# 波动方程

## 简介

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

## 标量形式的一维波动方程

### 波动方程的推导

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

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

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：

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

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

### 一般解

#### 代数方法

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

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

${\displaystyle u(x,0)=f(x)\,}$
${\displaystyle u_{t}(x,0)=g(x)\,}$

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

## 标量形式的三维波动方程

### 球面波

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

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

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

### 广义初值问题的解

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

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

u是这一族波函数的加权叠加，且权函数为φ，则

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

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

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

${\displaystyle u(0,x,y,z)=0,\quad u_{t}(0,x,y,z)=\phi (x,y,z).\,}$

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

${\displaystyle v(0,x,y,z)=\psi (x,y,z),\quad v_{t}(0,x,y,z)=0.\,}$

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

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

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

## 标量形式的二维波动方程

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

${\displaystyle u(0,x,y)=0,\quad u_{t}(0,x,y)=\phi (x,y),\,}$

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

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

## 边值问题

### 一维情形

${\displaystyle -u_{x}(t,0)+au(t,0)=0,\,}$
${\displaystyle u_{x}(t,L)+bu(t,L)=0,\,}$

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

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

${\displaystyle v''+\lambda v=0,\,}$
${\displaystyle -v'(0)+av(0)=0,\quad v'(L)+bv(L)=0.\,}$

### 多维情形

${\displaystyle {\frac {\partial u}{\partial n}}+au=0,\,}$

${\displaystyle u(0,x)=f(x),\quad u_{t}=g(x),\,}$

${\displaystyle \nabla \cdot \nabla v+\lambda v=0,\,}$

${\displaystyle {\frac {\partial v}{\partial n}}+av=0,\,}$

## 进一步推广

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

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

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

• ${\displaystyle \lambda }$${\displaystyle \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；
• ${\displaystyle \rho }$表示密度
• ${\displaystyle \mathbf {f} }$是源函数（即外界施加的激振力）；
• ${\displaystyle \mathbf {u} }$表示位移；

## 註釋

1. ^ Cannon, John T.; Dostrovsky, Sigalia. The evolution of dynamics, vibration theory from 1687 to 1742. Studies in the History of Mathematics and Physical Sciences 6. New York: Springer-Verlag: ix + 184 pp. 1981. ISBN 0-3879-0626-6. 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,页面存档备份，存于互联网档案馆） (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页面存档备份，存于互联网档案馆） (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页面存档备份，存于互联网档案馆） Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)
5. Speiser, David. Discovering the Principles of Mechanics 1600-1800页面存档备份，存于互联网档案馆, 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
• [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.