Wave equation

Japanese: 波動方程式 - はどうほうていしき(英語表記)wave equation
Wave equation

second-order linear hyperbolic equation

is called the wave equation. If f(x,y,z) is three times continuously differentiable and g(x,y,z) is two times continuously differentiable, then the initial condition in (1)

There is only one solution that satisfies

(Kirchhoff's formula). Here, S t is a sphere of radius t centered at (x, y, z) in the (ξ, η, ζ) space, and dS is a surface element on it. The solutions for two-dimensional and one-dimensional space are respectively

where D t is a circle of radius t centered at the origin.

[Yoshikazu Kobayashi]

Wave Equation in Physics

In general, in waves, a physical quantity u, such as the displacement of a medium or the components of an electric field or magnetic field, satisfies a second-order partial differential equation called the wave equation. A function u that satisfies the wave equation is called a wave function.

For a wave traveling in the x direction, the second partial differential coefficient ∂ 2 u/∂t 2 of the wave function u(x,t) with respect to time t is proportional to the second partial differential coefficient ∂ 2 u/∂x 2 with respect to the position coordinate x. In other words, ∂ 2 u/∂t 2 =v 2 (∂ 2 u/∂x 2 ), which is the (one-dimensional) wave equation. The v in the proportionality coefficient v 2 is the speed at which the wave travels. The general solution to this partial differential equation is
u=u 1 +u 2 ,u 1 =f(tx/v),u 2 =g(t+x/v)
Here, f and g are arbitrary functions. u1 represents a wave traveling in the positive x direction, and u2 represents a wave traveling in the negative x direction, both at a speed of v. In general, the wave function u(x,y,z,t) at a point with position coordinates (x,y,z) at time t is expressed by the wave equation ∂ 2 u/∂t 2
=v 2 (∂ 2 u/∂x 2 +∂ 2 u/∂y 2
+ ∂2u / ∂z2 )
Satisfy.

[Yoshiro Kainuma]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

二階線形双曲型方程式

を波動方程式という。f(x,y,z)が3回連続的微分可能、g(x,y,z)が2回連続的微分可能ならば、(1)の初期条件

を満たす解はただ一つ存在して

で与えられる(キルヒホッフの公式)。ここでStは(ξ,η,ζ)空間の(x,y,z)を中心とする半径tの球面、dSはその上の面素である。空間二次元、一次元の場合の解はそれぞれ

で与えられる。ただし、Dtは原点を中心とする半径tの円である。

[小林良和]

物理学における波動方程式

一般に波においては、媒質の変位や、電場、磁場の成分のような物理量uが波動方程式とよばれる二階の偏微分方程式を満足する。波動方程式を満足する関数uは波動関数とよばれる。

 x方向に進行する波については、波動関数u(x,t)の時間tに関する二階の偏微分係数∂2u/∂t2が、位置座標xに関する二階の偏微分係数∂2u/∂x2に比例する。すなわち∂2u/∂t2=v2(∂2u/∂x2)で、これが(一次元の)波動方程式である。比例係数v2のvは波の伝わる速度である。この偏微分方程式の一般解は、
 u=u1+u2,u1=f(t-x/v),u2=g(t+x/v)
と書ける。ここに、fとgは、任意の関数である。u1はプラスx方向に、u2はマイナスx方向に、いずれも速度vで伝わる波を表す。一般には、時刻tに、位置座標(x,y,z)の点における波動関数u(x,y,z,t)は、波動方程式
  ∂2u/∂t2
   =v2(∂2u/∂x2+∂2u/∂y2
   +∂2u/∂z2)
を満足する。

[飼沼芳郎]

出典 小学館 日本大百科全書(ニッポニカ)日本大百科全書(ニッポニカ)について 情報 | 凡例

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