TEMA (São Carlos) vol.14 número1

½

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G(r, t

|

r

′

_{, t}

′

₎

∇

2

G(r, t

|

r

′

, t

′

)

−

_c

1

2

∂

2

_G

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2

=

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−

r

′

₎

_δ(t

−

t

′

)

,

r

r

′

R

2

t

t

′

R

G(r, t

|

r

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′

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_.

! "# " $%

!

t

&

∇

2

G(r, ω

˜

|

r

′

, t

′

) + (ω/c)

2

G

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−

r

′

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e

i

ωt

′

/

√

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'

˜

G(r, ω

|

r

′

_{, t}

′

₎

_≡

_F

t

{

G(r, t

|

r

′

, t

′

)

}

=

1

√

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∞

−∞

dt e

i

ωt

_{G(r, t}

_|

_r

′

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′

₎

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!

G

=

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t

{

G

˜

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2

∞

−∞

dω e

−i

ωt

_G

_˜

# #!&&

(

−∞

,

0)

(0,

∞

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! &&&

ω

→ −

ω

" & &

G(r, t

|

r

′

, t

′

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√

1

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∞

0

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e

−i

ωt

G(r, ω

˜

|

r

′

, t

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) +

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˜

−

ω

|

r

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.

(

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ω

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x

y

r

∇

2

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′

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y

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2

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2

(x, k, ω

|

x

′

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′

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′

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−

k

2

−

ω

2

c

2

¯

˜

G

=

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i(

ky

′

₊

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′

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2π

δ(x

−

x

′

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,

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|

x

′

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y

{

G(x, y, ω

˜

|

x

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, y

′

, t

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}

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i

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′

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′

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ε

−

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ε

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d

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¯˜

x

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x

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G

¯

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(x

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_{, k, ω}

|

x

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|

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′

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=

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c

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(

a

+

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x

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2

e

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a

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b

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x

(x < x

′

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1

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a

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x

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2

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2

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2

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d

2

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3. 4

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|

x

′

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′

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′

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F

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y

{

G

¯

˜

}

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˜

G(x, y, ω

|

x

′

_{, y}

′

_{, t}

′

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_√

1

2π

∞

−∞

dk e

−i

ky

_{G(x, k, ω}

¯˜

_|

_x

′

_{, y}

′

_{, t}

′

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a

b

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ab

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a

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.

k

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a

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S

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˜

G(x, k, ω

|

x

′

_{, y}

′

_{, t}

′

₎

%()0

℄

a

= 0

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k

=

ω/c

a

=

b

= 0

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k

=

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% &'

( %

G

¯˜

) ! %

c

2

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1

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x

′

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′+

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_|

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′

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′−

_{, k, ω}

_|

_x

′

_{, y}

′

_{, t}

′

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%

d

2

¯

˜

G(x, k, ω

|

x

′

, y

′

, t

′

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1

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(

a

+

b

i)

x

_(x

_≤

_x

′

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c

1

e

2(

a

+

b

i)

x

′

e

−(

a

+

b

i)

x

_(x

_≥

_x

′

₎

_.

,$ - . %

c

1

=

−

e

−

(

a

+

b

i)

x

′

e

i(

ky

′

+

ωt

′

)

4

π

(

a

+

b

i)

%

+ '/

¯

˜

G(x, k, ω

|

x

′

, y

′

, t

′

) =

−

e

i(

ky

′

₊

_ωt

′

₎

4π(a

+

bi)

×

e

−(

a

+

b

i)(

x

′

₋

_x

₎

(x

≤

x

′

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e

−(

a

+

b

i)(

x

−

x

′

₎

(x

≥

x

′

₎

_,

¯

˜

G(x, k, ω

|

x

′

, y

′

, t

′

) =

−

e

−

√

k

2

₋

_ω

2

_/c

2

_|

_x

′

₋

_x

_|

_{+ i(}

_ky

′

₊

_ωt

′

₎

4π

k

2

₋

_ω

2

_/c

2

,

%

Re

k

2

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_ω

2

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2

_>

₀

_{a >}

₀

0%& $/

˜

G(x, y, ω

|

x

′

, y

′

, t

′

) =

−

1

4π

√

2π

∞

−∞

dk

k

2

₋

_ω

2

_/c

2

e

−|

X

|

√

k

2

₋

_ω

2

_/c

2

₋

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_kY

₋

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′

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,

1

%

X

≡

x

−

x

′

Y

≡

y

−

y

′

$ 1

##

k

=

k

x

+ ik

y

k

#

ζ

=

φ

+ iu

%

¾

/

k

=

−

ω

_c

cos

ζ

=

−

ω

_c

cos(φ

+ iu) =

−

ω

_c

cos

φ

cosh

u

k

x

+ i

ω

c

sin

φ

sinh

u

k

y

,

2

%

k

x

=

−

(ω/c) cos

φ

cosh

u

k

y

= (ω/c) sin

φ

sinh

u .

¾

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= (

ω/c

) cos(

φ

+ i

u

)

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φ

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φ

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1

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0

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φ

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φ

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u

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u

0

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2

u

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u

0

φ

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φ

0

π

# !

$

z(k)

≡

k

2

₋

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2

_/c

2

₌

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_|

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% &

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k

2

₋

_ω

2

_/c

2

₌

_±

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₌

_±

_{i (ω/c) sin(φ}

_{+ iu)}

_.

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dk

k

2

₋

_ω

2

_/c

2

=

(ω/c) sin

ζ dζ

±

i (ω/c) sin

ζ

=

∓

i

dζ

=

∓

i (dφ

+ idu) =

∓

(

−

du

+ idφ)

(

−|

X

|

k

2

₋

_ω

2

_/c

2

₋

_{i (kY}

₋

_ωt

′

_{) =}

−|

X

|

±

i

ω

c

sin(φ

+ iu)

−

i

−

ω

c

cos(φ

+ iu)Y

−

ωt

′

₌

∓

i

ω

c

|

X

|

(sin

φ

cosh

u

+ i cos

φ

sinh

u)

−

i

−

ω

_c

(cos

φ

cosh

u

−

i sin

φ

sinh

u)Y

−

ωt

′

=

ω

c

± |

X

|

cos

φ

+

Y

sin

φ

sinh

u

+ i

ω

c

(

∓|

X

|

sin

φ

+

Y

cos

φ) cosh

u

+

ct

′

_,

f

(φ, u)

≡ −|

X

|

k

2

₋

_ω

2

_/c

2

₋

_{i (kY}

₋

_ωt

′

_{) =}

ω

c

g(φ) sinh

u

+ i

ω

c

g

′

(φ) cosh

u

+

ct

′

,

g(φ)

≡ ±|

X

|

cos

φ

+

Y

sin

φ .

C

!" # "

E

1

$

H

E

2

%"

C

$$ &

˜

G(r, ω

|

r

′

, t

′

) =

−

1

4π

√

2π

E

1

+

H

+

E

2

e

f

(

φ,u

)

dk/

k

2

₋

_ω

2

_/c

2

∓(−

du

+i

dφ

)

=

±

1

4π

√

2π

i

φ

0

0

dφ e

f

(

φ,u

0

)

₋

−

u

0

u

0

du e

f

(

φ

0

,u

)

_{+ i}

π

φ

0

dφ e

f

(

φ,

−

u

0

)

.

φ

0

φ

C

R

= (ω/c) cosh

u

0

→ ∞

u

0

→ ∞

!

u

0

→ ∞

Ref

(φ,

±

u

0

) =

±

(ω/c)

g(φ) sinh

u

0

→ −∞

g(φ)

<

0

"

φ

g(φ)

>

0

#

g(φ

0

) = 0

g

′

_(φ

0

)

>

0

.

$

%& !

φ

0

g(φ

0

)

g

′

_(φ

0

)

g

′

_(φ

0

)

$ '

(

g(φ

0

) =

±|

X

|

cos

φ

0

+

Y

sin

φ

0

= 0

⇒

X

2

cos

2

φ

0

=

Y

2

sin

2

φ

0

⇒

X

2

₍₁

₋

_sin

2

_φ

0

) =

Y

2

sin

2

φ

0

⇒

X

2

= (X

2

+

Y

2

) sin

2

φ

0

,

sin

φ

0

)&

φ

0

∈

(0, π)

℄

cos

φ

0

sin

φ

0

=

|

X

|

/ρ

cos

φ

0

=

∓

Y

sin

φ

0

/

|

X

|

=

∓

Y /ρ ,

ρ

=

√

X

2

₊

_Y

2

₌

_(x

₋

_x

′

₎

2

_{+ (y}

₋

_y

′

₎

2

₌

_|

_r

₋

_r

′

_|

+(,

C

cos

φ

0

<

0

Å

0

W

C

COS

G

2

Å

X

K

Y

K

G

%

%

(Å

C

X

x

n2

Å

cos

φ

0

φ

0

> π/2

-

C

& +

4

g

′

_(φ

0

) =

∓|

X

|

sin

φ

0

+

Y

cos

φ

0

=

∓

X

2

_/ρ

_∓

_Y

2

_/ρ

=

∓

(X

2

₊

_Y

2

_)/ρ

₌

_∓

_{ρ .}

∓

±

∓

k

2

₋

_ω

2

_/c

2

!

g

′

_(φ

0

) =

ρ

" #

f

(φ

0

, u) =

ω

c

g(φ

0

)

0

sinh

u

+ i

ω

c

g

′

(φ

0

)

ρ

cosh

u

+

ct

′

= iω

ρ

c

cosh

u

+

t

′

_.

$

% & '

(

˜

G(r, ω

|

r

′

, t

′

) =

−

1

4π

√

2π

∞

−∞

du e

i

ω

[

ρ

c

cosh

u

+

t

′

]

.

)

* +

F

−1

t

G(r, t

|

r

′

, t

′

)

=

−

1

8π

2

∞

−∞

du

∞

0

dω

e

i

ω

[

ρ

c

cosh

u

−(

t

−

t

′

)

] +

e

−i

ω

[

ρ

c

cosh

u

−(

t

−

t

′

)

]

=

−

1

4π

∞

−∞

du

1

π

∞

0

dω

cos

ω

ρ

c

cosh

u

−

(t

−

t

′

₎

_.

,&

T

≡

t

−

t

′

' - .$℄0 $1

δ

[ (ρ/c) cosh

u

−

T

] =

δ

[ (ρ/c) (cosh

u

−

cT /ρ) ] = (c/ρ)

δ

(cosh

u

−

cT /ρ)

,

v

= cosh

u

2

G(r, t

|

r

′

, t

′

) =

−

1

4π

2

∞

0

du

c

ρ

δ

cosh

u

−

cT

_ρ

=

−

c

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