BAB VII TRANSFORMASI APACE Tujun Pmbljrn Slh mmpljr bb n, dhrpkn mhw mmlk kmmpun unuk mmbu bnuk-bnuk Trnform plc dr brbg jn fung. Dmkn jug dngn nvr Trnform plc yng dbuny. Slnjuny dhrpkn gr mhw mmpu mrubh PD k dlm bnuk prmn yng brkn unur-unur Trnform plc, dn mnylknny, u dngn yr b yng dkhu. A. Pndhulun Trnform plc T dlh uu mod unuk mnylkn prmn dffrnl PD dn mlh nl wl r yr b. Produr yng dmpuh rdr dr g lngkh, yu:. Mrubh PD mnjd prmn ljbr drhn prmn bnu, dngn mmnfkn bl T.. Prmn bnu dlkn cr ljbr drhn. 3. Mnrnformkn kmbl prmn bnu, hl pnyln k dlm bnuk wl u bl T, bg olu yng dmn. B. Trnform plc dr Fung f Suu fung f yng rdfn pd, bl d kl dngn - dn dngrkn rhdp, pd b < < ~, hlny brup fung u F, yng dnykn bg Trnform plc T. Dul {f} = F u: f F f.
blkny, rnform nvr dr F dul F mnghlkn f u: f F Cn: Fung wl dul dngn huruf kcl dn rnformny dngn huruf br. Conoh: Y dlh rnform dr y dn bgny. Sblkny F y bg nvr dr T. Conoh pmbun T :. f = {f} = = rny : Invrny dlh : dlh T dr f = = =.. f = {f} =. rny nvrny dlh dlh T dr f = nlog unuk f = n n n! n hngg n n n! ]
3. f =. rny dlh T dr f = nvrny dlh. f = co ω co co... dn co.. 5. f = n ω n. n.. dn n. 6. f = co h ω coh. coh.. dn coh.. 7. f = n h ω nh. nh.. dn nh..
Nl-nl rbu d blkn, dmkn jug nl-nl dr bnuk fung lnny. Opr Trnform plc brf lnr, unuk p f, g, h upun konn mbrng. Shngg dp dul : { f+b g+c h}= {f}+b {g}+c {h} Buk : { f+b g+c h} f f. bg ch b g. c h { f } b{ g } c{ h } Sf n mmprmudh pncrn bnuk-bnuk Trnform plc dr jumlh bbrp fung. Conoh :. Tnukn T dr coh. Jwb : Bnuk kponn dr coh. = Jd : coh. coh. Invrny dlh coh... Bl F nukn - {F} b Jwb: b = b b b b b b
3. ω u : ω = co..n. = co. + n. = C. Trnform plc dr Turunn Fung Sf lnr T dp dmnfkn unuk mrubh opr klkulu mnjd opr ljbr yng drhn dlm bnuk rnform. Scr kr dfrn uu fung f hny brhubungn dngn prkln rnform F dngn. Krn ngr mrupkn nvr dr dfrn, mk opr ng brhubungn dngn pmbgn rnform olh.. Trnform Turunn Fung Jk f konnu pd, unuk p γ dn M, dn mmlk urunn f yng konnu pd drh hl, mk T dr urunn f d bl γ dn : f = f f Buk : f f. f f. f f jd f = {f } f Prlun dr T urunn prm n dgunkn unuk T urunn yng lbh ngg. f `` = f f = f f f = f f f
f = f f = f f f f = 3 f f f f f n = n f n- f n- f n-3 f... f n- Mllu T urunn fung n, jug dp dcr bnuk-bnuk T uu fung. Conoh:. Tnukn =... Jwb : f = f ` = f `` = f = ; f ` ; f `` = jd : f `` f-f-f ` f = 3 f. = 3. Tnukn 3 =... Jwb : f = 3 ; f ` = 3 ; f `` = 6; f ``` = 6 f = ; f ` = ; f `` = ; f ``` = 6 jd : f ``` = 3 f f f ` f `` 6 3 f. 6 f 3 3! = 3. Tnukn co ω =... Jwb : f = co ω; f ` = -ω n ω; f `` = -ω co ω f = ; f ` = ; f `` = -ω ; f `` = -ω.f f `` = f f f ` = -ω. f. f = -ω f
+ω f = f co. dn - n.. Tnukn n =... Jwb : f = n f = f ` = n co = n f ` = f ` = n f-f = f dn - n. n 5. Tnukn.n ω =... Jwb: f = n ω f = f ` = n ω +. ω.co ω f ` = f `` = ω co ω-ω. n ω = ω co ω ω.f f `` = ω co ω ω S f f f = f = f = f f n ω =
dn - n. D. Pnggunn T unuk Pnyln Prmn Dffrnl Su dngn ujun mul, yu pmbhn T dgunkn unuk mmbnu pnyln prmn dffrnl, mk lnjuny kn d kmukkn bbrp conoh unuk pnyln Prmn Dffrnl rbu. Conoh :. Slkn dy y, bl y = Jwb : Prmn n dp dul dlm bnuk T yu Y= Y, hngg dp dul :.Y f + Y =.Y + Y = + Y = + Y = Y = y = - Y = - = coh u : y =, cob lkn dngn mod pnyln PD ordo u.. Slkn y``+ y`+3y =, bl y = 3 dn y`= Jwb : Prmn dp dul dlm bnuk T = y S Y f f ` + Y-f + 3Y =
S Y 3 + Y +3Y = + + 3 Y = 3S + 3 3 3 Y 3 5 Y 3 ddp : y = - - 5 3 3 5 pnyln PD ordo du. cob lkn dngn mod 3. Slkn y``+y = unuk y dn y ` Jwb : mbl T Y = Y Prmn dp dul Y f f ` Y Y y y` Y Y. y y` Y y y` y = y - ' y - + - y y y y co y` n y co y` n Aco B n n Solu umum Kond yng dkhu :
.. y y` A A B y` d dp : A = ; B = - B B A A An B co jd jwbn PD dlh y = co n +. Dk : rku lkrk = ; = B D : unuk > Jwb : d PD yng dp dbu: u d 5 Gunkn I= I, prmn dp dul :. I + + I = +5 I + = I 5 5 5 = 5 5 jd ku ru pd > dlh = - -5 5 5 Sol-ol :. Crlh Trnform plc dr fung brku:. + d. coω+ b. - +b. co ω c. n+b f. n ω g. co h. n. nh
j. coh l..co n..n ω k.. m.. - o. n ω. Dkhu F, pr d bwh n, crlh rnform nvr f=-f. 9 o. b. c. b c d. 3. f. 6 g. 3 h. 6. j. 3 k. l. m. 9 n.
3. Dngn mnggunkn T, lknlh!. Y``+ 9y = ; bl y = dn y` = b. y`` + π y = ; bl y = dn y` = c. y`` + 5y = ; bl y = dn y =, d. y`` - y` - 3y = ; bl y = dn y` = 7. y`` + y` - 8y = ; bl y = dn y` = 8. Tnukn knggn mxmum pluru yng dmbkkn vrkl k dngn kcpn wl 96 cm/, g = 98 m/. Slkn dngn T! 5. Tl rgnung pd pk pnng 8m dn m pd p ny. Bl m l m kg dn g = m/. Hunglh wku gr l lp dr pk. Gunkn T! Tbl 7. Trnform plc No f
3 3! 5 n! n n 6 7-8 Co ω 9 Sn ω Coh Snh. 3 n- n n! - b b b 5 - b b b b 6.n ω 7.co ω 8.n ω. 9 co co b b b n. nh