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A series of mathematical equations and formulas related to one-dimensional wave equations, boundary conditions, and temperature distribution. It also includes a problem-solving example related to the deflection of a vibrating string. suitable for students studying advanced mathematics and physics.
Typology: Exercises
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Dale Tulodal
of pte fovmulas CDT- One-Dirmensibra Wave^ E?uotion (vilbrallns ofa^ stolched (^) dorg) ue equation fos 04 il o Boundary Conditors yCO)= y (lD- Infla Cordtlins f y(z0)= a/0-f()-) ot Genesol Soluifo crit 4bo Col yCxyt)= Stn-
Sufloble Solutfon yCx= (apa +C2shpa) (Gcacpl c4sincpt
One-Dinmendona teat -fto uOale e9 uatfon
ot
u Cot)=o u (it)=O u (z/0) =f)
u Cxt)=CaCosprC sinp) PE
ron hoogeneaus boundary Condittons.
ae
(o,t)-o-> ond yClt)-0- -O ot-
and (^) yo)- Ye^ 5to )^ -> y
Sdutfo of (1) ts Cont cnMtbo Cas yat)= E^ Sto-X.(an Sto^ bo Cas^ ONE step Substthctfog (4)fnG) , get
ot
CrTt contan Cos 0 sthZan sin O- Sto AnCos COE-bnsonE/t t- O-0 (^0) stn (o|.o n
an-o fo^ 6),^
nTE
Substftuting6) n ), toe qal y(z0)-[ (z,+-0Jt-O COAt
C put n=
sin sofi))_nb (^) Sin (^) 6, (^) Sin 2 b (^) 3in Saa. (^) - Cowpau
b by (^) 0, b3- b4-^ br^ o
31 Cos 37E -yo sto 3Ho gfoTx_Cos
(x)ado(nM)
a (^) sto(on)d
ao-1 Cos(nx^ foDn O (^) Con (om L a -a- 40(1-(-1)") ® put e2n h y(xA)= aa(1-(-))sto^ (Om)^
(nt
du a Du ot the (^) Boudary candilbos^ ale u (o,t)=o > u (t)^ =0^ > ho (^) fnitio tempexctuse distibcfon h^ the^ kar Tho s ulzo)= P(X)>O The Vasious^ pocsble^ soludtfons (^) oF the^ heat Plouo (^) euaton by^ vahble Sepexable methomethao Sepesable cpt uCzt)= (1eaG a)ePt
(c4Cespz +^ C5^ npz)^ ee
The (^) only suttablo^ Solutfon s pt
Catpx +Cs^ dopx) g^ e
Substftute u^
100
a:
Now ot u (ot)= 90-(6) u C10,t)= 60>G)
distibutibn UsC)= uCot)^ +^ |^ u(107t)- u[ot) 90+ /Go- 10 0+ )x Us(=^ 90-3x^ (8) The tanslerst^ tempeyathuve^ dist^ olbttfoy
u(xA)- onnCospz + bnsfo Px) e (^) - he teropeahee u Cart) fro -the mteamecliate pe sbo &
u(xt) Us ()+^ ut^ (z,-4)
u Cxt) -^ 90-^ 3x 4E( Cotpx^ +bnshPX)e (10) sep
to ao) u C 90
+Sanospx +^ bo sinpx) eP^ 90 an+ bn(O)=
L0e get put an^ = (^) O h^10
90-3x
SioPx step- Subsitute C)o (U) = CuCxt)
10
Px) sn 10 (8x-46) sfo ol 40 U 8-40 (^) V sP^ NA
V= 10 Cos 10 Va 10 sf n2 10 10 2 (sx-46)^ (0. cos (^) +s s -(4) 10
u Cxt) = 9o-3 +
uCxr- (^) q0-3x - So(14G1)TO
10
fo tho bar act lime t fr the poblem ()
