio b i w g u sip p -p Sub p i w g > & p -p sip p si - p u si App odiio i w g 66

67 67 > & si > - Sub - i w g Th mos gra souio is App odiio d i 6 w g whih is of h form of Fourir si sris f b si whr si o u u si si sih si sih si sih si sih si u 6 sih si u d b si sih

68 os sih > Sub i 6 w g osh u osh sih sih To fid u Th orr souio is u osp sip p -p Appig h boudar odiios a b & d w g u osh sih sih Th souio of is u u u 68

69 osh sih sih si sih Tr oursf: A raguar pa is boudd b h is a & b h dg mpraur ar u ua ub u si a fid h mpraur disribuio. A raguar pa is boudd b h is a & b & h mpraurs a h dgs ar giv b. u i <<b/ b- i b/ <<b ; ua ; ub & pa. u 5si si a a. Fid h sad sa mpraur disribuio isid h Ifii pas A ifii og raguar pa has is surfas isuad & h wo sids as w as o of h shor sids ar maiaid a C. fid a prssio for h sad sa mp u if h shor sid is m og & is kp a u C. So :- Th mpraur disribuio is giv b u u wih boudar odiios i u ii u iii u iv uo u Th orr so is u osp sip p -p C C C u C 69

70 App odiio i i w g u p -p Hr p -p Sub i w g u sip p -p App odiio ii i w g u sip p -p Hr p -p & sip p si - p > p sub p i w g u si - App odiio iii i w g u si > ; si & Sub i w g u si - u - si u - si 7

71 7 Th mos gra souio is App odiio iv i 5 w g This is of h form of Fourir si sris whr Sub i 5 w g u si 5 si u o u b f si d f b si u si d u u u o o o os u u si u u si

72 A ifii og raguar pa wih isuad surfa is m.th wo og dgs & o shor dg ar kp a C. whi h ohr dg is kp a u for < < 5 - for 5<< Fid h sad sa mpraur disribuio i h pa. So Th mpraur disribuio is giv b u u wih boudar odiios C i u ii u iii u iv u < < / - / << for h ovi f O C C Th orr souio is u p -p osp sip App odiio i i w g uo p -p > sub i w g u p -p sip 7

73 7 App odiio ii i w g u p -p sip Hr sip i p > Sub p i w g App odiio iii i w g u - si > Sub i w g Th mos gra so is p u si si & si u si si si

74 7 App odiio iv i 5 w g whr f < </ - /< < This is form of h F.S.Sris u si 5 si f u si b f d f b si } si os si os si os si os si si ` d d si

75 8 si 6 Sub 6 i 5 w g u 8 si si Rpaig b w g whih is h rquird souio. si 8 u si Objiv Qusios Th suiab souio of ODWE is a A p B -p C pa D -pa b A p B -p C os pa D si pa A os pa B si pa C os pa D si pa d A B CD I o dimsioa wav quaio u a u a sads for a b d No of hs [T-Tsiom-Mass pr ui gh] I ODWE mass of h srig pr ui gh is a a b Cosa dvariab Amog h hr possib souio i ODWE h suiab souio is ak baus a Dispam is priodi i aur b I oais rigoomri rms Boh a & b d No of hs 75

76 5 Whih boudar odiio i ODWE shoud b ak as o ad usd afr fidig mos gra souio a. boudar odiio wih zro vau b. boudar odiio wih o zro vau. boh a ad b d. o of hs 6 Th sad sa mpraur of a rod of gh whos ds ar kp a ad is b d 7 Th gra souio of vibraor moio of a srig of gh wih fid d pois ad zro iiia voi is a b si л/ si лa/ b b os л/ os лa/ b os л/ si лa/ d b si л/ os лa/ 8 Two dimsioa sad sa ha oduio quaio is a U U bu a U U a U d No of hs. 9 Th quai of ha rquird o produ a giv mpraur hag i a bod is proporioa o h mass of h bod ad o h Chag atmpraur bdisa Tim dno of hs I o dimsioa ha fow quaio if h mpraur fuio u is idpd of im h h souio is acosa bsad Usad dno of Ths O of h possib souio o wo dimsioa ha fow quaio i Carsia ssm of oordias is au C C C C b u C C C C u C C C C dno of hs I wo dimsioa ha fow h ra of ha fow aross a ara is proporioa o a Ara ad h mpraur gradi para o h ara b Ara ad h mpraur orma o h ara Ara ad h mpraur gradi orma o h ara d Ara ad h mpraur para o h ara I D ha fow h mpraur a a poi is idpd of a Tim b X-Coordia Y-Coordia d Z-Coordia 76

77 I o dimsioa ha fow quaio u α u α sads for a Diffusivi of h maria b Thrma oduivi of h maria Spifi ha of h maria d Dsi of h maria 5 Wh h ds of a rod i o zro for o dimsioa ha fow quaio. Th mpraur fuio u is modifid as h sum of h sad sa ad rasi sa mpraur. Th rasi par of h souio whih a Iras wih iras of im b Dras wih iras of im Dras wih dras of im d Iras wih iras of im 6Th O Dimsioa Wav Equaio is au U bu a U U a U dno of hs. 7Th O Dimsioa Ha Equaio is au U bu a U U a U dno of hs. 8A Sod ordr Paria Diffria quaio is said o b ipi if ab -AC< bb -AC B -AC> dno of hs 9 A Sod ordr Paria Diffria quaio is said o b hprboi if ab -AC< bb -AC B -AC> dno of hs A Sod ordr Paria Diffria quaio is said o b paraboi if ab -AC< bb -AC B -AC> dno of hs I drivig O Dimsioa Wav Equaio w assum ha moio aks pa ir i pa. ao bwo hr dno of hs I drivig O Dimsioa Wav Equaio w assum ha h Tsio T is a a pois & a a im of h dfd srig. acosa bvaris Zro do of hs. I drivig O Dimsioa Ha Equaio w assum ha ha fows from o mpraur. ahigh o ow bow o high high o high dno of hs. W rquir umbr of boudar odiios o sov O Dimsioa Ha Equaio a b d 5 W rquir umbr of boudar odiios o sov O Dimsioa wav Equaio a b d 77

78 6Th suiab souio of h O Dimsioa Ha Equaio is auaos p B si p b UAos p B si p UAos p B si p dno of hs. PART-A. Cassif h paria diffria quaio z p z z z z si :. Wri dow a possib souios of o dimsioa wav quaio.. A au srig of gh 5 m fasd a boh ds is disurbd from is posiio of quiibrium b imparig o ah of is pois a iiia voi of magiud k for < < 5. Formua h probm mahmaia..wri dow a possib souios of h o dimsioa ha ow quaio. 5. If h mpraur a o d of a bar 5 m og ad wih isuad sids is kp a ±C ad ha h ohr d is kp a ±C ui sad sa odiios prvai d h sad sa mpraur i h rod. PART-B. A au srig of gh is fasd a boh ds. Th midpoi of his srig is ak o a high of b ad h rasd from rs i his posiio. Fid h dispam of h srig a a im.. A au srig of gh is fasd a boh ds. Th midpoi of h srig is ak o a high b ad h rasd from rs i ha posiio. Fid h dispam of a poi of h srig a a subsqu im.. A igh srhd srig wih fid d pois ad is iiia a rs i is quiibrium posiio. If i is s vibraig b givig ah poi a voi λ. fid h dispam of h srig a a disa from o d a a im.. Th d A ad B of a rod m og hav h mpraurs ad 9 a ui sad sa prvais. Th mpraur a A is sudd raisd o 9 ad a h sam im ha a B is owrd o.fid h mpraur disribuio i h rod a im.aso show ha h 78

79 mpraur a h mid poi of h rod rmais uard for a im rgardss of h marias of h rod. 5. A ma bar m og wih isuad sids has is ds A ad B kp a o ad o rspiv ui sad sa odiios prvai. Th mpraur a A is h sudd raisd o 5 o ad a h sam isa ha a B is owrd o o. Fid h subsqu mpraur a a poi a h bar a a im. 6. A raguar pa wih isuad surfa is m wid ad so og ompard o is widh ha i ma b osidrd ifii gh. If h mpraur aog shor dg is giv b Π u 8 si whr <<whi h wo og dgs ad as w as h ohr shor dg ar kp a.fid h sad sa mpraur fuio u. 79