Quiz4_Fall_2007_solutions

Quiz4_Fall_2007_solutions - Name 0N5 SID SO LU“ MAE 105...

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Unformatted text preview: Name: 0N5 . . . SID: SO LU“ MAE 105: Introduction to Mathematical Physxcs If,” QUIZ 4 December 4, 2007 Please state any assumptions clearly 1. (a) Write down all the boundary and initial conditions for a vertical rectangular plate (length = L, and m width = W, where L>W) with its shorter side placed on a surface heated to 400 units. Assume the ambient temperature tobeOunits. ASSumL SW43 5N1 9—1 a 0 Ed: wud‘ 42ij {Ma-Amman) w I. 4—) 3t: .3? -z-o at. [9 TQ’QTQ; W": 3 L 0 EM? mum 7 T(1=-o, ‘3) = O T(x,3=o) ”1.00 400 T(x=wva)=o T(w=l-) = o (b) If the shorter side was now insulated, write down the changed boundary! initial conditions. M17- 2. Evaluate the integral, I [sm(m7)sm(L)sm(fl)sm(—)dxdy (m 11,1), q areal: integers) for: G 0 0). . . " _-——>_ _5sw€ma>s-~ee)a L Rwanda 4- (11) maémprfiq =§ (0) (0) ® (0) a (a . a) => (0) 2 3. A circular membrane, of radius a units is clamped on all sides and initially given a velocity of 2 m/s. Write down all the relevant boundary and initial conditions for this case. @ Bimini? mm . “3331 we H“ “‘ M94 m'd’m“) “(31:69 9, t) = O 3 MM 3w “(4? (“Grant CWW m at a) gas? Mai. val .. 001:1 95(J2,Qt=€9=2! (@ 4. Mark as True or False. Justify briefly: 2M; x 3 : 31‘: S 2 (a) The radial differential equation, r2 Z—R + F? + (2911:»2 — n2 )R = o in circular or cylindrical geometry, r obtained for vibrationfheat conduction problemSJrepi-esents a Sturm-Liouville equation. 719.th CanLewnHGna/a: i£9téfi]+>\zn£‘flfk=0 J1. __, sam 65“.,“ M a Sth- Liouuille dLi'FQchhaL WM 471])‘37111‘ A&(G‘Jf 1- ‘Vi" 0 [M23 x mfmmtcfi: (b) The e1 genfimctions of the above equation (i. e, Bessel fimctions) are “orthogonal with a weight function of 1 (one). {FA LSEi New: {WM (5') : 3?. (o) The roots/”zeroes” of the Bessel fimction, say 1.,(x), are integers. E m Tut—5 am. m____i: Mayne ‘i’nf-dfifina aux! 2.405} 5-52,8.66‘,ll.‘-lfl... ...
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