ON RADIATION FIELD OF PULSE SOLUTIONS OF WAVE EQUATION .2.
ON RADIATION FIELD OF PULSE SOLUTIONS OF WAVE EQUATION .2.
复制标题
DOI：
10.1098/rspa.1964.0111
复制
发表时间：
1964-01-01
影响因子：
--
通讯作者：
FRIEDLANDER, FG
中科院分区：
文献类型：
--
作者：
FRIEDLANDER, FG
较早的论文表明，ifu(x1, x2, x3, t) =u(x, t) 在某个固定球体= │x│ =a 的外部满足波动方程 utt= Δu 并且消失 fort≤r，则 ru(rψ,t) ~f(ψ,t—r) asr→ ∞，前提是 ψ 是一个固定单位向量且 t—r 仍然有界。还表明，“辐射场”f(xi,s) 在 r ≥ a 中唯一确定 u(x,t)。在本文中，假设关于u的拉普拉斯变换存在。这表明关于 s 的拉普拉斯变换也存在，并且是 ψ 的解析函数，对于所有复数单位向量 ψ 都是正则的。由此可以推断，对于 all8 和（实）单位球体的任何开子集中的 allΨ 来说，iffitself 消失，则 f ≠ 0，因此在 r ≥ a 中，f ≠ 0。此外，还获得了拉普拉斯变换的积分表示，该表示将发散球面波的外尔积分表示推广为具有复传播矢量的平面波。
It was shown in an earlier paper that, ifu(x1, x2, x3, t) =u(x, t) satisfies the wave equationutt= ∆uin the exterior of some fixed spherer= │x│ =aand vanishes fort≤r, thenru(rξ,t) ~f(ξ,t—r) asr→ ∞, provided thatξis a fixed unit vector andt—rremains bounded. It was also shown that the 'radiation field' ’f(ξ,s) determinesu(x,t) uniquely inr ≥ a. In the present paper it is assumed that the Laplace transform ofuwith respect totexists. This is found to imply that the Laplace transform offwith respect tosalso exists, and is an analytic function ofξthat is regular for all complex unit vectorsξ. From this it can be inferred that, iffitself vanishes for all8, and for allξin any open subset of the (real) unit sphere, thenf≡ 0, and henceu≡ 0 inr ≥ a. Furthermore, an integral representation of the Laplace transform ofuin terms of the Laplace transform offis obtained, which generalizes Weyl’s integral representation of diverging spherical waves in terms of plane waves with complex propagation vectors.