7.4达朗贝尔公式(8)

数学物理方法课件

1 1 u ( x, t ) = [Φ ( x + at ) + Φ ( x at )] + ∫ Ψ (ξ )dξ 2 2a x atx + at 1 1 [ ( x + at ) + ( x at )] + ∫ ψ (ξ )dξ 2 2a x at u ( x, t ) x + at 1 1 [ ( x + at ) ( x at )] + ∫ xψ (ξ )dξ 2 2a at

x + at

(t ≤ x / a ) (t ≥ x / a)

(t ≥ x / a )

1 2a

x + at

x at

∫

Ψ (ξ ) dξ =x + at

1 2a

x + at

∫0

Ψ (ξ ) dξ +0

1 2a

x at

∫

0

Ψ (ξ ) dξ = 1 2ax + at

1 2a

x + at

∫0

ψ (ξ ) dξ +10

1 2a

x at

∫ ψ ( ξ ) dξ

0

=

1 2a

∫ ψ (ξ ) dξ + 2 a ∫0

1

ψ ( ξ ) d ( ξ ) =

( x at )

∫ ψ (ξ ) dξ + 2 a ∫ ψ (ξ ) d (ξ )0 at x

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