Généralités sur les phénomènes de propagation Chapitre 6 6.1 Propagation à une dimension

t x
∂2s
∂x2−1 V2
∂2s
∂t2= 0
V
η=t−x
V
ξ=t+x
V
t x
η ξ
∂η
∂t =∂ξ ∂t = 1
∂η
∂x =−∂ξ ∂x =−1 V

∂s
∂t =∂s ∂η
∂η
∂t +∂s ∂ξ
∂ξ
∂t =∂s ∂η +∂s ∂ξ
∂s
∂x =∂s ∂η
∂η
∂x +∂s ∂ξ
∂ξ
∂x =1 V∂s ∂η −∂s ∂ξ 
∂2s
∂η∂ξ =∂2s ∂ξ∂η
∂2s
∂t2=∂2s ∂η2−2∂2s ∂η∂ξ +∂2s ∂ξ2
∂2s
∂x2=1 V2∂2s ∂η2−2∂2s ∂η∂ξ +∂2s ∂ξ2
∂2s
∂t2
∂2s
∂x2
η ξ
∂2s
∂η∂ξ = 0
∂
∂ξ ∂s ∂η = 0
ξ
∂s
∂η =f(η)
f(η)η ξ
η
s(η, ξ) = F(η) + G(ξ)
F(η)η f (η)G(ξ)
ξ x t
s(x, t) = Ft−x
V+Gt+x V
Ft−x
VGt+x Vs(x, t)

Ft−x
VGt+x V
Ft−x
VFt−x V
Gt+x
V
t1x1s
s(x1, t1)t2t1(t2> t1)x2
s x1t1
s(x1, t1) = s(x2, t2)
Ft1−x1
V=Ft2−x2 V
t1−x1
V=t2−x2 V
x2
x2=x1+V(t2−t1)
t2> t1x2x1
x2−x1=V(t2−t1)
Ft−x
Vx
Ft−x
V
x1
x1
x2
x2
t=t1
t=t2>t1
Direction de
propagation
x2-x1=V(t2-t1)
x
x
x F t−x
V
Gt+x
VGt+x V
Ft−x
V
t1x1s
s(x1, t1)t2t1(t2> t1)x2
s x1t1
s(x1, t1) = s(x2, t2)
Gt1+x1
V=Gt2+x2 V
t1+x1
V=t2+x2 V

x2
x2=x1−V(t2−t1)
t2> t1x2x1
x1−x2=V(t2−t1)
Gt+x
Vx
Gt+x
Vx
x1
x1
x2
x2
t=t1
t=t2>t1
Direction de
propagation
x1-x2=V(t2-t1)
x
x
x G t+x
V
x,
x= 0
s(x= 0, t) = S0cos (ωt)
x > 0x= 0
x
V
s(x, t) = S0cos hωt−x
Vi
s(x, t) = S0cos [ωt −φ(x)]
φ(x) = ω
Vxx Vφ(x)
s(x, t) = S0cos 2πt
T−x λ
T=2π
ωλ=V T
s(x, t +nT ) = s(x, t)
s(x+nλ) = s(x, t)
n

s(x, t) = S0cos [ωt −kx]
k=ω
V=2π λm−1
s(x, t) = S0ei(ωt−kx)
s(x, t) = S eiωt
S=S0e−ikx
S0S−kx
S1S2φ1φ2
s(x, t) = S1ej(ωt−kx+φ1)+S2ej(ωt−kx+φ1)=S ej(ωt−kx+φ)
s(x, t) = Scos (ωt −kx +φ)
S=qS2
1+S2 2+ 2S1S2cos (φ1−φ2)
φ=Arctg S1sin (φ1) + S2sin (φ2)
S1cos (φ1) + S2cos (φ2)
S φ
s(x, t) = S1ej(ωt−kx+φ1)+S2ej(ωt+kx+φ1)=hS1ejφ1e−jkx +S2ejφ2e+jkx iejωt
S1=S2=S0
s(x, t) = 2S0cos kx +φ1−φ2
2ej ωt+φ1+φ2
2