w(c,)=f0+L5()-f] (x,)=(,+f)+L5)-] →，-iyn=4,-dle-e,-iwa)=-e,-aiw） =叫-w=p()-f0)-50)-f0]=() 以=4-=y0)-L0)-f]=(c) o=0,y以=0 26
26 ( ) ( ) ( ) ( ) 1 2 1 , x w x t f t f t f t l = + − ( ) ( ) ( ) ( ) ( ) 1 2 1 , , x u x t v x t f t f t f t l = + + − ( ) ( ) 2 2 2 2 tt xx tt xx tt xx tt xx v a v u a u w a w w a w − = − − − = − − ( ) ( ) ( ) ( ) ( ) 0 0 0 1 2 1 0 0 0 t t t x v u w x f f f x l = = = = − = − − − = ( ) ( ) ( ) ( ) ( ) ' ' ' 1 2 1 0 0 0 0 0 0 t t t t t t x v u w x f f f x l = = = = − = − − − = 0 0, 0 x x l v v = = = =
y.-a2y=-w,-aiw） 因此 =(x),e=Ψ(x) (0<x<l,t>0) 以e=0,ye=0 第二类非齐次边界条件：u=f),4=f() 可以设：w(,)=A()x2+B()x 第三类非齐次边界条件 (u-加=f),(a+hm以e=f(④ 可以设： w(,)=A()x+B() 27
27 (0 , 0 x l t ) ( ) ( ) ( ) 2 2 0 0 0 , 0, 0 tt xx tt xx t t t x x l v v w w v x v x v v = = = = − = − − = = = = 因此 第二类非齐次边界条件： ( ) ( ) 1 2 0 , x x x x l u f t u f t = = = = 可以设： ( ) ( ) ( ) 2 w x t A t x B t x , = + 第三类非齐次边界条件： ( ) ( ) ( ) ( ) 1 2 0 , x x x x l u hu f t u hu f t = = − = + = 可以设： w x t A t x B t ( , ) = + ( ) ( )
例弦的一端固定，另一端作谐振动Asin@t,弦的 初始位移和初始速度为零. un -a'uss =0 4g=0,4o=0 (0<x<l,t>0) ux=o =0,ux==Asinot 【解】 设(x,t)=(,)+w(x,) w(x,t)=f(t)+f(t)-f(t)=Asin@t -。=-gie.-on@侧月片onam 28
28 例 弦的一端固定，另一端作谐振动 ，弦的 初始位移和初始速度为零. A t sin 2 0 0 0 0 0, 0 0, sin tt xx t t t x x l u a u u u u u A t = = = = − = = = = = (0 , 0 x l t ) 【解】 设 u x t v x t w x t ( , , , ) = + ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 , sin x x w x t f t f t f t A t l l = + − = ( ) 2 2 2 2 sin sin tt xx tt xx x x v a v w a w A t A t l l − = − − = − − =
w(,=f()+汇5()-f(]=艺Asinot 。-vs=-(w,-aiw)=-气（4 osinat=54o2 sinot =4-w=0-0=0 儿=-0-40=月 以n=0,以-1=0 Va-a'va= ~Ao'sin ot 0 (0<x<l,t>0) x0=0, y以x1=0 29
29 ( ) ( ) ( ) ( ) 1 2 1 , sin x x w x t f t f t f t A t l l = + − = ( ) 2 2 2 2 tt xx tt xx sin sin x x v a v w a w A t A t l l − = − − = − − = 0 0 0 0 t t t t t t x x v u w A A l l = = = = − = − = − 0 0 0 0 0 0 t t t v u w = = = = − = − = 0 0, 0 x x l v v = = = = (0 , 0 x l t ) 2 2 0 0 0 sin 0, 0, 0 tt xx t t t x x l x v a v A t l x v v A l v v = = = = − = = = − = =
可以选择（化，）使得泛定方程、边界条件同时具备齐次条件. sinx w(x,t)=A-asinot w0w=Asinot sin-l a w,-aws=-Aoa sinot+a.4sm- 0-=0, sin →男，-。=-(w。-twa)=0,a=4a-州ln=0 Va-a'vs=O sin -x lmo-W=-40- →y以=0，w=-Ao sin- e lk0=0,=0 30
30 可以选择w (x, t)使得泛定方程、边界条件同时具备齐次条件. ( ) sin , sin sin x a w x t A t l a = 0 0, sin x x l w w A t = = = = 2 2 2 2 2 sin sin sin 0, sin sin tt xx x x a a w a w A t a A a l l a a − = − + = ( ) 2 2 0, tt xx tt xx v a v w a w − = − − = 0 x l x l x l v u w = = = = − = 0 0 0 sin sin t t t x a v u w A l a = = = = − = − 2 0 0 0 0 sin 0, sin 0, 0 tt xx t t t x x l v a v x a v v A l a v v = = = = − = = = − = =