地盤工学演習（第一部） 前田研究室 maedalab Partialdiff02

G _G G

E_EOEO_O--_-

D _{D D}

I_IIS_SASA_ASS_STT_TE_ERER_R

_P P _P

R_RRE_EEV_VVE_EEN_NNT_TTI_IIO_ONON_N

S _{S S}

Y_YYS_STST_TE_EMEM_{M 1/4}

G _G G

E_EEO_OO

S _{S S}

C_CICI_IEE_ENN_NCC_CE_EE&_{& &}

G _{G G}

E_EEO_OO

E _{E E}

N_NGNG_GII_IN_NENE_EEE_ERR_RII_INN_NG_{G G}L_LALA_ABB_B http://www.cm.nitech.ac.jp/maeda-lab/ 3.微分方程式の導出

3.1 一次元波動方程式の導入(Introduction to Wave Propagation in 1D)

・ 微分表記（Detonations for differential）

( ) ^{x t}

u

u ₌ ,

,

x

u

_x

u

∂ ∂

=

^,

t

u

_t

u

∂ ∂

=

^, 2

2

x

u

_xx

u

∂ ∂

=

^, 2

2

t

u

_tt

u

∂ ∂

=

・ 微小振幅（Infinitesimal Amplitude）

→ 0

θ

^⇒

^cos θ → ¹

^,

^sin θ → θ

^,

^tan θ → θ

^:

x

x

x

u

∂ ∂

θ ≈

・ 釣合い式・運動方程式（Equilibrium / Motion Equation）

( ) ^{cos 0}

cos _{+ + =}

− ^T θ

x

^{T dT} θ

x_+dx x-direction

( ) ^{sin 0}

sin _{+ + =}

−

⋅

⋅

− ρ ^{dx u}

tt

^T θ

x

^{T dT} θ

x_+dx y-direction

↓

≈ 0

dT

x-direction

( ⁻ ) ^{= 0}

−

⋅

⋅

− ρ ^{dx u}

tt

^{T u}

x

^u

x_+dx y-direction

↓

( ) ( ^{x t T x dx t} )

T , _{= +} ,

x-direction

→0 +

⎟⎟⎠ ^⎞

⎜⎜⎝ ⎛ −

=

=

dx x dx x xx

tt

dx

u

u

u T

u T

ρ

ρ

y-direction 振動する弦の小片（

_x

_,

_{x + ∆ x}

）と作用する力

Infinitesimal element ^（

x

,

x _{+ ∆} x

^）of bowstring in 1D vibration and forces

T

(internal tension) applied to it.

x _{x + dx}

dx

T

T d ₊

T ^u ( ^{x + dx , t} )

( ) ^{x t}

u ,

dx x₊

θ

x x

u

x

₌ tan _{θ ≈} sin _θ

θ

x

G _G G

E_EOEO_O--_-

D _{D D}

I_IIS_SASA_ASS_STT_TE_ERER_R

_P P _P

R_RRE_EEV_VVE_EEN_NNT_TTI_IIO_ONON_N

S _{S S}

Y_YYS_STST_TE_EMEM_{M 2/4}

G _G G

E_EEO_OO

S _{S S}

C_CICI_IEE_ENN_NCC_CE_EE&_{& &}

G _{G G}

E_EEO_OO

E _{E E}

N_NGNG_GII_IN_NENE_EEE_ERR_RII_INN_NG_{G G}L_LALA_ABB_B http://www.cm.nitech.ac.jp/maeda-lab/ 3.微分方程式の解析解：^解放

3.1 波動方程式の D’Alembert 解（D’Alembert’s Solution for Differential Equations for Wave Propagation）

Differential Equation

xx

tt

^{V u}

u ₌

² , (

_{− ∞ <} x _{< ∞}

,

0 _{< t < ∞}

) (1)

Initial Conditions: We must two initial conditions to solve wave propagation with second differential equation

( ) ( )

( ) ( )

⎩ ⎨

⎧

=

=

x

g

x

u

x

f

x

u

t

^{, 0}

0

,

, (

_{− ∞ <} x _{< ∞}

) (2)

D’Alembert’s Solution

1.- Replacement:

_{{ } { }} x , t _{⇒ ξ} , _η

⎩ ⎨

⎧

−

=

+

=

Vt

x

Vt

x

η ^ξ

⁽³⁾

= 0

u

ξη (4)

( )

( )

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

+

+

=

+

+

=

−

=

+

=

ηη ξη ξξ

ηη ξη ξξ

η ξ

η ξ

u

u

u

V

u

u

u

u

u

u

u

V

u

u

u

u

tt xx

t x

2

2

2

(5)

2.- Integration

( ) ( ) ξ η φ η

η

^{, =}

u

; Arbitrary Function

_{φ ( ) η}

as

_η

(6)

( ) ( ) ( ) ^{ξ , η = Φ η + θ ξ}

u

(7)

Arbitrary Function

_{θ ( ) ξ}

as

_ξ

( ) ⁼ _∫ ( )

Φ η φ η ^d η

G _G G

E_EOEO_O--_-

D _{D D}

I_IIS_SASA_ASS_STT_TE_ERER_R

_P P _P

R_RRE_EEV_VVE_EEN_NNT_TTI_IIO_ONON_N

S _{S S}

Y_YYS_STST_TE_EMEM_{M 3/4}

G _G G

E_EEO_OO

S _{S S}

C_CICI_IEE_ENN_NCC_CE_EE&_{& &}

G _{G G}

E_EEO_OO

E _{E E}

N_NGNG_GII_IN_NENE_EEE_ERR_RII_INN_NG_{G G}L_LALA_ABB_B http://www.cm.nitech.ac.jp/maeda-lab/ 3.- Replacement:

_{{ } { } ξ} , _{η ⇒} x, t

( ) ( ^{x t x Vt} ) ( ^{x Vt} )

u , _{= φ − + θ +}

(8)

4.- Initial Condition:

( ) ( ^{x t x Vt} ) ( ^{x Vt} )

u , _{= φ − + θ +}

( ) ( )

( ) ( )

⎩ ⎨

⎧ =

=

x

g

x

u

x

f

x

u

t

^{, 0}

0

,

( ) ( ) ( )

( ) ( ) ( )

⎩ ⎨

⎧

′ =

′ +

−

=

+

x

g

x

V

x

V

x

f

x

x

θ

φ ^θ

φ

(9)

( ) ⁺ ( ) ⁼ _∫ ( ) ⁺

−

^x

x

K

d

g

x

V

x

V

0

ξ

ξ

θ

φ

⁽¹⁰⁾

( ) ⁼ ( ) ⁻ _∫

^x

( ) ⁻

x

K

d

V g

x

f

x

0

2

1

2

1 ξ ξ

φ

⁽¹¹⁾

( ) ⁼ ( ) ⁺ _∫

^x

( ) ⁺

x

K

d

V g

x

f

x

0

2

1

2

1 ξ ξ

θ

⁽¹²⁾

( ) [ ( ) ( ) ]

⁺

_∫ ( )

−

+

+

+

−

=

^xVt

Vt x

d

V g

Vt

x

f

Vt

x

f

t

x

u _{ξ ξ}

2

1

2

, 1

(13) u

x 0

V: propagation velocity

1 2

t=0 t=1/V t=2/V

φ (η )=C

u

x 0 -1

t=0 t=1/V

t=2/V

θ (ξ )=C

-2

G _G G

E_EOEO_O--_-

D _{D D}

I_IIS_SASA_ASS_STT_TE_ERER_R

_P P _P

R_RRE_EEV_VVE_EEN_NNT_TTI_IIO_ONON_N

S _{S S}

Y_YYS_STST_TE_EMEM_{M 4/4}

G _G G

E_EEO_OO

S _{S S}

C_CICI_IEE_ENN_NCC_CE_EE&_{& &}

G _{G G}

E_EEO_OO

E _{E E}

N_NGNG_GII_IN_NENE_EEE_ERR_RII_INN_NG_{G G}L_LALA_ABB_B http://www.cm.nitech.ac.jp/maeda-lab/ - 特性曲線(Characteristic Curve)

Example-1

Initial condition

( ) ( )

( ) ( )

⎩ ⎨

⎧

=

=

=

0

0

,

0

,

x

g

x

u

x

f

x

u

t

(14)

( ) ^{x t} [ ^f ( ^{x Vt} ) ( ^{f x Vt} ) ]

u _{= − + +}

2

, 1

(15)

characteristic curve

⎩ ⎨

⎧

+

=

+

−

=

−

a a

a a

Vt

x

Vt

x

Vt

x

Vt

x

(16)

Example-2

( ) ( ) ^{( )}

⎪⎩ ( )

⎪ ⎨

⎧

⎩ =

⎨ ⎧ − ≤ ≤

=

=

0

0

,

int

0

1

1

0 1

,

x

u

s

po

other

the

x x

f

x

u

t

(17)

Characteristic curve x = -1 or 1 , t=0

⎩ ⎨

⎧

±

=

+

±

=

−

1

1

Vt

x

Vt

x

(18)

t

x

u(x

_a

, t

_a

)

(x

_a

-Vt

_a

, 0) 0 (x

_a

^＋ Vt

_a

, 0)

x-Vt=x

_a

-Vt

_a

x ^＋ Vt=x

_a

^＋ Vt

_a

x-Vt=0

x ^＋ Vt=0

u

x

-1 0 1

1

u

x

-1 0 1

1

1/2 1/2

t

x -1

u=0 _u=1/2 u=1/2 ^u=0

u=0 u=0

0 1

u=1

x-Vt=1 x-Vt=-1

x^＋Vt=1 x^＋Vt=-1