1 Introduction

(~ E, ~ B)
~
rot ~ B=µ0~ J+µ0ε0
∂~ E
∂t
∂~ B
∂t =−~
rot ~ E ,
div ~ B= 0,
~
E(t= 0) = E0,~ B(t= 0) = B0.
~
E= (Ex, Ey, Ez)~ B= (Bx, By, Bz)
~
J ε0=
1/(36π109)F.m−1µ0= 4π.10−7N.A−2
c= 1/√µ0ε0
Ω⊂R3
~
E×~n|∂Ω= 0,~ B·~n|∂Ω= 0.
~n ∂Ω
z
Ω
z
Ω = ω×R, ω ⊂R2,∂·
∂z = 0
ω
(0,0) (1,0) (1,1
2) (1 2,1 2) (1 2,1) (0,1)

~ν ∂ω ~τ (~τ, ~ν)
~
E ∂t~ E~ B ∂t~ B~ J
L2(R2)
(E0, B0)div(B0)=0 E0
ω
→(Ex, Ey, Bz)
→(Bx, By, Ez)
B=
Bx~ex+By~eyω
rot ~u =∂uy
∂x −∂ux ∂y ,~
rot u=∂u
∂y~ex−∂u ∂x~ey.
(Bx, By, Ez)
∂Ez
∂t −c2rot B=−1 ε0
Jz,∂B
∂t +~
rot Ez= 0 ω, t ∈R+.
Ez(0) = (E0)z,B(0) = B0ω.
Ez|∂ω = 0,B·~ν|∂ω = 0, t > 0.
B
∂2B
∂t2+c2~
rot rot B=1
ε0
~
rot Jz.
div(~ B) = 0 ~ A∈(L2(Ω))3
~
B=~ rot ~ A.
B=~ rot Az.
~
B Az
AzAz
Az
~
rot ∂2Az
∂t2−c2∆Az−1 ε0
Jz= 0.
∂2Az
∂t2−c2∆Az−1 ε0Jz
Az
∂2Az
∂t2−c2∆Az=1 ε0
Jz.
Az

Ez
∂2Ez
∂t2−c2∆Ez=−1 ε0
∂Jz
∂t .
t EzH1
0(ω)
EzAz
EzAz
(Bx, By, Ez)
1
3
AzEz
(Bx, By, Ez)
∂2Ez
∂t2−c2∂2Ez ∂x2= 0.
∆x= 1/(N+ 1)
∆t(tn, xj) = (n∆t, j∆x)j∈
{0, ..., N + 1}
Ez(j, n) = Aexp (i(kj∆x−wn∆t))
k w
sin2(w∆t/2) = C2sin2(k∆x/2)
C=c∆t/∆x
k w ∆x∆t C = 1
tan2(w∆t/2) = C2/4 sin2(k∆x).

Ez|0= cos(x) exp(−x2/20)
[−12π, 12π]
c= 1 ∆x= 0.1π
C= 0.7
t= 50 t= 100
t0= 0 t1= ∆t
2
(x, y)
t
E0=B0= 0 ~ J
EzAz
P1H1(ω)
∆t
H1(ω)
EzAz
hmin = minT∈ThhThTTTh
∆t
c∆t≤1
2hmin,
B
AzB=~ rot AzL2(ω)2
Zω
B·~ λ dω =Zω
~
rot Az·~ λ dω, ∀~ λ∈L2(ω)2.
++
Jz=f(x) sin(ζ t)e−ζ t
2.
f x = 1/4
[1/8,3/8] ζ ν = 4GHz ζ = 2πν

t= 0s
t= 3 10−9s
EzAz