Xem mẫu
- Bài tập Toán cho Vật Lý (Ôn thi Cao Học)
Bµi 1 : X¸c ®Þnh dao ®éng tù do cña d©y h÷u h¹n, g¾n chÆt t¹i c¸c mót x = 0 vµ x
4 x (l x )
(0 x l) cßn vËn tèc
= l, biÕt ®é lÖch ban ®Çu ®îc cho bëi u(x,0) =
l2
ban ®Çu b»ng 0.
Gi¶i :
Gäi u(x,t) lµ ®é lÖch cña thiÕt diÖn cã hoµnh ®é x ë thêi ®iÓm t.
2u 2u
a2 2
Ta cã ph¬ng tr×nh dao ®éng cña d©y : (1)
t 2 x
Theo bµi ra, ta cã :
4 x l x
u t 0 l2
®iÒu kiÖn ban ®Çu : (2)
u 0
x t 0
vµ ®iÒu kiÖn biªn : u x 0 0 u x l 0 (3)
Theo lý thuyÕt, ta cã nghiÖm riªng cña ph¬ng tr×nh (1) tho¶ m·n ®iÒu kiÖn biªn
kat kat kx
(3) cã d¹ng : u(x,t) = u k ( x, t ) (a k cos (4)
bk sin ). sin
l l l
k 1 k 1
Ta x¸c ®Þnh ak, bk sao cho u(x,t) tho¶ m·n ®iÒu kiÖn ban ®Çu (2)
kx 4 x (l x )
Thay (4) vµo (2) : u t 0 a k sin (5)
l2
l
k 1
ka kx
u
bk (6)
0
sin
t l l
k 1
t 0
4 x (l x )
Gi¶i (5) : NhËn thÊy ak lµ hÖ sè trong khai triÓn thµnh chuçi Fourier theo
l2
hµm sin trong kho¶ng (0, l).
kx
råi lÊy tÝch ph©n 2 vÕ tõ 0 l ta cã :
Nh©n 2 vÕ cña (5) víi sin
l
l l
2 kx kx
4 x (l x )
(7)
ak sin l dx l 2 sin l dx
0 0
kx
1 cos l
l l
l dx a k x l sin kx = a l
kx
VT = a k sin 2 dx a k
2 l 0
k
k 2
l 2 2
0 0
l
VT = a k (8)
2
- l l
kx
kx
4
dx x 2 . sin
VP = l. x. sin dx
l2 l l
0
0
l l
l
l2 l2
kx kx kx
l
Ta cã : I1 = x. sin cos k
dx 2 2 sin
.x. cos
k k
l o k
l lo
0
l
l l
kx kx kx
l2 2l
2
I2 = x .sin dx
.x . cos x. cos l dx
k l o k 0
l
0
l3 2l 3 2l 3
I2 = - cos k 3 3 cos k 3 3
k k k
3 3 3
2l 3
4 l 2l l
Nªn VP = 2 cos k 3 3 cos k cos k 3 3
l k k
k k
2l 3 2l 3
4
VP = (9)
3 3 3 3 cos k
l2 k k
Thay (8) (9) vµo (7) ta cã :
8 2l 3
ak = 3 . 3 3 (1 cos k )
l k
nÕu k 2n
0
16
= 3 3 (1 cos k ) (n=0,1,2...)
32
k nÕu k 2n 1
2n 13 3
Tõ (6) bk = 0
do ®ã, nghiÖm cña bµi to¸n ®· cho :
32 (2n 1)at (2n 1)x
1
3
u(x,t) = .
cos sin
3
n 0 2n 1 l l
Bµi 2 : X¸c ®Þnh dao ®éng tù do cña d©y h÷u h¹n, g¾n chÆt t¹i c¸c mót x= 0
x = 1 biÕt ®é lÖch ban ®Çu b»ng 0, vËn tèc ban ®Çu ®îc cho bëi :
v0 cos( x c) nÕu x c /2
u
( x,0)
t 0 nÕu x c /2
víi v0 lµ h»ng sè d¬ng vµ /2 c l - /2.
Gi¶i :
Gäi u(x,t) lµ ®é lÖch cña d©y cã hoµnh ®é x ë thêi ®iÓm t .Ta cã ph¬ng tr×nh
2u 2
2 u
trong miÒn (0
- u t 0 0
(0 x l) (3)
v cos x c nÕu x c /2
u
0
t t 0 0 nÕu x c /2
T¬ng tù bµi 1) ta cã nghiÖm cña ph¬ng tr×nh (1) tho¶ m·n ®iÒu kiÖn biªn (2) :
ka ka kx
a
u(x,t) = (4)
t bk sin
cos t sin
k
l l l
k 1
kx
Tõ ®iÒu kiÖn ban ®Çu ta cã : u t 0 a k sin (5)
0 ak 0
l
k 1
ka kx
u
F x
bk sin
t t 0 k 1 l l
NhËn thÊy bk lµ hÖ sè trong khai triÓn F(x) thµnh chuçi Fourier theo hµm sin trong
l l
ka 2 kx kx
sin l dx F x sin l dx
kho¶ng (0, l) bk
l0 0
c / 2
kx
2vo
bk / 2 cos( x c) sin l dx
ka c
v0 c / 2 k c / 2
k
= 1.x c dx sin 1.x c dx
sin
ka c / 2 l l
c / 2
c / 2 c / 2
v0 1
k k
1
= 1 x c 1 x c
cos cos
k
c / 2 k 1 l
ka l c / 2
1
l
l
v0 1
k k
= 1 c c cos 1 c c
cos
k
ka l 2 l 2
1
l
1 k
k
1 c c cos 1 c c
cos
k l 2 l 2
1
l
v0 1 kc k 2 kc k 2
= cos cos l 2l 2
2
ka k l 2l
1
l
kc k 2
1 kc k 2
cos l 2l 2 cos l 2l 2
k
1
l
-
v 0 1 kc k 2
kc k 2 1 kc k 2 kc k 2
sin sin sin sin
l 2l l 2l l 2l l 2l
ka k k
1 1
l
l
v0 1 1 k 2 2
= 4v 0 . 2 1 sin . kc cos k
kc
= 2 sin cos
ka k 2
ka k k l 2l
l 2l
1
1 1
l 2
l
l
2
4v 0 kc k
bk = . sin . cos
22
l 2l
k
ka1 2
l
Do ®ã nghiÖm cña bµi to¸n ®· cho lµ :
k 2
kc
sin . cos
u(x,t) = 4v 0 . 2l sin kat sin kx .
l
22
a l l
k
k 1
k 1 2
l
Bµi 3 : X¸c ®Þnh dao ®éng däc cña thanh nÕu 1 mót g¾n chÆt cßn 1 mót tù do, biÕt
u
c¸c ®iÒu kiÖn ban ®Çu : u t 0 f ( x) , F ( x)
t t 0
Gi¶i :
Gäi u(x,t) lµ ®é lÖch cña thiÕt diÖn cã hoµnh ®é x ë thêi ®iÓm t
2u 2u
a2 2
Ph¬ng tr×nh : (1)
t 2 x
u
Tho¶ m·n ®iÒu kiÖn ®Çu : u t 0 f ( x) , (2)
F ( x)
t t 0
u
Tho¶ m·n ®iÒu kiÖn biªn : u x 0 0 , (3)
0
x x l
NghiÖm cña ph¬ng tr×nh cã d¹ng : U(x,t) = X(x).T(t) (4)
X " X 0 (5)
Tõ (1) vµ (4) ta cã : 2
T " a T 0 ( 6)
Tõ (3)&(4) X(0) = 0 ; X’(l) = 0 (7)
Gi¶i (5) :
* = - c2 X(x) = c1.e-cx + c2.ecx nªn theo (7) :
X(x) = c1 + c2 = 0 c1 + c2 = 0 c1 = 0
X’(l) = -c.c1.e-cl + c.c2.ecl = 0 c2.ecl – c1e-cl = 0
c2 = 0 (lo¹i)
- X 0 c1 0
* = 0 X(x) = c1 + c2x Theo (7) : (lo¹i)
X ' l c 2 0
* = c2 X(x) = c1cos cx + c2sin cx
X (0) c1 0
Tõ (7)
X ' (l ) c2 c cos cl 0
2k 1 = 2k 1 2
§Ó c2 = Ak cos cl = 0 cl k c
2 2l 2l
NghiÖm cña ph¬ng tr×nh (5) tho¶ m·n ®iÒu kiÖn biªn (7) lµ :
2k 1x
X k x Ak sin
2l
2k 1at D 2k 1at
Tk t Bk
Gi¶i (6) : cos sin
k
2l 2l
Nªn nghiÖm riªng cña ph¬ng tr×nh (1) lµ :
2k 1at b sin 2k 1at sin 2k 1x
u ( x, t ) ak cos (8)
k
2l 2l 2l
k 0
2k 1x f ( x)
Tõ (2) ta cã : u t 0 a k sin (9)
2l
k 0
2k 1a sin 2k 1x F ( x)
u
bk (10)
t t 0 k 0 2l 2l
NhËn thÊy ak lµ hÖ sè trong khai triÓn chuçi Fourier nh©n 2 vÕ cña (8) víi
2k 1x nªn : l
2k 1x dx l 2k 1x dx
a k sin 2
sin f ( x) sin
2l 2l 2l
o o
l
2k 1x dx a k x l sin 2k 1x a k l
al
k 1 cos
2
2 o 2k 1
l l 2
0
l
2k 1x dx
2
a k f ( x) sin (11)
lo 2l
2k 1a l sin 2 2k 1x l F ( x) sin 2k 1x dx
(10) bk
2l 2l 2l
o o
2k 1a l 1 cos 2k 1x dx b a 2k 1
bk F ( x)
k
2l 2l 4
o
l
2k 1x
4
bk (12)
F ( x) sin 2l dx
2k 1a o
VËy (8) lµ nghiÖm cña bµi to¸n trong ®ã ak vµ bk ®îc x¸c ®Þnh tõ (11),(12)
Bµi 4 : Còng nh bµi 3 nhng c¶ 2 mót ®Òu tù do
- Gi¶i :
2u 2u
a2 2
Ta cã ph¬ng tr×nh dao ®éng cña d©y (1)
t 2 x
u
Tho¶ m·n ®iÒu kiÖn ®Çu : u t 0 f ( x) , (2)
F ( x)
t t 0
u
Tho¶ m·n ®iÒu kiÖn biªn : u x 0 0 , (3)
0
x x l
NghiÖm cña (1) cã d¹ng : U(x,t) = X(x).T(t)
X " X 0 ( 4)
Nªn '' 2
T a T 0 (5)
Gi¶i(4) :
u
c c1 c c 2 0
x
c1 0
* =-c2 X(x)= c1.e-cx+c2.ecx x 0
c 2 0
u c c1 e cl c c 2 e cl 0
x
x l
u
c c1 c c 2 0
x
c 2 0
* = 0 X(x) = c1.x + c2 x 0
c1 0
u c c1 c c 2 0
x
x l
c2 = A0
øng víi trÞ riªng = 0 th× ta cã hµm riªng t¬ng øng X0(x) = A0
(5) cã nghiÖm : T0(t) = B0.t + D0 u0(x,t) = a0 + b0t b0 A0 B0
a A D
0 0 0
u
* =c2 X(x) = c1cos cx + sin cx c.c 2 0
x x 0
u
c.c1 . sin cl 0
x x l
k
§Ó cã nghiÖm kh«ng tÇm thêng th× sin cl = 0 cl = k c = khi ®ã
l
kx
c1=Ak nªn X ( x) Ak cos
l
kat kat
vµ T (t ) B k cos Dk sin
l l
do ®ã nghiÖm riªng cña ph¬ng tr×nh (1) :
kat kat kx
u k x, t a k cos bk sin cos
l l l
-
kat kat kx
nghiÖm cña pt (1) : u ( x, t ) a 0 b0 t a k cos bk sin cos
l l l
k 1
kx
Tõ (2) u t 0 a 0 a k cos (6)
f ( x)
l
k 0
ka kx
u
b0 bk (7)
F ( x)
cos
t t 0 l l
k 0
ka
NhËn thÊy a0, ak vµ b0, bk lµ c¸c h»ng sè trong khai triÓn f(x),F(x) thµnh chuçi
l
Fourier theo hµm cosin trong kho¶ng (0,l).
l l l
kx
Tõ (6) a 0 dx a k cos dx f ( x)dx
l
0 0 0
l l l
ka kx
(7) b0 dx bk dx F ( x)dx
cos
l l
0 0 0
u 0 x,0 f ( x)
V× u0(x,t) lµ 1 nghiÖm riªng cña (1) nªn u 0
t x,0 F ( x)
l l l
1
a 0 dx f ( x)dx a0 (8)
l
f ( x )dx
0 0 0
l l l
1
(9)
b0 dx F ( x)dx b0 l F ( x)dx
0 0 0
u k x,0 f x
T¬ng tù uk(x,t) lµ nghiÖm riªng cña (1) u k
t x,0 F ( x )
l l l
kx kx kx
2
a k cos 2 dx a k f ( x) cos (10)
dx f ( x ) cos dx
x x l0 l
0 0
l l l
ka kx kx kx
2
cos 2 (11)
bk dx F ( x ) cos F ( x) cos l dx
dx bk
ka 0
l l l
0 0
VËy nghiÖm cña bµi to¸n :
kat kat kx
a
u(x,t) = a0 + b0t + .
bk sin
cos cos
k
l l l
k 1
Trong ®ã : a0, b0 , ak , bk ®îc x¸c ®Þnh bëi (8) , (19) , (10) , (11)
Bµi 5 : Mét thanh ®ång chÊt cã ®é dµi 2l bÞ nÐn cho nªn ®é dµi cña nã cßn l¹i lµ
2l(1-). Lóc t = 0, ngêi ta bu«ng ra. Chøng minh r»ng ®é lÖch cña thiÕt diÖn cã
hoµnh ®é x ë thêi ®iÓm t ®îc cho bëi:
- 8l (1) n1 (2n 1)x (2n 1)at
2
nÕu gèc hoµnh ®é ®Æt ë t©m cña
u ( x, t ) sin cos
2
n 0 (2n 1) l l
thanh.
Gi¶i:
Chän hÖ trôc to¹ ®é cã gèc trïng víi t©m cña thanh . Trôc ox däc theo thanh
Theo bµi ra, thanh ®ång chÊt cã ®é dµi 2l bÞ nÐn
th× ®é dµi cßn l¹i cña nã lµ 2l(1-) Do ®ã khi trôc
dÞch chuyÓn 1 ®o¹n lµ x th× thanh bÞ nÐn x(1-)
®é lÖch u(x,0) = x(1-) – x = - x
Gäi u(x,t) lµ ®é lÖch cña mÆt c¾t x ë thêi ®iÓm t
XÐt tiÕt diÖn cã hoµnh ®é x, do thanh ®ång chÊt
nªn ë thêi ®iÓm t nã bÞ nÐn ®Õn vÞ trÝ x(1 - ) vµ
cã ®é lÖch u(x,0) = - .x = f(x).
2u 2u
a2 2
Ph¬ng tr×nh dao ®éng cña thanh : (1)
t 2 x
Theo bµi ra, t¹i thêi ®iÓm t = 0 ngêi ta bu«ng ra tøc vËn tèc ban ®Çu = 0 chøng tá
hai ®Çu mót cña thanh ®Òu tù do
u u
ta cã ®iÒu kiÖn biªn : 0; (2)
0
x x
x 0 x l
u
vµ ®iÒu kiÖn ban ®Çu : u t 0 .x f ( x ) ; (3)
0
t t 0
T×m nghiÖm cña ph¬ng tr×nh (1) díi d¹ng u(x,t) = X(x).T(t) (4)
X " x X ( x ) 0 (5)
Tõ (4) vµ (1) ta cã :
T " t a T (t ) 0
2
( 6)
B©y giê ta ®i t×m nghiÖm cña ph¬ng tr×nh (5) tho¶ m·n ®iÒu kiÖn :
X’(-l) = 0 ; X’(l) = 0 (7)
rx
Gi¶i (5) : §Æt X = e ta cã ph¬ng tr×nh ®Æc trng cña (5) : r2 + = 0
= -c2 X(x) = c1e-cx + c2ecx
Tõ (7) c1 = c 2 = 0 (lo¹i)
= 0 X(x) = c1x + c2
X ' (l ) c1 0
c2 0 vµ c2 = A0
Theo (7) :
X ' (l ) c1 0
Nªn X0(x) = A0
øng víi trÞ riªng = 0 th× (6) cã nghiÖm : T0(t) = B0t + D0
nªn ta cã nghiÖm riªng cña (1) u0(x,t) = a0 + b0t (a0 = A0D0; b0= A0B0) (8)
= c2 X(x) = c1cos cx + c2sin cx
Theo (7) :
- u
c1cc sin( cl ) cc 2 cos(cl ) 0
x
c1 sin cl c 2 cos cl 0 c1 sin cl 0
x l
c1 sin cl c 2 cos cl 0 c 2 cos cl 0
u cc1c sin( cl ) cc 2 cos(cl ) 0
x
xl
§Ó (4) cã nghiÖm kh«ng tÇm thêng th× sincl = 0 hoÆc coscl = 0
k
+ XÐt sincl = 0 cl = k c = vµ c1 = Ak
l
kx
ph¬ng tr×nh (5) cã nghiÖ m : X k x Ak sin
l
2
k
øng víi k ph¬ng tr×nh (6) cã nghiÖm tæng qu¸t :
l
kat kat
Tk t Bk cos Dk sin
l l
Ta cã nghiÖm riªng cña (1) tho¶ m·n ®iÒu kiÖn biªn (2) :
a k Ak B k
kat kat kx
uk x, t ak cos (9)
bk sin cos
bk Ak Dk
l l l
(2n 1) (2n 1)
+ XÐt coscl = 0 cl c
2 2l
2n 1at D sin 2n 1at
(2n 1)x
X n x An sin vµ Tn t Bn cos n
2l 2l 2l
Nªn nghiÖm riªng cña (1) tho¶ m·n ®iÒu kiÖn biªn (2) :
(2n 1)x a n An Bn
(2n 1)at (2n 1)at
un x, t an cos (10)
bn sin sin
bn An Dn
2l 2l 2l
Tõ (8),(9),(10) ta cã nghiÖm cña ph¬ng tr×nh (1) tho¶ m·n ®iÒu kiÖn biªn (2)
chÝnh lµ tæng cña c¸c nghiÖm riªng cña u(x,t) :
kat kat kx
u ( x, t ) a 0 b0 t a k cos bk sin
cos
l l l
k 1
2n 1at b sin 2n 1at sin 2n 1 x
a n cos
n
2l 2l 2l
n 0
Tõ ®iÒu kiÖn ban ®Çu (3) :
kx (2n 1)x
u t 0 a 0 a k cos a n sin (11)
.x
l 2l
k 1 n 0
kx
ka (2n 1)a (2n 1)x
u
b0 bn (12)
0
bk cos sin
t k 1 l l 2l 2l
n 0
t 0
Tõ (12) b0 = bk = bn = 0 (13)
LÊy tÝch ph©n 2 vÕ cña (11) theo x cËn tõ (-l l)
l l l l
kx (2n 1)x
l a0 dx l ak cos l dx l an sin 2l dx l .xdx
- v× b0 = 0 u0(x,t) = a0
v× u0(x,t) lµ 1 nghiÖm riªng nªn u0(x,o) = -x
l l
a0 = -x lÊy tÝch ph©n 2 vÕ a 0 dx xdx
l l
l
x2 22
l
a0 x l 2a0l = (l - l ) = 0 a0 = 0 (14)
2 2
l
kat kx
v× bk = 0 uk(x,t) = akcos cos
l l
v× uk(x,t) lµ 1 nghiÖm riªng cña (1) nªn uk(x,0) = - x
kx
vµ lÊy tÝch ph©n 2 vÕ cËn tõ (-l l)
Nh©n 2 vÕ víi cos
l
l l
kx kx
2
dx . x cos
a k cos dx
l l
l l
l
l
a a
k 2x k 2
l
VT = k (1 cos )dx k x ak l
sin
l l
2k
2 l 2
l
l l
l
kx kx
kx l l
VP = x. cos dx =
x sin l sin l dx
k k
l
l l
l
l
2 2
kx
l l
cos k cos k 0 a k (15)
0
cos
2 2
k 2
2
k l l
2n 1at sin 2n 1x
V× bn = 0 u n x, t a n cos
2l 2l
V× un(x,t) lµ 1 nghiÖm riªng cña (1) nªn un(x,0) = - .x
l l
(2n 1)x (2n 1)x
2
a n sin dx .x sin dx
2l 2l
l l
l
l
an (2n 1)x
a (2n 1)x l
VT = n l 1 cos 2l dx 2 x (2n 1) sin 2l
2 l
an l l
l (2n 1) sin( 2n 1) l (2n 1) sin( 2n 1)
2
l
2n 1x dx
x sin 2l
l
- => VT = an.l
VP =
l l
(2n 1)x (2n 1)x
2l 2l
VP
x cos l cos 2l dx
(2n 1) (2n 1)
2l l
l
4l 2
(2n 1) (2n 1) (2n 1)x
2l
l cos
l cos (2n 1) 2 2 sin
(2n 1) 2 2 2l
l
4l 2 8l 2
(2n 1) (2n 1) (2n 1)
sin
sin (2n 1) 2 2 sin
(2n 1) 2 2 2 2 2
8l 2
VP = 1n
(2n 1) 2 2
8. .l 2 8. .l
n 1
(1) n a n 1
an l (16)
(2n 1) 2 2
22
(2n 1)
Tõ (14), (15), (16) ta cã nghiÖm cña (1) :
8. .l (1) n1 (2n 1)at (2n 1)x
2
u ( x, t ) cos sin
2
n 0 (2n 1) 2l 2l
Bµi 6 : B»ng ph¬ng ph¸p t¸ch biÕn, t×m nghiÖm cña ph¬ng tr×nh :
2u 4u
a 2 4 0 tho¶ m·n c¸c ®iÒu kiÖn biªn vµ ®iÒu kiÖn ban ®Çu sau :
t 2 x
u x 0 0
u x l 0
u t 0 Ax(l x )
2
u
;
0 u
2
0
x x 0 t
2 t 0
u 0
x 2
x l
Gi¶i :
2u 4u
a2 4 0
Ta t×m nghiÖm cña ph¬ng tr×nh : (1)
t 2 x
- u x 0 0
u x l 0
2
tho¶ m·n c¸c ®iÒu kiÖn biªn : u 0 (2)
2
x x 0
2
u 0
x 2
x l
u t 0 Ax(l x )
vµ ®iÒu kiÖn ban ®Çu : u (3)
0
t
t 0
díi d¹ng : u(x,t) = X(x).T(t) (4)
Thay (4) vµo (1) : T”(t).X(x) + a2X(4)(x).T(t) = 0
X ( 4 ) ( x)
T " (t )
a 2T (t ) X ( x)
T " (t ) a 2T (t ) 0 (5)
( 4)
(6)
X ( x ) X ( x) 0
X (0) X (l ) 0
Tõ (2) vµ (4) ta cã : (7)
X " (0) X " (l ) 0
Gi¶i (6) : §Æt X(x) = erx th× ph¬ng tr×nh (6) r4 – = 0 r4 =
*NÕu = 0 X(4)(x) = 0 X(x) = c1x3 + c2x2 + c3x + c4
Nªn tõ (7) ta cã :
X ( 0) c 4 0
2
X (l ) l (c1l c 2 l c3 ) 0
c1 = c2 = c3 = c4 = 0
X " ( 0) 2c 2 0
X " (l ) 6c1l 0
* NÕu < 0 r4 = < 0 : ph¬ng tr×nh v« nghiÖm
* NÕu > 0 r4 = : ph¬ng tr×nh cã 4 nghiÖm :
r1 4 ; r2 4 ; r3 i.4 ; r4 i.4
§Æt 4 th× nghiÖm tæng qu¸t cña ph¬ng tr×nh (6) :
- X x c1e x c 2 e x c3 cos x c 4 sin x
X ' x c1e x c 2 e x c 3 sin x c 4 cos x
X " ( x) 2 c1e x 2 c 2 e x 2 c3 cos x 2 c 4 sin x
Tõ (7) ta cã hÖ 4 ph¬ng tr×nh :
X(0) c1 c 2 c 3 0
l l
X (l ) c1e c 2 e c3 cos cl c 4 sin cl 0
2 2 2
X " (0) c1 c 2 c 3 0
X " (l ) 2 c e l 2 c el 2 c cos cl 2 c sin cl 0
1 2 3 4
c1 c 2 c3 0
c 4 sin cl 0
4
k
k
= (k = 1,2 ...)
§Ó c4 0 sin cl = 0 c =
l l
kx
X k x Ak sin
ph¬ng tr×nh (6) cã nghiÖm : (8)
l
4
k 2 2 a 2
k
Thay = vµo ph¬ng tr×nh (5) : T " t T t 0
l2
l
k 2 2 at k 2 2 at
Tk t Bk cos (9)
Dk sin
l2 l2
Thay (8), (9) vµo (4) ta cã :
k 2 2 at k 2 2 at
kx
u ( x, t ) a k cos (10)
sin
bk sin
2 2
l
l l
k 1
Víi ak = Ak.Bk ; bk = Ak.Dk
Tõ ®iÒu kiÖn ban ®Çu (3) :
kx
u t 0 a k sin (11)
Ax(l x)
l
k 1
k 2 2 a
kx
u
bk (12)
0
sin
2
t l
l
k 1
t 0
NhËn thÊy ak lµ hÖ sè trong khai triÓn Ax(l - x) thµnh chuçi Fourier theo hµm sin
kx kx
l l
nªn : a k 0 sin 2 dx A x (l x) sin dx
l l
0
- kx kx
l l
a k 0 sin 2 dx A x (l x) sin dx
l l
0
l
kx kx kx
l l
l l
(lx x 2 ) cos
Ta cã : I 0 x(l x ) sin dx (l x) cos dx
k l 0 k
l l
0
l l
kx
2l 2
kx
l
= (l 2 x) sin cos
l 0 k 2 2
k l 0
nÕu k=2n
0
2l 3
I 3 3 cos k 1
4l 3
k nÕu k=2n+1
2n 13 3
8l 2 A
ak (13)
(2n 1) 3 3
Tõ (10, (12), (13) ta cã nghiÖm cña bµi to¸n :
(2n 1) 2 2 at (2n 1)x
cos sin
8l 2 A 2
l
l
u ( x, t ) 3 3
n 0 (2n 1)
Bµi 7 : XÐt dao ®éng tù do cña mét d©y g¾n chÆt ë c¸c mót x = 0, x = l trong 1 m«i
trêng cã søc c¶n tû lÖ víi vËn tèc, biÕt c¸c ®iÒu kiÖn ban ®Çu :
u
u t 0 f ( x) ; F ( x)
t t 0
Gi¶i :
Gäi u(x,t) lµ ®é lÖch cña thanh cã hoµnh ®é x t¹i thêi ®iÓm t. Do d©y g¾n chÆt t¹i 2
mót chÞu 1 lùc t¸c dông g(x,t) nªn ph¬ng tr×nh dao ®éng cña d©y cã d¹ng:
2u 2
2 u
(1)
a g ( x, t )
t 2 x 2
u
V× trong m«i trêng cã søc c¶n tØ lÖ víi vËn tèc nªn g(x,t) = k
t
2u 2
0 x l
u 2u
u
®Æt k = 2h g(x,t) = 2h nªn (1) 2 2h a 2 víi (1’)
t 0 t T
t t t
B©y giê ta t×m nghiÖm cña (1’)
- u
Tho¶ m·n ®iÒu kiÖn ®Çu : u t 0 f ( x) ; (2)
F ( x)
t t 0
Tho¶ m·n ®iÒu kiÖn biªn : u x 0 0 ; u x l 0 (3)
Ta t×m nghiÖm díi d¹ng u(x,t) = X(x).T(t) (4)
thay (4) vµo (1’) ta cã :T”(t).X(x) + 2hT’(t).X(x) = a2X”(x).T(t)
T " (t ) T ' (t ) X " ( x)
a2
Chia 2 vÕ cho T(t).X(x) : 2h
T (t ) T (t ) X ( x)
T " (t ) 2hT ' (t ) a 2T (t ) 0 (5)
T " (t ) 2h T ' (t ) X " ( x)
2
a 2T (t ) a T (t ) ( 6)
X ( x) X " ( x ) X ( x ) 0
Tõ (3) vµ (4) ®Ó cã nghiÖm kh«ng ®ång nhÊt b»ng 0 th×
(7)
X 0
X x 0 x l
Ta ph¶i t×m nghiÖm cña (6) tho¶ m·n (7)
X ( x) c1 c 2 0 c1 0
* = - c2 X(x) = c1e-cx + c2ecx (lo¹i)
cl cl
c 2 e 0 c 2 0
X (l ) c1e
X (0) c 2 0 c1 0
* = 0 X(x) = c1x + c2 (lo¹i)
X (l ) c1l 0 c 2 0
X (0) c1 0 c 2 Ak
* = c2 X(x) = c1cos cx + c2sin cx
X (l ) c 2 sin cl 0 sin cl 0
k
®Ó cã nghiÖm kh«ng tÇm thêng th× cl k c
l
kx
pt (6) cã nghiÖm : X ( x) Ak sin
l
vµ gi¶i (5) : §Æt T = et th× (5) cã pt ®Æc trng :
2 + 2h + a2 = 0
Ta cã :
2
ka 2 2
2
ka ka
2 2 2 2
’ = h – a = h – h .i
= ' i h
l l l
= - h ' = - h qk.i
nghiÖm cña ph¬ng tr×nh (5) lµ : T (t ) e ht c1 cos q k t c 2 sin q k t
2
ka
víi q k 2
h
l
kx
Nªn nghiÖm riªng cña (1) : u k x, t e a k cos q k t bk sin q k t sin
ht
l
k 1
-
kx
u t 0 a k sin
Tõ ®iÒu kiÖn ®Çu (8)
f ( x)
l
k 1
kx
u
ha k bk q k sin (9)
F ( x)
t l
k 1
t 0
NhËn thÊy ak lµ hÖ sè trong khai triÓn hµm f(x) thµnh chuçi Fourier nªn nh©n 2 vÕ
l l
kx kx kx
2
cña (8) víi sin ®îc : ak sin dx f ( x) sin dx
l l l
0 0
l
l
l l
k 2x kx
a k 2x kx a l
k 1 cos dx k x f ( x) sin
sin dx
dx f ( x) sin
2k
2 0 l l 2 l 0 0 l
0
l
kx
2
a k f ( x) sin (10)
dx
l0 l
l l
kx kx
(9) ha k bk q k sin 2 dx F ( x) sin dx
l l
0 0
l l
kx kx
l 2
ha k bk q k F ( x ) sin dx ha k bk q k F ( x ) sin dx
20 l l0 l
l
ha k kx
2
bk (11)
F ( x) sin
dx
qk lq k l
0
VËy nghiÖm cña bµi to¸n :
kx
a cos q k t bk sin q k t sin
ht
u ( x, t ) e k
l
k 1
trong ®ã ak ,bk ®îc x¸c ®Þnh bëi (10) vµ (11)
2u 2u
a 2 2 bshx
Bµi 8: T×m nghiÖm cña ph¬ng tr×nh t 2 x
Víi ®iÒu kiÖn ban ®Çu ban ®Çu b»ng 0 vµ ®iÒu kiÖn biªn u x 0 0 ; u x l 0
Gi¶i :
2u 2u
a 2 2 bshx
Ta cã ph¬ng tr×nh : (1)
t 2 x
u
tho¶ m·n ®iÒu kiÖn : u t 0 0 ; (2)
0
t t 0
vµ tho¶ m·n ®iÒu kiÖn biªn : u x 0 0 ; u x l 0 (3)
Ta t×m nghiÖm cña ph¬ng tr×nh (1) díi d¹ng : u(x,t) = V(x) + W(x,t) (4)
- 2W 2W
a2
Trong ®ã : W(x,t) tho¶ m·n ph¬ng tr×nh (5)
t 2 x 2
tho¶ m·n ®iÒu kiÖn biªn W 0; W (6)
0
x 0 x l
2V
V(x) tho¶ m·n ph¬ng tr×nh a 2 (7)
bshx
x 2
tho¶ m·n ®iÒu kiÖn biªn V 0; V (8)
0
x 0 x l
b b
Gi¶i (7) : V " x shx V ' ( x) 2 chx c1
2
a a
b
V ( x) shx c1 x c 2
a2
Tõ (8) ta cã :
V c2 0 c 2 0
x 0
b
b
2 shl c1l 0 c1 2 shl
V x l al
a
bx
V(x) = ( shl – shx) (9)
a2 l
b x
V t 0 a 2 l shl shx
®iÒu kiÖn ban ®Çu cña (7) (10)
V 0
t t 0
mµ theo lý thuyÕt ph¬ng tr×nh (5) cã nghiÖm :
kat kat kx
W ( x, t ) a k cos (11)
bk sin sin
l l l
k 1
b x
W t 0 V t 0 a 2 shl shx
l
Tõ (2), (4), (10)
W 0
t t 0
kx b x
a
2 shl shx
sin
k
l l
a
k 1
ka kx
b (12)
0 bk 0
sin
k
l l
k 1
- kx kx
2b l x 2b
l
dx 2 I 1 I 2
Ta cã a k (13)
dx shx. sin
shl. sin
2 0
la l l l la
0
l
kx
x
víi I 1 shl sin dx
l l
0
l l
(1) k 1 l 2
l2 l2
kx kx
l
I1 cos k
2 2 sin
x. cos
k l 0 k k
l 0 k
1k 1 l 2 (14)
I1
k
l
kx
vµ I 2 shx sin dx
l
0
l
l2
kx kx
l l l
l
I 2 shx. cos shl cos k
dx
chx cos I3
k l 0 k k k
l
0
l
kx
mµ I 3 chx cos dx
l
0
l
kx kx
l l l
l
I 3 chx. sin dx
shx sin I2
k l 0 k k
l
0
(1) k 1 l.shl
l2 l2 l2
shl (1) k 2 2 I 2 1 2 2
nªn I 2 I 2
k
k k
k
(1) k 1 l.shl.k
(15)
I2
l 2 k 2 2
Thay (14), (15) vµo (12) :
2b (1) k 1 l.shl (1) k 1 l.shl.k (1) k 1 2b.shl (1) k 1 2b.shl.k
(16)
ak 22
a 2l l 2 k 2 2 a 2 k a l k 2 2
k
Thay (12) vµ (16) vµo (11) ta cã :
(1) k 1 2b (1) k 1 2b.shl.k
kat kx
W ( x, t ) (17)
22 cos sin
2 22
k 1 a k a l k l l
Tõ (4), (9), (17) ta cã nghiÖm cña (1) :
kx 2bshl (1) k 1 k
(1) k
kat kat kx
b x 2b
k (l 2 k 2 2 ) cos l sin l
u ( x, t ) shl shx 2
cos sin
2 2
a
a l l l a
k 1 k 1
- 2u 2u
a 2 2 bx( x l )
Bµi 9: T×m nghiÖm cña ph¬ng tr×nh
t 2 x
Víi ®iÒu kiÖn ban ®Çu ban ®Çu b»ng 0 vµ ®iÒu kiÖn biªn u x 0 0 ; u x l 0
Gi¶i :
2u 2u
a 2 2 bx( x l )
T¬ng tù bµi 8) ta t×m nghiÖm cña pt (1)
t 2 x
díi d¹ng : u(x,t) = V(x) + W(x,t) (2)
2
V
Víi V(x) tho¶ m·n ph¬ng tr×nh : (3)
bx( x l )
t 2
tho¶ m·n ®iÒu kiÖn biªn : V 0; V (4)
0
x 0 x l
2W 2W
Víi W(x) tho¶ m·n ph¬ng tr×nh : (5)
t 2 x 2
tho¶ m·n ®iÒu kiÖn biªn : W 0; W (6)
0
x 0 x l
b bl 2
Gi¶i (3) : V " ( x) bx 2 blx V ' ( x ) x 3 x c1
3 2
b 4 bl 3
V ( x) x x c1 x c 2
12 6
V (0) c 2 0
Tõ (4) b 4 bl 4
b
c1l 0 c1 l 3
V (l ) l
12 6 12
b 4 bl 3 bl 3
V ( x) (7)
x x x
12 6 12
b 4 bl 3 b 3
V t 0 12 x 6 x 12 l x
Ta cã ®iÒu kiÖn ban ®Çu cña pt (3) : (8)
V 0
t t 0
Mµ ph¬ng tr×nh (5) cã nghiÖm :
kat kat kx
w x, t a k cos (9)
bk sin sin l
l l
k 1
b
3 2 3
W t 0 V t 0 12 x( x 2lx l )
Tõ (2) vµ (8) (10)
W 0
t t 0
- kx b 3 2 3
a k sin l 12 x( x 2lx l )
Tõ (9) vµ (10) k1
ka b sin kx 0 b 0
l k k
l
k 1
kx
b b
l
x
4
2lx 3 l 3 x sin
ak dx I1
6l l 6l
0
kx
l
I 1 x 4 2lx 3 l 3 x sin dx
l
0
l
kx kx
l l l
4 x
x 4 2 x 3l l 3 x cos 3
6 x 2 l l 3 cos
= dx
k l 0 k l
0
kx
l l
4 x
3
6 x 2 l l 3 cos
= dx
k l
0
l
l kx
kx
l l l
12 x
3 2 3 2
= 4 x 6 x l l sin 12 xl sin dx
k k l 0 k l
0
l2 kx
l
12 x
2
= 12 xl sin dx
k 2 2 l
0
l
12l 2 kx
l2 kx 12l l
2 x l cos
= 2 2 x xl cos dx
k l 0 k
k l
0
12l 3 kx
l
2 x l cos
= 3 3 dx
k l
0
l l
12l 3 kx
2l 2
kx
= 3 3 2 x l sin cos
l 0 k 2 2
k l 0
nÕu k=2n
0
24l 5
= 48l 5
I 1 5 5 cos(k 1)
k nÕu k=2n+1
2n 15 5
8bl 4
ak (11)
(2n 1) 5 5
(2n 1)at (2n 1)x
cos sin
4
8bl l l
Thay (11) vµo (9) : W ( x, t ) (12)
5 (2n 1) 5
n 0
nguon tai.lieu . vn