# 波动方程

## 简介

${ \partial^2 u \over \partial t^2 } = c^2 \nabla^2u$

## 标量形式的一维波动方程

### 波动方程的推导

$F_{Newton}=m \cdot a(t)=m \cdot {{\partial^2 \over \partial t^2}u(x+h,t)}$
$F_{Hooke} = F_{x+2h} + F_x = k \left [ {u(x+2h,t) - u(x+h,t)} \right ] + k[u(x,t) - u(x+h,t)]$

$m{\partial^2u(x+h,t) \over \partial t^2}= k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]$

N个质点间隔均匀地固定在长度L = N h的弹簧链上，总质量M = N m，链的总体劲度系数K = k/N，我们可以将上面的方程写为：

${\partial^2u(x+h,t) \over \partial t^2}={KL^2 \over M}{u(x+2h,t)-2u(x+h,t)+u(x,t) \over h^2}$

${\partial^2 u(x,t) \over \partial t^2}={KL^2 \over M}{ \partial^2 u(x,t) \over \partial x^2 }$

### 一般解

#### 代数方法

$\left[ \frac{\part}{\part t} - c\frac{\part}{\part x}\right] \left[ \frac{\part}{\part t} + c\frac{\part}{\part x}\right] u = 0.\,$

$u(x,t) = F(x-ct) + G(x+ct) \,$

$u(x,0)=f(x) \,$
$u_t(x,0)=g(x) \,$

$u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c} \int_{x-ct}^{x+ct} g(s) ds$

## 标量形式的三维波动方程

### 球面波

$u_{tt} - c^2 \left( u_{rr} + \frac{2}{r} u_r \right) =0. \,$

$(ru)_{tt} -c^2 (ru)_{rr}=0; \,$

$u(t,r) = \frac{1}{r} F(r-ct) + \frac{1}{r} G(r+ct), \,$

### 广义初值问题的解

$r^2 = (x-\xi)^2 + (y-\eta)^2 + (z-\zeta)^2. \,$

$U(t,x,y,z;\xi ,\eta ,\zeta ) = \frac{{\delta (r - ct)}}{{4\pi cr}}$

u是这一族波函数的加权叠加，且权函数为φ，则

$u(t,x,y,z) = \frac{1}{4\pi c} \iiint \varphi(\xi,\eta,\zeta) \frac{\delta(r-ct)}{r} d\xi\,d\eta\,d\zeta; \,$

$u(t,x,y,z) = \frac{t}{4\pi} \iint_S \varphi(x +ct\alpha, y +ct\beta, z+ct\gamma) d\omega, \,$

$u(t,x,y,z) = t M_{ct}[\phi]. \,$

$u(0,x,y,z) = 0, \quad u_t(0,x,y,z) = \phi(x,y,z). \,$

$v(t,x,y,z) = \frac{\part}{\part t} \left( t M_{ct}[\psi] \right), \,$

$v(0,x,y,z) = \psi(x,y,z), \quad v_t(0,x,y,z) = 0. \,$

$u_{tt} - c^2 (u_{rr} + \frac{1}{r}u_r ) = 0$

$U(t,x - \xi ,y - \eta )=\begin{cases} \frac{1}{{2\pi c}}\frac{1}{{\sqrt {c^2 t^2 - r^2 } }}, & r \le ct \\ 0, & r > ct \end{cases}$

$r = \sqrt {(x - \xi )^2 + (y - \eta )^2 }$

## 标量形式的二维波动方程

$u_{tt} = c^2 \left( u_{xx} + u_{yy} \right). \,$

$u(0,x,y)=0, \quad u_t(0,x,y) = \phi(x,y), \,$

$u(t,x,y) = tM_{ct}[\phi] = \frac{t}{4\pi} \iint_S \phi(x + ct\alpha,\, y + ct\beta) d\omega,\,$

$u(t,x,y) = \frac{1}{2\pi c} \iint_D \frac{\phi(x+\xi, y +\eta)}{\sqrt{(ct)^2 - \xi^2 - \eta^2}} d\xi\,d\eta. \,$

## 边值问题

### 一维情形

$-u_x(t,0) + a u(t,0) = 0, \,$
$u_x(t,L) + b u(t,L) = 0,\,$

$u(t,x) = T(t) v(x).\,$

$\frac{T''}{c^2T} = \frac{v''}{v} = -\lambda. \,$

$v'' + \lambda v=0, \,$
$-v'(0) + a v(0) = 0, \quad v'(L) + b v(L)=0.\,$

### 多维情形

$\frac{\part u}{\part n} + a u =0, \,$

$u(0,x) = f(x), \quad u_t=g(x), \,$

$\nabla \cdot \nabla v + \lambda v = 0, \,$

$\frac{\part v}{\part n} + a v =0, \,$

## 进一步推广

$v_\mathrm{p} = \frac{\omega}{k}$

${ \partial^2 u \over \partial t^2 } = c(u)^2 \nabla^2u$

$\rho \ddot {\bold{u}} = \bold{f} + ( \lambda + 2\mu )\nabla(\nabla \cdot \bold{u}) - \mu\nabla \times (\nabla \times \bold{u})$

• $\lambda$$\mu$被称为弹性体的拉梅常数（也叫“拉梅模量”，英文Lamé constants或Lamé moduli），是描述各向同性固体弹性性质的参数；
• $\rho$表示密度
• $\bold{f}$是源函数（即外界施加的激振力）；
• $\bold{u}$表示位移；

## 註釋

1. ^ GRAY, JW. BOOK REVIEWS. BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY. July 1983, 9 (1). (retrieved 13 Nov 2012).
2. ^ Gerard F Wheeler. The Vibrating String Controversy, (retrieved 13 Nov 2012). Am. J. Phys., 1987, v55, n1, p33-37.
3. ^ For a special collection of the 9 groundbreaking papers by the three authors, see First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
4. ^ For de Lagrange's contributions to the acoustic wave equation, can consult Acoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
5. ^ 5.0 5.1 Speiser, David. Discovering the Principles of Mechanics 1600-1800, p. 191 (Basel: Birkhäuser, 2008).

## 参考文献

• 严镇军编，《数学物理方程》，第二版，中国科学技术大学出版社，合肥，2002，第210页~第224页，ISBN 7-312-00799-6/O·177
• [英]胡·普賴斯著，肖巍譯，《時間之矢與阿基米德之點—物理學時間的新方向》，上海科學技術出版社，上海，2001，ISBN 7-5323-5737-6
• M. F. Atiyah, R. Bott, L. Garding, Lacunas for hyperbolic differential operators with constant coefficients I, Acta Math., 124 (1970), 109–189.
• M.F. Atiyah, R. Bott, and L. Garding, Lacunas for hyperbolic differential operators with constant coefficients II, Acta Math., 131 (1973), 145–206.
• R. Courant, D. Hilbert, Methods of Mathematical Physics, vol II. Interscience (Wiley) New York, 1962.