89.2二维线性波动080808080808388888微幅波速度场adacosh[k(z + d)]ag I cos(kx - ot)uaxax0cosh kdagk cosh[k(z + d)]sin(kx - ot)cosh kd0adaag cosh[k(z + d)]/cos(kx - ot)WazOzcosh kd0agksinh[k(z +d)]0cos(kx - ot)cosh kd0R18BeeeccndGngineeing
College of Marine Science and Engineering 18 微幅波速度场 ( ) ( ) ( ) ( ) cosh[ ( )] [ cos ] cosh cosh[ ( )]sin cosh cosh[ ( )] [ cos ] cosh sinh[ ( )] cos cosh ag k z d u kx t x x kd agk k z d kx t kd ag k z d w kx t z z kd agk k z d kx t kd + = = − − + = − + = = − − + = − − §9.2 二维线性波动
89.2二维线性波动1808080808388188质点运动轨迹dxagk cosh[k(z+d)] sin(kx - ot)dtcoshkd0dzagk sinh[k(z+d)]cos(kx -ot)dtcosh kd0由于是小振幅，所以在积分时，被积函数中的坐标可用平衡位置的坐标近似代替，所以有：dx =[agk cosh[k(2。 +d)], sin (kx -ot)]dtcoshkd0agk sinh[k(o + d) cos(kx-ot)]dtdz:coshkd0agk cosh[k(zo + d)]cosh[k(zo +d)]cos (kx。-ot)cos(kx。- otX-Xo1?x-Xo=o?cosh kdsinh kd02=kgthkdagk sinh[k(zo + d)]sinh[k(zo +d)]sin(kx - ot)sin(kx-ot)Z-Zoz-zo=ao?cosh kdsinh kdABoue19incerin
College of Marine Science and Engineering 19 质点运动轨迹 ( ) ( ) cosh[ ( )]sin cosh sinh[ ( )] cos cosh dx agk k z d u kx t dt kd dz agk k z d w kx t dt kd + = = − + = = − − 由于是小振幅，所以在积分时，被积函数中的坐标可用平衡位置 的坐标近似代替，所以有： ( ) ( ) ( ) ( ) 2 0 0 0 0 0 0 0 0 0 2 0 0 0 2 cosh[ ( )] [ sin ] cosh sinh[ ( )] [ cos ] cosh cosh[ ( )] cosh[ ( )] cos cosh sinh sinh[ ( )]sin cosh kgthkd agk k z d dx kx t dt kd agk k z d dz kx t dt kd agk k z d k z d x x kx t x x a kd agk k z d z z kx t kd = + = − + = − − + + − = − − = ⎯⎯⎯⎯→ + − = − ( ) ( ) 0 0 0 0 cos sinh[ ( )]sin sinh kx t kd k z d z z a kx t kd − + − = − §9.2 二维线性波动
89.2二维线性波动0质点运动轨迹(x-x)(z-zo)2x-xoab2cosh k(zo +d)sinh k(zo +d)[a-[asinh kdsinh kd上式表明：水质点的运动轨迹为椭圆，椭圆的水平轴和铅直轴随着离开自由表面向下而逐渐减小，于水底处，铅直轴变为零，质点只做水平运动。质点走向：若波自左向右传播，则均为顺时针，如图所示。椭圆的短轴就是表面波的振幅。OBouege20nineen
College of Marine Science and Engineering 20 上式表明：水质点的运动轨迹为椭圆，椭圆的水平轴和铅直轴 随着离开自由表面向下而逐渐减小，于水底处，铅直轴变为零， 质点只做水平运动。 质点走向：若波自左向右传播，则均为顺时针，如图所示。 椭圆的短轴就是表面波的振幅。 ( ) ( ) ( ) ( ) 2 2 2 2 0 0 0 0 2 2 0 0 2 2 1 1 cosh ( ) sinh ( ) [ ] [ ] sinh sinh x x z z x x z z k z d k z d a b a a kd kd − − − − + = + = + + 质点运动轨迹 §9.2 二维线性波动
89.2二维线性波动0808083838080808080808388微幅波压力场ad把= _ cosh[k(z+ d)]cos(kx-ot)代入P-Pogz得atcoshkd0p[- ag cosh[k(2+d)]I cos(kx-ot)cosh kd0p=po-ppgzatcosh[k(z + d)]5 - pgz= pg(k. -2)p. =p-po =pgcosh kdzAn(xt)Dynamic pressureDynamicpressureStaticpressureStatic pressureABoueg21igineening
College of Marine Science and Engineering 21 ( ) ( ) 0 0 0 cosh[ ( )] cos cosh cosh[ ( )] [ cos ] cosh cosh[ ( )] ( ) cosh z z ag k z d p p kx t gz kd t ag k z d kx t kd p p gz t k z d p p p g gz g k z kd + − = − − = − − + − − = − − + = − = − = − 把 代入 得 微幅波压力场 Static pressure Dynamic pressure η (x,t) Static pressure Dynamic pressure Z X §9.2 二维线性波动
89.2二维线性波动80808080808083888水深的影响>深水波（d>L/2）2元d→8kd :元ktanh kd 1, sinh kd cosh kd 2所以0? = gk tanh kd →? = gk8T2gT2tanh kd -→ L = L2元2元gTgTtanh kd 2元2元(x-x)?Z2 +(z-z0)~ =(ae't)(x-xcosh k(zo +d)sinh k(zo +d)C[asinh kdsinh kdABole22gineeing
College of Marine Science and Engineering 22 水深的影响 ( ) ( ) ( ) ( ) ( ) 0 2 2 2 2 2 2 2 0 0 2 2 0 0 0 0 2 2 2 1 tanh 1,sinh cosh 2 tanh tanh 2 2 tanh 2 2 1 cosh ( ) sinh ( ) [ ] [ ] sinh sinh kd kz kd d kd kd kd e gk kd gk gT gT L kd L gT gT c kd c x x z z x x z z ae k z d k z d a a kd kd = → = → = = → = = → = − − + = → − + − = + + 所以 ➢ 深水波（d>L/2） §9.2 二维线性波动