MATH Midterm Examination Victor Matveev October 26, 2016

Transcription

1 MATH 33- Midterm Examiati Victr Matveev Octber 6, 6. (5pts, mi) Suppse f(x) equals si x the iterval < x < (=), ad is a eve peridic extesi f this fucti t the rest f the real lie. Fid the csie series fr this eve peridic extesi. Make a rugh sketch f this peridic fucti, as well as the sum f the first tw -zer Furier terms (i.e. graph the sum f the cstat plus the ext -zer term), fr <x<. si cs si si Hit: cs A = f x dx= si x dx cs x x A = f xcs dx=sixcsx dx si xsi xdx si x Nte: if =, bth itegrals are zer cs x csx cs csx, dd 4, eve First tw -zer terms are: cs x... cs x f x A A Sketch des t require exact result: just sketch the mst bvius peridic apprximati f f : x

2 . (pts, 4mi) Csider a rd/cable f legth = with cstat thermal prperties (c =, K =): x ut x t u x t x t ux (, t) u, t ux, t, xx, cs (, ) a) Determie the equilibrium temperature distributi x x 4 cs cs dx dx x x si c u x4cs c xc dx Nw use budary cditis: eq u ' eq ueq c c c c x ueq x x u' c c eq 4cs 4 b) Make a rugh sketch f the equilibrium temperature distributi (hit: yu d t have t slve part a t prduce a rugh sketch) ueq(x) ueq ()> ()< ueq ()= ()= x= x c) Explai the equilibrium: where des the eergy eter the rd, ad where des it leave the rd? Eergy eters frm the surce heatig term Qx cs x /, which is psitive at all pits, but decreases frm left t right. Eergy leaves thrugh the left ed ly, which beys Newt s law f clig (clig ut-flw is prprtial t the temperature). The right ed is sealed: heat escapes.

3 3. (5pts, mi) Separate variables i the fllwig differetial equati, ad simplify the tw equatis yu btai (i.e. get rid f ay qutiets i yur equatis). D t slve the equatis! u u u r r r t r r r t () rgt () urt (, ) ( rgt ) ( ) r ( rgt ) ( ) r r ( rgt ) ( ) t r r r t d g dg g t d d r() r r() r r dt dt r dr dr ( rg) d g dg d d 3 r g() t dt dt rr dr dr d g dg gt () dt dt d g dg gt () d d dt dt 3 r () r r dr dr d d r r ( r) dr dr r d d r dr dr

4 4. (3pts, 3mi) Use separati f variables t slve the fllwig wave equati prblem. Yu d t have t prvide mre detail tha yu eed, but make sure t idicate briefly all the mai steps. Whe separatig variables, recall that yur sluti will be simpler if yu mve the cstat c away frm the budary value part f the prblem. Make sure t check yur sluti agaist the give equati, the budary cditis, ad the iitial cditis. u u c ( x, t ) t x u(, t) u(, t) u ux (,) ; ( x,) t dh d dh d ( x) c h() t c ( x) h() t dt dx c h() t dt ( x) dx '' () ( ) h c h h() '' : ( x) Csi x, : ( x) (trivial slutis) : ( x) (trivial slutis) : h ( ) si x C c t i The full sluti is uxt (, ) Csi xs t where c c Fid Furier cefficiets frm the iitial velcity ("kick"): u ( x, ) C si x t Furier cefficiet, eve x x c C si dx cs 4 where c, dd m Odd, 3,5,... m m x c m t si si si si 4 x t 4 uxt (, ) c m

5 5. (pts, 6mi) Aswer e f the fllwig tw questis (i.e. cmplete part a r part b ): e a) I the eergy cservati equati, write dw the physical uits f e ad t x E : eergy desity: eergy per uit vlume : eergy flux: eergy per uit area per uit time b) I the wave equati f a strig, utt Tuxx, write dw the physical uits f ρ ad T T : strig tesi (frce): mass accelerati = mass legth / time : liear mass desity: mass per uit legth 6. (pts, 6mi) Which f the fllwig fucti(s) d/des t agree with the aplace s equati, u xx + u yy =? Explai very briefly (Hit: examie the curvatures with respect t x ad y) Third plt (right pael) bviusly disagrees: equilibrium temperature ca t have a lcal max (maximum priciple): heat will always flw t lwer temperature util there s maximum. First plt (left pael) als disagrees: ear the peak, curvature is dwward i bth directis Aswer: ly the middle plt agrees with the aplace s equati, u xx = -u yy : te that the curvature i x-directi is dwward, ad the curvature i the y-directi is upward, s the sum f the tw is zer.