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0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.1 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.2 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.3 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.4 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.5 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.6 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.7 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.8 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 0.9 cek
0.25 0.5 0.75 1
x
- 3
- 2
- 1
1
2
3
u
t= 1 cek
v = 2 / , l = 1 , h =
(4 − π)/π
3
α
n
=
2
l
Z
l/4
0
x
2
h
sin
πx
l
(n +
1
2
)
dx
+
2
l
Z
l/2
l/4
(x −
l
2
)
2
h
sin
πx
l
(n +
1
2
)
dx
=
8l
2
hπ
3
(n +
1
2
)
3
π
4
(n +
1
2
) − sin
π
4
(n +
1
2
)
sin
π
4
(n +
1
2
)
,
β
n
=
2
vπ
Z
l
0
F (x) sin(
πnx
l
)dx = 0.
u(x, t) =
8l
2
hπ
3
∞
X
n=0
sin
π
4
(n +
1
2
)
(n +
1
2
)
3
π
4
(n +
1
2
) − sin
π
4
(n +
1
2
)
× cos(ω
n+
1
2
t) sin
πx
l
(n +
1
2
)
.
t
t = 2
ν = 1/2
∂
2
u
∂t
2
− v
2
∂
2
u
∂x
2
= G
sin(
πnx
l
)
G(x, t) =
∞
X
n=1
γ
n
(t) sin(
πnx
l
),
γ
n
(t) =
2
l
Z
l
0
G(x, t) sin(
πnx
l
)dx,
u(x, t) =
∞
X
n=1
c
n
(t) sin(
πnx
l
)
u u u l
2 πnx
Z
3 3 3
2
t=0 cek 2
t=0.1 cek 2
t=0.2 cek βn = F (x) sin( )dx = 0.
1 1 1
vπ 0 l
x x x
-1
0.25 0.5 0.75 1
-1
0.25 0.5 0.75 1
-1
0.25 0.5 0.75 1
Òàêèì îáðàçîì, ïîäñòàâëÿÿ ïîëó÷åííûå âûðàæåíèÿ â (11), ïîëó÷àåì
-2 -2 -2 çàêîí êîëåáàíèÿ ñòðóíû ñ òàêèìè íà÷àëüíûìè äàííûìè
-3 -3 -3
u u u
8l2 X sin π4 (n + 21 )
∞
3
t=0.3 cek
3
t=0.4 cek
3
t=0.5 cek π 1 π 1
2 2 2 u(x, t) = (n + ) − sin (n + )
1 1 1 hπ 3 n=0 (n + 12 )3 4 2 4 2
0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x
-1 -1 -1 πx 1
-2 -2 -2 × cos(ωn+ 21 t) sin (n + ) .
-3 -3 -3
l 2
u u u
3
2
t=0.6 cek
3
2
t=0.7 cek
3
2
t=0.8 cek Íà ðèñóíêå 5 èçîáðàæåíî äâèæåíèå ñòðóíû â òå÷åíèè ïîëóïåðè-
1 1 1 îäà. Ïðè äàëüíåéøåì óâåëè÷åíèè t îðìà ñòðóíû áóäåò ìåíÿòüñÿ â
0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x îáðàòíîì ïîðÿäêå, è â ìîìåíò t = 2ñåê ñòðóíà âåðíåòñÿ â èñõîäíîå
ïîëîæåíèå. Ëèíåéíàÿ ÷àñòîòà êîëåáàíèé òàêîé ñòðóíû ν = 1/2 ö.
-1 -1 -1
-2 -2 -2
-3 -3 -3
u u
3
t=0.9 cek
3
t=1 cek 2.4 Âûíóæäåííûå êîëåáàíèÿ ñòðóíû êîíå÷íîé äëèíû.
2 2
1 1 Îäíîðîäíûå ãðàíè÷íûå óñëîâèÿ (2a).
0.25 0.5 0.75 1
x 0.25 0.5 0.75 1
x
-1 -1  äàííîì ñëó÷àå íåîáõîäèìî ðåøèòü íåîäíîðîäíîå óðàâíåíèå
-2 -2
-3 -3
∂2u ∂ 2u
2
− v2 2 = G
èñ. 5: Íà ðèñóíêàõ èçîáðàæåíà îðìà ñòðóíû â ðàçëè÷íûå ìîìåíòû âðåìå-
∂t ∂x
íè. Ëåâûé êîíåö ñòðóíû çàêðåïëåí, à ïðàâûé ñâîáîäíî äâèæåòñÿ, ïðè÷åì óãîë
ñ íåíóëåâîé ïðàâîé ÷àñòüþ.  ñëó÷àå ñâîáîäíûõ êîëåáàíèé (8) òàêèå
íàêëîíà ïðàâîãî êîíöà ðàâåí íóëþ âî âñå ìîìåíòû âðåìåíè; âíåøíèå ñèëû îò-
ñóòñòâóþò. Âûáðàíû ñëåäóþùèå çíà÷åíèÿ ïàðàìåòðîâ: v = 2ì/ñåê, l = 1ì, h = ãðàíè÷íûå óñëîâèÿ ïðèâîäèëè ê ñîáñòâåííûì óíêöèÿì sin( πnx l ).
(4 − π)/π 3 ì. Ïî ýòîé ïðè÷èíå ðàçëîæèì âûíóæäàþùóþ ñèëó â ðÿä ïî ýòèì ñîá-
ñòâåííûì óíêöèÿì
Íà÷àëüíàÿ îðìà ñòðóíû èçîáðàæåíà íà èñ. 3. Ïî îðìóëàì ∞
X πnx
(16) ïîëó÷àåì: G(x, t) = γn (t) sin( ),
n=1
l
l/4
x2 ãäå
2 πx 1
Z
αn = sin (n + ) dx 2 l
Z
πnx
l 0 h l 2 γn (t) = G(x, t) sin( )dx, (18)
Z l/2 l 2
(x − 2 )
l 0 l
2 πx 1
+ sin (n + ) dx è ïðåäñòàâèì íàøå ðåøåíèå â âèäå ñëåäóþùåãî ðÿäà:
l l/4 h l 2
∞
2 πnx
8l π 1 π 1 π 1 X
= (n + ) − sin (n + ) sin (n + ) , u(x, t) = cn (t) sin( ) (19)
hπ 3 (n + 12 )3 4 2 4 2 4 2 n=1
l
15 16
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