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 哈尔滨工程大学学报  2020, Vol. 41 Issue (2): 161-165  DOI: 10.11990/jheu.201905128 0

### 引用本文

YANG Shi'e. Characteristics of sound propagation in sea with 3-dimensional irregular elastic bottom[J]. Journal of Harbin Engineering University, 2020, 41(2): 161-165. DOI: 10.11990/jheu.201905128.

### 文章历史

1. 哈尔滨工程大学 水声技术重点实验室, 黑龙江 哈尔滨 150001;
2. 海洋信息获取与安全工业和信息化部重点实验室(哈尔滨工程大学), 黑龙江 哈尔滨 150001;
3. 哈尔滨工程大学 水声工程学院, 黑龙江 哈尔滨 150001

Characteristics of sound propagation in sea with 3-dimensional irregular elastic bottom
YANG Shi'e 1,2,3
1. Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China;
2. Key Laboratory of Marine Information Acquisition and Security(Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China;
3. College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Related formulae for solving sound propagation in sea with three-dimensional elastic bottom are given, and the result for one special case had been solved. It shows the horizontal diffraction of sound wave due to inclined bottom, and characteristics of the interface wave.
Keywords: successive approximation    three-dimensional(3D)    elastic ocean bottom    inclined bottom    horizontal diffraction    the interface wave    sound wave propagation    wave equation

1 基本思路与公式

 ${\nabla ^2}{\varphi _1} + k_1^2{\varphi _1} = - 4{\rm{ \mathsf{ π} }}\delta \left( {0,0,z - {z_s}} \right)$ (1)
 ${\nabla ^2}{\varphi _2} + k_2^2{\varphi _2} = 0$ (2)
 $\nabla \times \nabla \times \mathit{\boldsymbol{\psi }} - {\chi ^2}\mathit{\boldsymbol{\psi }} = 0$ (3)
 $\nabla \cdot \mathit{\boldsymbol{\psi }} = 0$ (4)

 $\frac{\mathit{\boldsymbol{T}}}{{ - {\rm{i}}\omega }} = \left[ {\begin{array}{*{20}{l}} {{\tau _{xx}}}&{{\tau _{xy}}}&{{\tau _{xz}}}\\ {{\tau _{yx}}}&{{\tau _{yy}}}&{{\tau _{yz}}}\\ {{\tau _{zx}}}&{{\tau _{zy}}}&{{\tau _{zz}}} \end{array}} \right]$ (5)

 $\left\{ {\begin{array}{*{20}{l}} {{M_1} = \frac{{\partial {\varphi _2}}}{{\partial x}} + \frac{{\partial {\psi _z}}}{{\partial y}} - \frac{{\partial {\psi _y}}}{{\partial z}}}\\ {{M_2} = \frac{{\partial {\varphi _2}}}{{\partial y}} + \frac{{\partial {\psi _x}}}{{\partial z}} - \frac{{\partial {\psi _z}}}{{\partial x}}}\\ {{M_3} = \frac{{\partial {\varphi _2}}}{{\partial z}} + \frac{{\partial {\psi _y}}}{{\partial x}} - \frac{{\partial {\psi _x}}}{{\partial y}}} \end{array}} \right.$ (6)

 $\left\{ \begin{array}{l} {\tau _{xx}} = - \lambda k_2^2{\varphi _2} + 2\mu \frac{{\partial {M_1}}}{{\partial x}}\\ {\tau _{{\rm{yy}}}} = - \lambda k_2^2{\varphi _2} + 2\mu \frac{{\partial {M_2}}}{{\partial y}}\\ {\tau _{zz}} = - \lambda k_2^2{\varphi _2} + 2\mu \frac{{\partial {M_3}}}{{\partial z}}\\ {\tau _{yx}} = {\tau _{xy}} = \mu \left\{ {\frac{{\partial {M_1}}}{{\partial x}} + \frac{{\partial {M_2}}}{{\partial y}}} \right\}\\ {\tau _{zy}} = {\tau _{yz}} = \mu \left\{ {\frac{{\partial {M_2}}}{{\partial y}} + \frac{{\partial {M_3}}}{{\partial z}}} \right\}\\ {\tau _{zx}} = {\tau _{xz}} = \mu \left\{ {\frac{{\partial {M_3}}}{{\partial z}} + \frac{{\partial {M_1}}}{{\partial x}}} \right\} \end{array} \right.$ (7)

 $X = \varepsilon x,Y = \varepsilon y,Z = z$ (8)

 $\left\{ {\begin{array}{*{20}{l}} {{\phi _1} = {{\rm{e}}^{{\rm{i}}\frac{{W\left( {X,Y} \right)}}{\varepsilon }}}\sum\limits_{m = 0}^\infty {{A_m}} (X,Y,Z){{({\rm{i}}\varepsilon )}^m}}\\ {{\phi _2} = {{\rm{e}}^{{\rm{i}}\frac{{W\left( {X,Y} \right)}}{\varepsilon }}}\sum\limits_{m = 0}^\infty {{B_m}} (X,Y,Z){{({\rm{i}}\varepsilon )}^m}}\\ {\vec \psi = {{\rm{e}}^{{\rm{i}}\frac{{W\left( {X,Y} \right)}}{\varepsilon }}}\sum\limits_{m = 0}^\infty {{\mathit{\boldsymbol{C}}_m}} (X,Y,Z){{({\rm{i}}\varepsilon )}^m}} \end{array}} \right.$ (9)

 $\left\{ {\begin{array}{*{20}{l}} {{f_x} = \frac{{\partial f}}{{\partial x}},{f_y} = \frac{{\partial f}}{{\partial y}}}\\ {\sigma = {{\left( {1 + f_x^2 + f_y^2} \right)}^{1/2}}}\\ {{w_x} = \frac{{\partial W}}{{\partial X}},{w_y} = \frac{{\partial W}}{{\partial Y}}} \end{array}} \right.$ (10)

 ${\left( {{w_x}} \right)^2} + {\left( {{w_y}} \right)^2} = {\xi ^2}$ (11)

 $\begin{array}{*{20}{c}} { - {\xi ^2}{A_0} + {A_{0zz}} + k_1^2{A_0} = - 4{\rm{ \mathsf{ π} }}\delta \left( {0,0,Z - {Z_s}} \right)}\\ { - {\xi ^2}{A_1} + {A_{1zz}} + k_1^2{A_1} = - {\xi ^2}{A_0} - 2\left( {{w_x}{A_{0x}} + {w_y}{A_{0y}}} \right)}\\ { \cdots \cdots ,} \end{array}$ (12)
 $\begin{array}{*{20}{c}} { - {\xi ^2}{B_0} + {B_{0zz}} + k_2^2{B_0} = 0}\\ { - {\xi ^2}{B_1} + {B_{1zz}} + k_2^2{B_1} = - {\xi ^2}{B_0} - 2\left( {{w_x}{B_{0x}} + {w_y}{B_{0y}}} \right)}\\ { \cdots \cdots ,} \end{array}$ (13)
 $\begin{array}{*{20}{c}} { - {\xi ^2}{\mathit{\boldsymbol{C}}_0} + {\mathit{\boldsymbol{C}}_{0zz}} + {\chi ^2}{\mathit{\boldsymbol{C}}_0} = 0}\\ { - {\xi ^2}{\mathit{\boldsymbol{C}}_1} + {\mathit{\boldsymbol{C}}_{1zz}} + {\chi ^2}{\mathit{\boldsymbol{C}}_1} = - {\xi ^2}{\mathit{\boldsymbol{C}}_0} - 2\left( {{w_x}{\mathit{\boldsymbol{C}}_{0x}} + {w_y}{\mathit{\boldsymbol{C}}_{0y}}} \right)}\\ { \cdots \cdots ,} \end{array}$ (14)
 $\begin{array}{*{20}{c}} {{w_x}{C_{x0}} + {w_z}{C_{y0}} + {C_{z0z}} = 0}\\ { \cdots \cdots ,} \end{array}$ (15)

 $\mathit{\boldsymbol{n}} = \frac{{\left( {{f_x}\mathit{\boldsymbol{i}} + {f_y}\mathit{\boldsymbol{j}} - \mathit{\boldsymbol{k}}} \right)}}{\sigma }$ (16)

 $\nabla {\phi _1} \cdot \mathit{\boldsymbol{n}} = \left( {\nabla {\phi _2} + \nabla \times \mathit{\boldsymbol{\psi }}} \right) \cdot \mathit{\boldsymbol{n}}$ (17)
 ${\rm{i}}\omega {\rho _1}{\phi _1} = \left( {\mathit{\boldsymbol{T}} \cdot \mathit{\boldsymbol{n}}} \right) \cdot \mathit{\boldsymbol{n}}$ (18)
 $\left( {\mathit{\boldsymbol{T}} \cdot \mathit{\boldsymbol{n}}} \right) \times \mathit{\boldsymbol{n}} = 0$ (19)

 ${A_0} = \left\{ \begin{array}{l} D{S_0}V_0^{1/3}{N_0},\;\;\;\;0 \le Z < {Z_s}\\ D{S_0}V_0^{1/3}{N_0} + P\left( {{V_0}} \right),\;\;\;\;{Z_s} \le Z < h\\ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over S} }_0}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} _0^{1/3}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over N} }_0} + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0}} \right),\;\;\;\;h \le Z < f \end{array} \right.$ (20)

 $\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{l}} {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over D} = D{{\rm{e}}^{{\rm{i}}\left( {{v_{0h}} - {v_{1h}}} \right)}}}\\ {{N_0} = {\rm{H}}_{1/3}^{(1)}\left( {{V_0}} \right){\rm{H}}_{1/3}^{(2)}\left( {{V_{00}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{V_{00}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{V_0}} \right)}\\ {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over N} }_0} = {\rm{H}}_{1/3}^{(1)}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0}} \right){\rm{H}}_{1/3}^{(2)}\left( {{V_{00}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{V_{00}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0}} \right)} \end{array}} \right.}\\ {\left\{ \begin{array}{l} P\left( {{V_0}} \right) = \frac{{{\rm{ \mathsf{ π} i}}}}{2}{\left( {\frac{{{V_0}{V_{0s}}}}{{{Q_0}{Q_{0s}}}}} \right)^{1/2}}{N_{0s}}\\ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0}} \right) = \frac{{{\rm{ \mathsf{ π} i}}}}{2}{{\rm{e}}^{{\rm{i}}\left( {{v_{0h}} - {v_{1h}}} \right)}}{\left( {\frac{{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0}{V_{0s}}}}{{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over Q} }_0}{Q_{0{\rm{s}}}}}}} \right)^{1/2}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over N} }_{0{\rm{s}}}} \end{array} \right.} \end{array}$ (21)
 ${S_0} = \frac{{V_0^{1/6}}}{{Q_0^{1/2}}},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over S} }_0} = \frac{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} _0^{1/6}}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over Q} _0^{1/2}}}$ (22)
 ${Q_0} = \sqrt {k_1^2 - {\xi ^2}} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over Q} }_0} = \sqrt {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over k} _1^2 - {\xi ^2}}$ (23)
 $\left\{ {\begin{array}{*{20}{l}} {{V_0}(Z) = \int {{Q_0}} (Z){\rm{d}}Z}\\ {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0}(Z) = \int {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over Q} }_0}} (Z){\rm{d}}Z} \end{array}} \right.$ (24)

 ${B_0} = FS_0^\prime V{'}_0^{1/3}H_{1/3}^{(1)}\left( {V_0^\prime } \right),S_0^\prime = \frac{{V{'}_0^{1/6}}}{{Q{'}_0^{1/2}}}$ (25)
 $Q_0^\prime = \sqrt {k_2^2 - {\xi ^2}} ,V_0^\prime = \int {Q_0^\prime } {\rm{d}}Z$ (26)
 ${\mathit{\boldsymbol{C}}_0} = \mathit{\boldsymbol{G}}S_0^{\prime \prime }V_0^{\prime \prime 1/3}{\rm{H}}_{1/3}^{(1)}\left( {V_0^{\prime \prime }} \right),S_0^{\prime \prime } = \frac{{V_0^{\prime \prime 1/6}}}{{Q_0^{\prime \prime 1/2}}}$ (27)
 $Q_0^{\prime \prime } = \sqrt {{\chi ^2} - {\xi ^2}} ,V_0^{\prime \prime } = \int {Q_0^{\prime \prime }} {\rm{d}}Z$ (28)
 ${w_x}{G_{x0}} + {w_y}{G_{y0}} + Q_0^{\prime \prime }{G_{z0}} = 0$ (29)

 ${\left( {\frac{{\partial W}}{{\partial X}}} \right)^2} + {\left( {\frac{{\partial W}}{{\partial Y}}} \right)^2} = {\xi ^2}$ (30)

 ${\tau _{xx}} = F{U_x} - 2\mu \frac{{{w_x}}}{{{Q^{\prime \prime }}}}{P_y}$ (31)
 ${\tau _{yy}} = F{U_y} + 2\mu \frac{{{w_y}}}{{{Q^{\prime \prime }}}}{P_x}$ (32)
 ${\tau _{zz}} = F\left( {\lambda k_2^2 + 2\mu {{Q'}^2}} \right) - 2\mu {Q^{\prime \prime }}\kappa$ (33)
 ${\tau _{xy}} = \mu F{\xi ^2} + \mu {Q^{\prime \prime }}\kappa$ (34)
 ${\tau _{yz}} = \mu F{u_y} + \mu \frac{{{w_x}}}{{{Q^{\prime \prime }}}}{P_y}$ (35)
 ${\tau _{zx}} = \mu F{u_x} - \mu \frac{{{w_y}}}{{{Q^{\prime \prime }}}}{P_x}$ (36)

 $\left\{ \begin{array}{l} {U_x} = \lambda k_2^2 + 2\mu w_x^2,\;\;\;\;{U_y} = \lambda k_2^2 + 2\mu w_y^2\\ \begin{array}{*{20}{l}} {{u_x} = {{Q'}^2} + w_x^2,{u_y} = {{Q'}^2} + w_y^2}\\ {{P_x} = {G_y}{w_x}{w_y} + {G_x}\left( {{Q^{\prime \prime 2}} + w_x^2} \right)}\\ {{P_y} = {G_x}{w_x}{w_y} + {G_y}\left( {{Q^{\prime \prime 2}} + w_y^2} \right)}\\ {\kappa = {G_x}{w_y} - {G_y}{w_x}} \end{array} \end{array} \right.$ (37)
 ${\Re _{11}} = \mathit{\Omega }\sin \left( {v - {v_0}} \right) - {Q_{0f}}\cos \left( {v - {v_0}} \right)$ (38)
 ${\Re _{12}} = \left( {{f_x}{w_x} + {f_y}{w_y} - Q_{0f}^\prime } \right)$ (39)
 ${\Re _{13}} = - \frac{1}{{{Q^{\prime \prime }}}}\left( {{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \Omega } }}_2} - {{\bar \omega }_1}} \right)$ (40)
 ${\Re _{14}} = - \frac{1}{{{Q^{\prime \prime }}}}\left( {{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \Omega } }}_1} - {{\bar \omega }_1}} \right)$ (41)
 ${\Re _{21}} = {\omega ^2}{\rho _1}\sin \left( {v - {v_0}} \right)$ (42)
 ${\Re _{22}} = \lambda k_2^2{\sigma ^2} + 2\mu {\zeta _ - }{\mathit{\Omega }^\prime }$ (43)
 ${\Re _{23}} = - \frac{{2\mu {w_y}{\zeta _ - }}}{{{Q^{\prime \prime }}}}\left( {\mathit{\Omega }_1^{\prime \prime } - \bar \omega _1^\prime } \right)$ (44)
 ${\Re _{24}} = - \frac{{2\mu {w_x}{\zeta _ + }}}{{{Q^{\prime \prime }}}}\left( {\mathit{\Omega }_2^{\prime \prime } - \bar \omega _2^\prime } \right)$ (45)
 ${\Re _{31}} = 0$ (46)
 ${\Re _{32}} = \mu \left( {{\zeta _ - }{\vartheta ^\prime } + {F_x}\vartheta _1^\prime } \right)$ (47)
 ${\Re _{33}} = - \frac{{\mu {w_y}}}{{{Q^{\prime \prime }}}}\left\{ {{\gamma _1}{Q^{\prime \prime 2}} + {\zeta _1}w_x^2} \right\}$ (48)
 ${\Re _{33}} = - \frac{{\mu {w_y}}}{{{Q^{\prime \prime }}}}\left\{ {{\gamma _1}{Q^{\prime \prime 2}} + {\zeta _1}w_y^2} \right\}$ (49)
 ${\Re _{41}} = 0$ (50)
 ${\Re _{42}} = \mu \left\{ {{\zeta _ - }{\vartheta ^{\prime \prime }} + {F_y}\vartheta _1^{\prime \prime }} \right\}$ (51)
 ${\Re _{43}} = - \frac{{\mu {w_y}}}{{{Q^{\prime \prime }}}}\left\{ {{\gamma _3}{Q^{\prime \prime 2}} - \left( {{f_y}{\zeta _ - } - {F_y}} \right)w_x^2} \right\}$ (52)
 ${\Re _{44}} = \frac{{\mu {w_x}}}{{{Q^{\prime \prime }}}}\left\{ {{\gamma _4}{Q^{\prime \prime 2}} + {\zeta _2}w_y^2} \right\}$ (53)

 $\mathit{\Omega } = {f_x}{w_x} + {f_y}{w_y},{\mathit{\Omega }^\prime } = {f_x}w_x^2 + {f_y}w_y^2 - {{Q'}^2}$
 ${{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \Omega } }}_1} = {w_x}\left( {{f_y}{w_y} - {Q^{\prime \prime }}} \right),{{\mathit{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \Omega } }}_2} = {w_y}\left( {{f_x}{w_x} - {Q^{\prime \prime }}} \right)$
 $\mathit{\Omega }_1^{\prime \prime } = \left( {{f_x} - {f_y}} \right)w_x^2,\mathit{\Omega }_2^{\prime \prime } = \left( {{f_x} - {f_y}} \right)w_y^2$
 ${{\bar \omega }_1} = {f_x}\left( {{Q^{\prime \prime 2}} + w_y^2} \right),{{\bar \omega }_2} = {f_y}\left( {{Q^{\prime \prime 2}} + w_x^2} \right)$
 $\bar \omega _1^\prime = \left( {{f_x} - 1} \right){Q^{\prime \prime 2}},\bar \omega _2^\prime = \left( {{f_y} + 1} \right){Q^{\prime \prime 2}}$
 ${\vartheta ^\prime } = {f_x}{Q^{\prime 2}} + w_x^2,\vartheta _1^\prime = {f_x}w_x^2 + {f_y}w_y^2 - {Q^{\prime 2}}$
 ${\vartheta ^{\prime \prime }} = {f_y}{Q^{\prime \prime 2}} + w_y^2,\vartheta _1^{\prime \prime } = {f_x}w_x^2 + {f_y}w_y^2 - {Q^{\prime \prime 2}}$
 ${\gamma _1} = f_x^2 - {f_x} - {\zeta _ + },{\gamma _2} = 4{f_x} + {f_y} - {f_x}{f_y}$
 ${\gamma _3} = {f_x}{f_y} - {f_x} - 4{f_y},{\gamma _4} = f_y^2 - {f_y} - {\zeta _ + }$
 ${\zeta _1} = f_x^2 + 2{f_x} - {f_x}{f_y} - 1,$
 ${\zeta _2} = f_y^2 + 2{f_y} - {f_x}{f_y} - 1$
2 仿真算例

 ${c_1} = \left\{ {\begin{array}{*{20}{l}} {\frac{{1500}}{{{c_{11}}}},}&{0 \le Z < 1000{\rm{m}}}\\ {\frac{{1500}}{{\sqrt {1.1} }}{c_{12}},}&{Z > 1000{\rm{m}}} \end{array}} \right.$

 ${c_{11}}(Z) = \sqrt {1 + {{10}^{ - 4}}Z[1 + 10\alpha (1000 - Z)]}$
 $\alpha = \sin \frac{{30}}{{1 + {{(X - 1)}^2}{{(Y - 2)}^2}/{{10}^4}}}$
 ${c_{12}}(Z) = \left[ {1 + 0.4 \times {{10}^{ - 4}}(Z - 1000)} \right]$

 $Z = 1200 + 1000\tanh (X/100 + Y/70)$

 ${\rho _2} = 1.5(1 + 0.0005Z)$

 $\lambda = 2 \times {10^4}\left( {1 + {{10}^{ - 5}}X + {{10}^{ - 5}}Y} \right),$
 $\mu = 3 \times {10^6}\left( {1 + {{10}^{ - 5}}X - {{10}^{ - 6}}Y} \right),$

 $Z < 1000,$
 $\begin{array}{*{20}{c}} {{V_0} = \int {\sqrt {{{\left( {\frac{{\rm{ \mathsf{ π} }}}{{7.5}}} \right)}^2}{c_{11}} - {\xi ^2}{\rm{d}}Z} } = }\\ {\int {\sqrt {a + bZ + c{Z^2}} } {\rm{d}}Z = \frac{{(2cZ + b)}}{{4c}}\sqrt {a + bZ + c{Z^2}} + }\\ {\left\{ {\begin{array}{*{20}{l}} { - \frac{{4ac - {b^2}}}{{{{(2\sqrt { - c} )}^3}}}\arcsin \frac{{2cZ + b}}{{\sqrt {{b^2} - 4ac} }}}&{c < 0}\\ {\frac{{4ac - {b^2}}}{{{{(2\sqrt c )}^3}}}\ln (2\sqrt {cR} + 2cZ + b)}&{c > 0} \end{array}} \right.} \end{array}$

 $a = {\left( {\frac{{\rm{ \mathsf{ π} }}}{{7.5}}} \right)^2} - {\xi ^2},$
 $b = {\left( {\frac{{\rm{ \mathsf{ π} }}}{{7.5}}} \right)^2}\left( {{{10}^{ - 4}} + \alpha } \right),$
 $c = - {\left( {\frac{{\rm{ \mathsf{ π} }}}{{7.5}}} \right)^2} \times {10^{ - 3}}\alpha ;$
 $Z > 1000,\;令\;u = 0.96 + 0.4 \times {10^{ - 4}}Z$
 $\begin{array}{l} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over V} }_0} = \int {\sqrt {\frac{{0.1930}}{{{{\left( {0.96 + 0.4 \times {{10}^{ - 4}}Z} \right)}^2}}} - {\xi ^2}} } {\rm{d}}Z = \\ 5 \times {10^4}\int {\frac{{u{\rm{d}}u}}{{\sqrt {0.1930 - {\xi ^2}{u^2}} }}} = \\ - \frac{{2.5 \times {{10}^4}}}{{{\xi ^2}}}\sqrt {0.1930 - {\xi ^2}{{\left( {0.96 + 0.4 \times {{10}^{ - 4}}Z} \right)}^2}} \end{array}$

 $\begin{array}{l} k_2^2 = \frac{{{\rho _2}{\omega ^2}}}{{\lambda + 2\mu }} - \frac{3}{4}{\left( {\frac{{\partial {\rho _2}}}{{\partial z}}/{\rho _2}} \right)^2} = \\ \frac{{0.0984\left( {1 + 5 \times {{10}^{ - 4}}Z} \right)}}{{1 + {{10}^{ - 2}}X - {{10}^{ - 3}}Y}} - \frac{{1.875 \times {{10}^{ - 7}}}}{{{{(1 + 0.0005Z)}^2}}} \end{array}$
 $\begin{array}{l} {\chi ^2} = \frac{{{\rho _2}{\omega ^2}}}{\mu } - \frac{3}{4}{\left( {\frac{{\partial {\rho _2}}}{{\partial z}}/{\rho _2}} \right)^2} = \\ \frac{{0.1974\left( {1 + 5 \times {{10}^{ - 4}}Z} \right)}}{{1 + {{10}^{ - 2}}X - {{10}^{ - 3}}Y}} - \frac{{1.875 \times {{10}^{ - 7}}}}{{{{(1 + 0.0005Z)}^2}}} \end{array}$

 $\beta = \frac{{0.0984}}{{1 + {{10}^{ - 2}}X - {{10}^3}Y}},$
 $\eta = \frac{{0.1974}}{{1 + {{10}^{ - 2}}X - {{10}^{ - 3}}Y}},$
 ${u^\prime } = 1 + 5 \times {10^{ - 4}}Z;$
 $\begin{array}{l} V_0^\prime = 2000\int {\sqrt {\beta {u^{\prime 3}} - {\xi ^2}{u^{\prime 2}} - 1.875 \times {{10}^{ - 7}}} } \frac{{{\rm{d}}{u^\prime }}}{{{u^\prime }}} \approx \\ 2000\int {\left\{ {\sqrt {\beta {u^{\prime 3}} - {\xi ^2}{u^{\prime 2}}} - \frac{{0.938 \times {{10}^{ - 7}}}}{{\sqrt {{\beta ^2}{u^{\prime 3}} - {\xi ^2}{u^{\prime 2}}} }}} \right\}} \frac{{{\rm{d}}{u^\prime }}}{{{u^\prime }}} = \\ \frac{{4000}}{{3\beta }}{\left( {\beta {u^\prime } - {\xi ^2}} \right)^{\frac{3}{2}}} + \frac{{1.875 \times {{10}^{ - 4}}\sqrt {\beta {u^\prime } - {\xi ^2}} }}{{{\xi ^2}{u^\prime }}} + \\ \frac{{1.875 \times {{10}^{ - 4}}\beta }}{{{\xi ^3}}}\arctan \frac{{\sqrt {\beta {u^\prime } - {\xi ^2}} }}{\xi } \end{array}$

 $\begin{array}{l} V_0^{\prime \prime } = \frac{{4000}}{{3\eta }}{\left( {\eta {u^\prime } - {\xi ^2}} \right)^{3/2}} + \frac{{1.875 \times {{10}^{ - 4}}\sqrt {\eta {u^\prime } - {\xi ^2}} }}{{{\xi ^2}{u^\prime }}} + \\ \frac{{1.875 \times {{10}^{ - 4}}\eta }}{{{\xi ^3}}}\arctan \frac{{\sqrt {\eta {u^\prime } - {\xi ^2}} }}{\xi } \end{array}$

W取二次函数近似，设

 $W = {a_1}X + {a_2}Y + {a_3}{X^2} + {a_4}{Y^2}$

 ${w_x} = {a_1} + 2{a_3}X,$
 ${w_y} = {a_2} + 2{a_4}Y,$
 ${\xi ^2} = {\left( {{a_1} + 2{a_3}X} \right)^2} + {\left( {{a_2} + 2{a_4}Y} \right)^2}$

 ${\varphi _1} = {{\rm{e}}^{{\rm{i}}\frac{{W(x,y)}}{\varepsilon }}}\sum\limits_{m = 0}^\infty {{A_m}} (x,y,z){({\rm{i}}\varepsilon )^m}$ (A1)

Am为不同阶局地简正波。不难直接写出其零阶形式解为：

 $A = \left\{ {\begin{array}{*{20}{l}} {{C_0}{{\left( {\frac{{{v_0}}}{{{Q_0}}}} \right)}^{1/2}}{l_0}}&{0 \le z < {z_s}}\\ {{{\left( {\frac{{{v_0}}}{{{Q_0}}}} \right)}^{1/2}}{E_1}}&{{z_s} < z < h}\\ {{{\left( {\frac{{{v_1}}}{{{Q_1}}}} \right)}^{1/2}}{E_2}}&{h < z} \end{array}} \right.$ (A2)

 ${Q_j} = \sqrt {k_j^2 - {\xi ^2}} ,{v_j} = \int {{Q_j}} {\rm{d}}z,{S_j} = \frac{{v_j^{1/6}}}{{Q_j^{1/2}}},j = 0,1;$
 $\begin{array}{*{20}{c}} {{l_0} = {\rm{H}}_{13}^{(1)}\left( {{v_0}} \right){\rm{H}}_{13}^{(2)}\left( {{v_{00}}} \right) - {\rm{H}}_{13}^{(1)}\left( {{v_{00}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_0}} \right)}\\ {{E_1} = {C_1}{\rm{H}}_{1/3}^{(1)}\left( {{v_0}} \right) + {C_2}{\rm{H}}_{1/3}^{(2)}\left( {{v_0}} \right)}\\ {{E_2} = {C_3}{\rm{H}}_{1/3}^{(1)}\left( {{v_0}} \right) + {C_4}{\rm{H}}_{1/3}^{(2)}\left( {{v_0}} \right)} \end{array}$ (A3)

 $\begin{array}{*{20}{l}} {{C_0}\left\{ {{\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{v_{00}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right)} \right\} = }\\ {{C_1}{\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right) + {C_2}{\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right)} \end{array}$ (A4)
 $\begin{array}{*{20}{l}} {{C_0}\left\{ {{\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0s}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{v_{00}}} \right){\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{0s}}} \right)} \right\} - }\\ {\left\{ {{C_1}{\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0s}}} \right) + {C_2}{\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{{0_s}}}} \right)} \right\} = 2} \end{array}$ (A5)

 $\begin{array}{*{20}{l}} {{S_{0h}}v_{0h}^{1/3}\left\{ {{C_1}{\rm{H}}_{1/3}^{(1)}\left( {{v_{0h}}} \right) + {C_2}{\rm{H}}_{1/3}^{(2)}\left( {{v_{0h}}} \right)} \right\} = }\\ {{S_{1h}}v_{1h}^{1/3}\left\{ {{C_3}{\rm{H}}_{1/3}^{(1)}\left( {{v_{1h}}} \right) + {C_4}{\rm{H}}_{1/3}^{(2)}\left( {{v_{1h}}} \right)} \right\}} \end{array}$ (A6)
 $\begin{array}{*{20}{l}} {{{\left( {{Q_{0h}}{v_{0h}}} \right)}^{1/2}}\left\{ {{C_1}{\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0h}}} \right) + {C_2}{\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{0h}}} \right)} \right\} = }\\ {{{\left( {{Q_{1h}}{v_{1h}}} \right)}^{1/2}}\left\{ {{C_3}{\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{1h}}} \right) + {C_4}{\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{1h}}} \right)} \right\}} \end{array}$ (A7)

$\dot{S}_{0 s}, \dot{S}_{0 h}, \dot{S}_{1 h}$均远小于1上式中一律忽略不计，式(A4)乘以H-2/3(2)(vs)减去式(A5)乘H1/3(2)(vs)，可得到：

 $\begin{array}{l} \left\{ {{C_0}{\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) - {C_1}} \right\}{L_{0s}} = - 2{\left( {{Q_{0s}}{v_{0s}}} \right)^{ - 1/2}}{\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right)\\ {L_{0s}} = {\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right){\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{0s}}} \right) - {\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0s}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right)\\ {C_0}{\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) = {C_1} - \frac{{{\rm{ \mathsf{ π} }}i}}{2}{\left( {\frac{{{v_{0s}}}}{{{Q_{0s}}}}} \right)^{1/2}}{\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right) \end{array}$ (A8)

 $\begin{array}{*{20}{c}} {\left\{ {{C_0}{\rm{H}}_{1/3}^{(1)}\left( {{v_{00}}} \right) - {C_2}} \right\}{L_{0s}} = 2{{\left( {{Q_{0s}}{v_{0s}}} \right)}^{ - 1/2}}{\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right)}\\ {{C_0}{\rm{H}}_{1/3}^{(1)}\left( {{v_{00}}} \right) = {C_2} + \frac{{{\rm{ \mathsf{ π} i}}}}{2}{{\left( {\frac{{{v_{0s}}}}{{{Q_{0s}}}}} \right)}^{1/2}}{\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right)} \end{array}$ (A9)

 ${C_0}{\left( {\frac{{{v_{0h}}}}{{{Q_{0h}}}}} \right)^{1/2}}{l_{0h}} + \frac{{{\rm{ \mathsf{ π} }}i}}{2}{\left( {\frac{{{v_{0h}}{v_{0s}}}}{{{Q_{0h}}{Q_{0s}}}}} \right)^{1/2}}{l_{0hs}} = {\left( {\frac{{{v_{1h}}}}{{{Q_{1h}}}}} \right)^{1/2}}{E_{2h}}$ (A10)
 ${C_0}{\left( {{Q_{0h}}{v_{0h}}} \right)^{1/2}}{L_{0h}} + \frac{{{\rm{ \mathsf{ π} }}i}}{2}{\left( {\frac{{{Q_{0h}}{v_{0h}}{v_{0s}}}}{{{Q_{0s}}}}} \right)^{1/2}}{L_{0hs}} = {\left( {{Q_{1h}}{v_{1h}}} \right)^{1/2}}{E_{41h}}$ (A11)

 ${l_{0hs}} = {\rm{H}}_{1/3}^{(1)}\left( {{v_{0h}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{0h}}} \right)$
 ${L_{0h}} = {\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0h}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{v_{00}}} \right){\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{0h}}} \right)$
 ${L_{0hs}} = {\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0h}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right) - {\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right){\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{0h}}} \right)$
 ${E_{41h}} = {C_3}{\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{1h}}} \right) + {C_4}{\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{1h}}} \right)$

 $\begin{array}{*{20}{c}} {\left\{ {{C_0}{\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) + \frac{{{\rm{ \mathsf{ π} }}i}}{2}{\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right){{\left( {\frac{{{v_{0s}}}}{{{Q_{0s}}}}} \right)}^{1/2}}} \right\} \cdot }\\ {\left. {{\rm{H}}_{1/3}^{(1)}\left( {{v_{0h}}} \right){\rm{H}}_{ - 2/3}^{(2)}\left( {{v_{1h}}} \right) - {\rm{H}}_{ - 2/3}^{(1)}\left( {{v_{0h}}} \right){\rm{H}}_{1/3}^{(2)}\left( {{v_{1h}}} \right)} \right\} = }\\ {{C_3}\frac{{ - 4i}}{{{\rm{ \mathsf{ π} }}\sqrt {{v_{0h}}{v_{1h}}} }}} \end{array}$ (A12)

 ${C_3} \approx \left\{ {{C_0}{\rm{H}}_{1/3}^{(2)}\left( {{v_{00}}} \right) + \frac{{{\rm{ \mathsf{ π} }}i}}{2}{\rm{H}}_{1/3}^{(2)}\left( {{v_{0s}}} \right)\sqrt {\frac{{{v_{0s}}}}{{{Q_{0s}}}}} } \right\}{{\rm{e}}^{{\rm{i}}\left( {{v_{oh}} - {v_{1h}}} \right)}}$

 ${C_4} \approx - \left\{ {{C_0}{\rm{H}}_{1/3}^{(1)}\left( {{v_{00}}} \right) - \frac{{{\rm{ \mathsf{ π} }}i}}{2}{\rm{H}}_{1/3}^{(1)}\left( {{v_{0s}}} \right)\sqrt {\frac{{{v_{0s}}}}{{{Q_{0s}}}}} } \right\}{{\rm{e}}^{{\rm{i}}\left( {{v_{0h}} - {v_{1h}}} \right)}}$

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