Формула Кирхгофа

u u

I

u

u

I

u

ut

I

It

2

t

t

t

tt

t

t

t

2

t

t

t

t

u

1 u

I

I

It

It

u

It

2

2

t

t

t

t

tt

t

t t

t

I

at2

y

ξ= y

4 |ξ| 1

( (x atξ),ξ)dSξ

x

at2

( ( y),ξ)dSy

4

(at)

2

| y x | at

|y x| at

1

( y)dSy

1

( y)dy

4 a

|y x| at

4 a |y x| at

I

1

at

r2dr2 d

(x rξ( , ))sin d

4 a

0

0

0

1

2 2

I

(at)

d (x atξ( , ))sin d

a

4 a

t

0

0

a2t2 2 d

(x atξ( , ))sin d

4

0

0

a2t

2

(x atξ)dSξ

4

|ξ| 1

I

a2t

t

u

(x atξ)dS

ξ

tt

t

4

|ξ| 1

u(x,t)

t

(x atξ)dS

u a2 u

ξ

tt

4 |ξ| 1

t

u(x,t)

4

(x atξ)dSξ

|ξ| 1

u (x,t)

1

(x atξ)dS

at

( ,ξ)dS

t

ξ

ξ

4 |ξ| 1

4

|ξ| 1

u (x,0)

1

(x a 0 ξ)dS

a 0

( ,ξ)dS

t

ξ

ξ

4 |ξ| 1

4

|ξ| 1

1

(x)

(x)

4 (x)

4

(x)dSξ

4

dSξ

4

|ξ| 1

|ξ| 1

(x) C3(R3 )

tM [ ]

t

u(x,t)

(x atξ)dSξ

t

4

t

|ξ| 1

v(x,t) tM [ ]

t

(x atξ)dSξ

4

|ξ| 1

u(x,t) vt (x,t)

v(x,t) C3 (x R3,t 0) u(x,t) C2 (x R3,t 0)

v a2

v (v )

a2 ( v)

t

(v )

a2 (v )

t

u a2

u

tt

tt t

t tt

t

tt

v(x,0) 0,

vt (x,0) (x) u(x,0) (x)

v (x,t)

a

2t

u (x,0) v (x,0) 0

(x atξ)dS

ξ

tt

4 |ξ| 1

t

tt

vtt a2 v, x R3, t

t

v

t

0

u(x,t) v(x,t, )d

f (x, )

0

vt

t

2

3

, t 0

utt a

u f (x,t), x R

u(x,0) 0,

x R3

u (x,0) 0,

x R3

t

2

u2

a2 u, x R3

, t 0

t

t

u(x,t)

(x atξ)dS

u(x,0) 0, x R3

4

ξ

|ξ| 1

u (x,0) (x), x R3

t

v a2 v, x R3, t

v(x,t, )

t

f (x a(t )ξ, )dSξ

tt

0

v t

4

|ξ| 1

v

f (x, )

t

t