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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 {MaAmman) w I.
4—) 3t: .3? zo 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 problemSJrepiesents a SturmLiouville 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’nfdﬁﬁna aux! 2.405} 552,8.66‘,ll.‘lﬂ... ...
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 Fall '07
 NeimanNassat

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