ВУЗ:
Составители:
Рубрика:
M5
dρ
dt
+ ρ div = 0, ρ
d
dt
= −
~
∇p + ρ ,
p = p(ρ, s),
ds
dt
= 0.
M5
ρ = ρ
0
, s = s
0
, p
0
= p(ρ
0
, s
0
)
M5
M6 :
∂
∂t
+
1
ρ
0
~
∇p = ,
1
c
2
∂ρ
∂t
+ ρ
0
div = 0.
c
2
c
2
=
∂p
∂ρ
¯
¯
¯
¯
ρ=ρ
0
> 0.
c
M6
∂
2
p
∂t
2
− c
2
∆p = F, F = −c
2
ρ
0
div ,
c
p = Re[Φ(x)e
−ikct
], = Re[ (x)e
−ikct
], F = Re[ϕ(x)e
−ikct
].
Φ
∆Φ + k
2
Φ = g, g = −
ϕ
c
2
,
k = ω/c
22 Ìåõàíèêà Ñïëîøíûõ Ñðåä
íåòåïëîïðîâîäíîãî ãàçà (óðàâíåíèÿ ãàçîâîé äèíàìèêè)
dρ dv ~ + ρf,
+ ρ div v = 0, ρ = −∇p
M5 dt dt
ds
p = p(ρ, s), = 0.
dt
Êîëåáàòåëüíûå äâèæåíèÿ ñ ìàëûìè àìïëèòóäàìè ñïëîøíîé ñðåäû, îïè-
ñûâàåìîé ñèñòåìîé M 5, íàçûâàþòñÿ çâóêîâûìè âîëíàìè. Â îêðåñòíîñòè
íåêîòîðîãî ðàâíîâåñíîãî ñîñòîÿíèÿ ρ = ρ0 , s = s0 , p0 = p(ρ0 , s0 ) ìîæíî ïðî-
âåñòè ëèíåàðèçàöèþ ìîäåëè M 5.  ðåçóëüòàòå ïîëó÷àåòñÿ ñèñòåìà ëèíåéíûõ
óðàâíåíèé àêóñòèêè
∂v 1 ~ 1 ∂ρ
M6 : + ∇p = f, 2 + ρ0 div v = 0.
∂t ρ0 c ∂t
Çäåñü ÷åðåç c2 îáîçíà÷åíà âåëè÷èíà
¯
2 ∂p ¯¯
c = > 0. (4.4.1)
∂ρ ¯ρ=ρ0
Çíàê ïðîèçâîäíîé â (4.4.1) îáóñëîâëåí òåì, ÷òî äàâëåíèå âñåãäà ðàñòåò ñ óâå-
ëè÷åíèåì ñêîðîñòè. Âåëè÷èíà c íàçûâàåòñÿ ñêîðîñòüþ çâóêà. Ñìûñë òàêîãî
íàçâàíèÿ ñòàíîâèòñÿ ïîíÿòíûì, åñëè â M 6 èñêëþ÷èòü ñêîðîñòü è äëÿ äàâëå-
íèÿ ïîëó÷èòü ïîëó÷èòü âîëíîâîå óðàâíåíèå
∂2p
2
− c2 ∆p = F, F = −c2 ρ0 div f, (4.4.2)
∂t
â êîòîðîì c èãðàåò ðîëü ñêîðîñòè ðàñïðîñòðàíåíèÿ âîçìóùåíèé.
Âåñüìà èíòåðåñíûìè äëÿ ïðèëîæåíèé ÿâëÿþòñÿ ãàðìîíè÷åñêèå ïðîöåññû,
äëÿ êîòîðûõ çàâèñèìîñòü õàðàêòåðèñòèê çâóêîâîãî ïîëÿ îò âðåìåíè èìååò
âèä:
p = Re[Φ(x)e−ikct ], v = Re[V(x)e−ikct ], F = Re[ϕ(x)e−ikct ].
 ýòîì ñëó÷àå èç (4.4.2) äëÿ ôóíêöèè Φ ïîëó÷àåòñÿ ñòàöèîíàðíîå óðàâíåíèå
Ãåëüìãîëüöà
ϕ
∆Φ + k 2 Φ = g, g = − , (4.4.3)
c2
ãäå k = ω/c âîëíîâîå ÷èñëî.
