數(shù)學(xué)物理方程習(xí)題解答案

理解習(xí)題一1，驗證下面兩個函數(shù)：u)ln

1y

u(x,y

x

siny

uxxu(,

12yx

y

)

(

y

)

x

xy

x(22)(x2y2)

y

(2

)

x

yy

x

y2y22(y2)(x2y2)2

x22(22)2(x)

u(,ln

12y

xx

yy

uysinyusinx

x

uyxx

xux,uyyyyuxsinxsinu)

x

sinxxyy

uf()()

uuuxyxy

f

證明：因為uf()g(y)ugy)x

f(xy

ufxyuuu(x)(y)xy

g

()f

x)

u(x,)(

xy)

uuxx

(y)f(

)

ufx

u

xx

f

uxyyyyff

-4

1

2

u(,y)fx),)fy)122

f,1

2

u(,(3)fy)1

4，試導(dǎo)出均勻等截面的彈性桿作微小縱振動的運動方程（略去空氣的阻力和桿的重量解性桿的假設(shè)直于桿的每一個截面上的每一點受力與位移的情形都是相同的，取桿的左端截面的形心為原點，桿軸為

[,x

(x,(xt)

(x

(xt(x(t)

SE(x

(x)(x(x)

xx2tt13t2ttx2tx11

(x,t

(xt)xtsExxxsx()[x]

x

E

u2

EEldQ()dSdt1C,Ku

x2

2

1

t

(u1

ldxdtx12Q2

t

2K(t)dtQ

t

K(x)dtQ12

x

K(t)K(u4tx

xxctxtxt11tttttxxctxtxt11tttttt

)(t)21tx

tx

c

2

t

[

c(u]44

x,x,,121

2

k4k1(u)c

t

(,yz)

u(xyzt)

u

(y)dsdt

u

dt

D(xz)

m

tt

D[(D)()()]

t

D(

)dxdydzdt

t1

t

2

[u(yzt)(y,z,t)]2t

dtdxdydzt

D()dxdydzdtdtdxdydzt

tl0xtl0xt

[D(

)]dxdydzdtt12D()

l

x

x

utt

u,0xxx

u

x

0,

x

u

,uttt

l

度quxttxx

(l)2

u

x

u

xx

qtu

t

l)2

,0xl處ux00u(x

u2u2ut

a

2

kc

b

2

4k

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xt

,()

u(xt)1

u)2

u,0x,t,0xttttt0,u(x),tttttt2t),tuu1xx

u1

2

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fi

u

u

x,

u

2

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u

xx

i

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0,t)x

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t

()

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f

f

f

u

t

u

x,t

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u1

2

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ii

ut

ux

u

ut

x

u

t

tt

u12習(xí)題二1，

用分離變量法解齊次弦振動方程Utt

U

xx

0

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l

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X(x

t)

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TaT(t)(x)

X(xTt)

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a

tX

X(xU(0,)T()U(lt)(l)T()

t)

X(l)

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X(x)

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r2r

()

X(0)(l)

A

1ee

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rr

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sin

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0)k

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k

l

),k

X

k

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nl

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n

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X(x)

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X(l)

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X)X(0)(l

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()cos

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A

sin

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l

)2,k1,2,

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l

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t)

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