物理

基本三维线性波动方程

三维线性波动方程比一维线性波动方程增加一项表示波源耗散的动力学量 $F(x,t)$，写成

$\frac{\partial^2 \xi}{\partial t^2} - u^2 \frac{\partial^2 \xi}{\partial x^2} = F(x,t)$

那么它在二维方向传播时，

$\frac{\partial^2 \xi}{\partial t^2} - u^2 \left( \frac{\partial^2 \xi}{\partial x^2} + \frac{\partial^2 \xi}{\partial y^2} \right) = F(x,y,t)$

在极坐标下，

$\frac{\partial^2 \xi}{\partial t^2} - u^2 \left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \xi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \xi}{\partial \phi^2} \right] = F(r,\phi,t)$

在三维方向传播时，

$\frac{\partial^2 \xi}{\partial t^2} - u^2 \left( \frac{\partial^2 \xi}{\partial x^2} + \frac{\partial^2 \xi}{\partial y^2} + \frac{\partial^2 \xi}{\partial z^2} \right) = F(x,y,z,t)$

在球坐标中，

$\frac{\partial^2 \xi}{\partial t^2} - u^2 \left\{ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \xi}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial \xi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 \xi}{\partial \phi^2} \right\} = F(r,\theta,\phi,t)$

对球对称的系统，$ \xi = \xi(r,t) $，有

$\frac{\partial^2 \xi}{\partial t^2} - u^2 \left[ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \xi}{\partial r} \right) \right] = 0$

利用公式

$\frac{\partial^2 (r\xi)}{\partial r^2} = \frac{1}{r} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \xi}{\partial r} \right)$

得

$\frac{\partial^2 (r\xi)}{\partial t^2} - u^2 \frac{\partial^2 (r\xi)}{\partial r^2} = 0$

将 $ r\xi $ 视为 $ \xi $，其通解为

$r\xi(r,t) = \phi_1(t - \frac{r}{u}) + \phi_2(t + \frac{r}{u})$

得球面波的运动方程

$\xi(r,t) = \frac{1}{r} \left[ \phi_1(t - \frac{r}{u}) + \phi_2(t + \frac{r}{u}) \right]$

简谐球面波

$\xi(r,t) = \frac{f_0}{r} A_0 \cos[\omega(t - \frac{r}{u}) + \varphi]$

可将其分解为三个标量波动方程

$\frac{\partial^2 \vec{\xi}_x}{\partial t^2} - u^2 \left( \frac{\partial^2 \vec{\xi}_x}{\partial x^2} + \frac{\partial^2 \vec{\xi}_x}{\partial y^2} + \frac{\partial^2 \vec{\xi}_x}{\partial z^2} \right) = F_x(x,y,z,t)$

y, z 同上，得

$\frac{\partial^2 \vec{\xi}}{\partial t^2} - u^2 \left( \frac{\partial^2 \vec{\xi}}{\partial x^2} + \frac{\partial^2 \vec{\xi}}{\partial y^2} + \frac{\partial^2 \vec{\xi}}{\partial z^2} \right) = \vec{F}(x,y,z,t)$

引入 $ \nabla $：

$\nabla = \frac{\partial}{\partial x} \hat{i} + \frac{\partial}{\partial y} \hat{j} + \frac{\partial}{\partial z} \hat{k}$

引入 Laplace 算符：

$\nabla^2 = \nabla \cdot \nabla = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$

可改写为$\frac{\partial^2 \vec{\xi}}{\partial t^2} - u^2 \nabla^2 \vec{\xi} = \vec{F}(x,y,z,t)$