Ondes acoustiques Notes de cours - Alain Le Rille

•P=P0+p|p|  P0
•µ=µ0+ρ|ρ|  µ0
•~ V=~ 0 + ~v |~v|  cs
p ρ ~v
ν λ cs=λ.ν
ν < 20kHz
` v ≈`.ν |~v|  cs
`λ
`
`λ⇔νcs
l≈109Hz
◦

•
•
χS=−1
V∂V ∂P S⇒
χS=+1
µ∂µ ∂P S
∂2ψ
∂t2=c2
s.∆ψ
ψ=p µ vxvyvzφ
cs=1
√µ0.χS
∂−→ v
∂t =−1 µ0−−→
grad(p)
ρ χS.µ0.p
∂p
∂t =1 χs.µ0
∂ρ
∂t =−µ0.div~v χs.µ0
=−1
χS
div (−→ v)
~uz−−→ grad(p) = ∂p
∂z ~uzdiv (−→
v) = ∂vz
∂z ~uz
(∂vz
∂t =−1 µ0
∂p
∂z
∂p
∂t =−1 χS
∂vz
∂z
◦

χS=+1
µ∂µ ∂P S≈+1 µ0∂µ ∂P S
cs=s∂P
∂µ S
P=P0.V γ
0
Vγ=P0.µγ
µγ
0
cs=sγ. P0.µγ−1
µγ
0≈sγ. P0
µ0
=rγ. R.T
M
⇒
cs=rγ. R.T
M
csm.s−1
O2317
331
N2339
H21270
0◦C
χS
cs(liquides)> cs(gaz)
csm.s−1
1500
Hg 1450
◦

φ(z, t)
∂2φ
∂t2=c2
s.∂2φ
∂z2
•ψ+t−z
cs=−1 csf0t−z csz
•ψ−t+z
cs=+1 csg0t+z csz
~v =hψ+t−z
cs+ψ−t+z csi~uz
p=µ0.cs.hψ+t−z
cs−ψ−t+z csi
z=cste
~v//~uz⇒
ω~ k
φ=<˜ φ
˜
φ=˜ φ0.ej.(ω.t−~
k.~r)
~v =−−→ grad(φ)p(z, t) = −µ0∂φ
∂t
(˜
~v =−j.~ k. ˜ φ0.ej.(ω.t−~
k.~r)
˜p=−j.µ0.ω. ˜ φ0.ej.(ω.t−~
k.~r)
~uz
•~ k= + ω
cs~uzz
•~ k=−ω
cs~uzz
◦

z
vz=ψ+t−z
cs
p=µ0.cs.ψ+t−z
cs
⇒
p=Zc.vzz
Zc=µ0.cs=1
χs.cs
=rµ0
χs
cs=1
√µ0.χSkg.m−2.s−1
Zc= 400 kg ·m−2·s−1
z
vz=ψ−t+z
cs
p=−µ0.cs.ψ−t+z
cs
⇒
p=−Zc.vzz
ec=1
2µ.~
V2
ec=1
2µ0.v2
ep=1
2χS.p2
es=ec+ep=1
2χS.p2+1 2µ0.v2
Zc=µ0.cs=1
χs.cs=qµ0 χs
◦