HW2Sols - OPT287 MTH287 Due Fri Feb 13 2 pm Wilmot 303...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: OPT287, MTH287 Due: Fri. Feb. 13, 2 pm, Wilmot 303 Homework 2 1 a 21 12: )0? t 1. Show that (9r f(r))- r fir, [rf(r)]. 2. Evaluate the line integral I= fund/1, for F(r) = (2)9251} + (x2z2)y + (2x2yz)2, r0 = (1,1,1), and r, = (2,2,1). Do this integral explicitly along two different paths: 7; a) first integrate along the straight line from (1,1,1) to (2,1,1) and then along the straight line from (2,1,1) to (2,2,1); 2; b) first integrate along the straight line from (1,1,1) to (1,2,1) and then along the straight line from (1,2,1) to (2,2,1). 7; c) Do your answers for a) and b) agree? (They should! Otherwise you made an error.) Can you explain why? (Hint: a result from your previous homework can be useful.) [9 3. D0 problem 2 again but with F(r) = (2xy2z)x + (x230)? + (2xyz)i. Do the e. integrals along the two different paths agree? Why? \_____ 60‘, _ 4. Consider the sum of two counter-propagating monochromatic waves in 1D: A(x,t) = cos(kx — cot) + cos(kx + cot). (i) [O a) Show by using angle sum and difference formulas (or you can also use complex notation if you want) that this sum gives what is known as a “standing wave”, where A equals zero at all times at certain values of x. What are these values? (It) b) If we have instead the difference of the waves: {o d) .1; g) A(x,t) = cos(kx — wt) — cos(kx + out), (ii) what are now the values of x at which A is zero at all times? Consider the string of a musical instrument. The ends of this string are attached to the instrument at x = 0 and x = D. We know that the amplitude of the string satisfies the wave equation, as do the linear combinations of traveling waves in Eqs. (i) and (ii). Which of the formulas (i) or (ii) can be used to describe the movement of this string? (Recall that the ends of the string are fixed.) Recall that the velocity of the waves in Eqs. (i) and (ii) are determined by the linear mass density and the tension of the string, which are fixed. Recall also that the ratio of a) and k gives precisely the velocity. What are the permitted frequencies a) at which the string can oscillate? How do these frequencies change when we “tune” the string, that is, when we change the tension? Draw a sketch of the string at several times for the lowest permitted frequency. (Note: in music and physics, this frequency is called the “fundamental”. The other allowed frequencies are called the “harmonics”.) Draw also the corresponding sketch for the next allowed ' frequency. Based on the appropriate equation (i) or (ii), and recalling the “string experiment” in class, what physical interpretation can you give to the motion of a string fixed at the ends? Le}? u: ex/qluoé-e/ fie (8%" haud {1052‘- _ afii‘un ): ' 'Aar' Lifité’iflc) all“ LHs’fagg 5r r) Ar“? “er )f’ffz ar ’ r ar 1Lari; Nw evamm “We. mahch Mud 2 HHS»: L El Quad) :2 J. 3:» irw5(r)lx: if g? [£(rxd‘, r37[/r):( ;: r Brz r 33202? 7 > Br F ‘3 Mr) . r (r : ,2“ ear) 3% V: ‘ 3 +19%: + “57‘” at? f 3r + in; MS :1. A HFMP «car 433(5): 2x313"? + x253+2x§%’2‘ / aw (“l/l) W0! ?\=(2/2,‘). be ‘flMS MJf’fiforti er‘td’hfl QIOpg dyi‘FFC’feu‘f MS. a] Firm" 'm4eflrode abvfla “he (inc: flow (bl/i) +5 (ZN/Q (W01 "New C(le we. grrm‘rUW (me ha (2,1,0 40 (2,2/s)‘ 3:2) OI\/\)£31>(Z,\fl) (2/2») C\§CL C3”; 3513’1 :2; A I d7fl%Z/O / Xéfi/Zx X C‘Z.‘ X: 2/ Z;\ / dr:0L%;Q/ 390,2] \lf‘)?(?)wde +J Z disfix dj‘d’t) CK CZ / ‘ 2 AL?“ 6b ”’ 223A 2i}?A r" r, j{2xfi>?+x¢§+2% “:2 a a»? +J «m 8-22053 3: 1 2:03: 2: 3:0 52 2 ‘ I Zr «2 :: } 2xdx+j 413% a 2;: ‘++3\‘ : AVAJrXflAk—Lj Firfl' 'Mk'i’jmJQ aim/«3 ’Me madam: 7W6 «from O/U‘) +0 62/9 QMJ “luau alogfl “he (1,146 7%!” 6/2,!) 1’0 @221 CA : X: {L I %:i / szLOld 3 §CC~_§%2) C2? XEGIZ] 3:1. %=fl_ @flgyo 10,.) X 7— 2 I; } “VJ é—Xg’k xQ§+4¥§> «520W : 4. l - V Z, Z - 9 X11015 *J‘quxdzx: 5\\+ 4:215“: Z»\+3»~2,7 171% 635%,: In HomeworK :L/ Pf‘oblew "50A; we 14mm} ’K’kcfi“ 2.. 2A 2 »\ ‘7 @2563 5 2X3? X + 3‘53 + 25317:? “L “‘3’ dew W253?) m Wfim. @"QOQ‘QJ S’WVQS‘ WW ~, J \7x7\3 000; :i: E a J “£11? S COPS S COL: 4/ M HoWuofK {L/ $rOblew 4 WQ flat, k -> *3; vawzo & PeM :0 j CoCS ¥ ~ fiér/Z) so SMCQ» C5) (02250 ‘m park JD) > mtg“: T a 3 2 (M) é,2) - WW Y qua Ca (US$19 M (364+ 0Q) mg: ‘ “08, cicgeci cow’mur Czqvcg \ E :49 3 Row so :J 290% «J 4:000 2$J¥d2 :Jf-dfl 3, Helmet“ 2, 934m wH’U “x? (F): 245% Q + {33+ 233% g! be 4m mkfirmg molipfme, m dixéoerafi W: 558“??? M3? 13 (2:2) Q32 K 6/ ‘ > Z Z 1:: ) @xcwgflxg f)» sou + l A ’7 ., Q 0K ‘: /"><ZZ 21 «w " vjizxmgjdj éii‘f jflféww 2J1 T9117“) :F;J(i§9€fi+?$%)wfi3*J Kx§+2§3+4x%)wb3 : _ Z 2 ‘ \ :Jijdfl’vji‘ngdw: 'g:\l\+ g2; Ca) {‘4 Pmb‘e/M 2’ M2» Vkfi LO I, Chad'- qi‘ y cw“me 33062 :5 m, , Ca Cb f“ 2 \M ’h/ui (QR, home/{IQI/ 0X3: #0 In #204» ,3, g Ct /\ A l V)???» i I} 3/ _, >< KZY%“O>+5(2*J‘ZB%>+ ¢>b )Svflg‘cigj %O a» Mjfdi 94C) <A§9+43§+433\23%3‘ , Z_L+,%y4:\3% I 4. Cot/{Edda 4049, Sun WON/€51 1]; -, 0; “Mo Cal/ww‘ PrquSrq‘hn/E? moMOC/hfiomq'fic Aer/t): 6mm,th Rogaawzr). 0’) CA Show 135 «mug mate gnaw di‘éfleuce/ wénmuias (or 300 am afio age Couple/v (Adv-fwbu (“x-C {Sou wqu-J {hai— -{1¢Lg Sum 3N2: wké‘t’ is Known CLS‘ a geamug wave, where k0 values of k.(}Wd(” W61» ‘v‘kese, Vamps . ‘v M 0d“ 40 flue: 0+ Ceflfifk USING SUM ANB NWWCg 59:12MUMQ: ‘3! ‘ : Gaga + mg 33.26b3(d+’2.é/CO Z 603009603 ZaanKx)&:g(u 4:8 M’qfl: COS [’KX “00%) + Cos; [Kx+ wt) 2 “’27” ¢[ ‘ " COS —6L$’ ‘; v. USING COMPLW NOW/WW 2; (d 44 3 GOSH/ct) N (Logéz) : e +9 [email protected]) -{fla’wfl {(KM'WQ ~i(kx+w AWH : COS (Rx 40%) +7 co5( Kx+w+) 2» i 46 + e (e \ . A Z “a / ‘ "W’JC{ {Rx ~“C‘<> (w‘t ‘ m‘gkx {La V 7: E e € +6 ‘\’ e (8/ —{- e : wkw—‘C «wt :: a @3603 + 6 C051 (4*) 5 2604MB a” ‘ W UJCHAJC :: 200$ :0 V ,, __ “ %er<:> {5r 5° “’3 “w COSQOQ V0 3 gjmioigfii‘es <30 7‘72 KX ~ @144) I: m-Hfi M 1% 7 2 x: _L 2”“ Tr . K 2 K we *hde QC Win/63; 7 MW) t, cos. (Rx—aft) -aosgkxmfl m) am, new “We Wakes op x cfi' Maid A (NS $60 QJC a“ iYLWCTCS? Use, ‘fle, fic‘bl‘MUla C0305 ’ COS & :7 “23M (fiEJS‘N/I(§:§J I r v Z ’2. “Mk/fl 3 __ 2 SM (M3 swam) 294W) swear) WK Eg :20 ‘éfl'fl :—S\M(W‘C) \MUK‘HPRiS 0‘9 'TT " (Mil/Heft) K H ,___. C) Consider “We, o?! a mfg-m Mfl‘wm/‘f. The 944019 1143 SW59 0V9» afiacued? ~60 we ‘Mfi‘fv‘vuaqf' a}: 2:10 and )(423. We 14x2“) Mr “be amfali’mdz mt We: ‘Sq'h‘i’fi‘egx We: wave 639mb“ / 43 do We Q1499,» Oomh‘uqfious 0C “Rm/euufl cum/94 from a) mm! B) . Lama/4 01C 414‘s, 2 “gm” ‘0? Q0“ 414d (2:14 be (15659 "+29 désm cfiC “(1463 340'»? 7 We Head 40 i€ he m expresgioui $013373; ‘be Cburkqm+ A20 at” X30 cal/0° )VZYS . @9181qu A; ZCOSQQ wgw’c) (6‘3 44» ark Kco : A; ngcwt) ._>> Camus? be eero qJ‘“ ail ‘hmes/ + I a . N K 30 “his wave dog: “of “f X ‘5 ' A" 2605 (Kb)CoSCUW) swing “he: aqqgwqu Now) OOME‘OQ: A; '2 Simfl‘ixB S‘Mfw't) (H) «v 6% xao: A30 fir all W9 "’*’ at“ *‘B ‘» A: 253%005) SMQM‘) ’49 can be, izero 4” 4“ ‘Hmes 7%( cer‘ftarm vcflueS ‘ 01C ‘3 W‘C make, Sit/106(3) :C Were-(‘0 re} fig/‘6) x CGS(KA(~W‘C) --Co$(K)(+ 06%) Ccm be used? +0 aegmgg We, mm,wa o? ~We Sfiw‘a . a) 230:1“ W1” “We; «docflg (.7\C “We waves ti"? ad are bj Jme Wear mass dewfifl and ’We, mum 0“: ‘W‘Mi mug, wiest are 6X€dfl aIS‘o “’Wa‘t’ “We. rafio go and? K \cjxiugg “fl/r: Vefoam. Mod” one. “(he {EU/“Md 'VEGQWQ‘QA w (1+ whoa W sdflvg (cm 030‘ Kcn‘e T) K e W AS we showed? 3m Part“ C3,, We nea¢°+o Comzy‘cfief (NB #0 maid “We: EN (NB 4’0 We, (buf‘mifly A10 14x kzO/ %/ if” mos‘r be. A: ngadx) Smaxtx :0 *6r all “B'Meg. 14>: M'T' (h mam} 8) H000 do 4148.39 cbmuge we“ We: +0148 "We W'v/‘Wa‘ {3/ (0146,01 we Chaucae “We 4644532001? The magma are frbfor‘hbwcfl 40 we, 89(1ch r006 cut—{us mam '0 NW? 4 gxkeerA oé‘ we, qt 36V€1h(’fi%{€_9‘743f fie booed“ QEbfi/Wg (caled ’QM&qM€q~{q(/v vfue o‘flaer alfowed WQGQUwCr’pS (We C‘dféd "We; haWoMCg) . Dram) qf-So “file QEDVQXPOKlefl Skiffdt {5r 7% l4th aflowed L, ' k%”dq’“m( ‘ééfiWaCfl 1 Us]: I 4, b Kb-zT :> ht; > ‘V 2 140547) 7/ 2 simCKQ 8114(th ‘ 21> ngnmkk‘ Sink/V at”: my A30 Basred We“, q no (“146" equod‘o“ 03 or CH) and recalwg r 3 { 3) made/:‘waW C(CLSS’ weir? finifimcat {W'flmo‘ (@4300 3N8 flu We mafia“ a {moi q‘f’ W5? “7 is a wavefirwe/ngl bouua‘m bat/K Crud %m #bmvfue. “6 X941 944043 and “Fan/Ming q gfom M wave, , ...
View Full Document

This document was uploaded on 03/23/2011.

Page1 / 8

HW2Sols - OPT287 MTH287 Due Fri Feb 13 2 pm Wilmot 303...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online