Uniform asymptotic expansions at a caustic
Uniform asymptotic expansions at a caustic
复制标题
DOI：
10.1002/cpa.3160190207
复制
发表时间：
1966-05
影响因子：
3
通讯作者：
D. Ludwig
中科院分区：
文献类型：
--
作者：
D. Ludwig
Au+kzu=0对于较大的k值是有效的，它可以通过几何光学的射线来构造(例如，参见J.B.Keller[lo])。这种解由形式为u=Eik+‘z的项之和组成，其中p和z取决于自变量，p独立于k，z是KL的正整数幂的形式级数。焦散是几何光学的光线有包络的轨迹；在焦散处，振幅t有奇点，并且上面给出的几何光学的渐近展开式是不值的。我们给出了一个形式的渐近级数，它有焦散线，但仍然一致有效，即在一个大小与k无关且包含焦散线的区域内，满足K1中约化到任意高阶的波动方程。在最简单的光滑凸焦散线的情况下，解被表示为艾里函数及其导数的线性组合。展开式中的系数与几何光学的输运系数(2)密切相关，但我们展开式中的系数在焦散处不是奇异的。在远离焦散线的点上，艾里函数可以用它的渐近展开式代替：在焦散线的一侧，我们得到几何光学给出的展开式，而在焦散线的另一侧，我们得到指数阻尼解。在焦散线附近，我们获得了振荡和指数衰减行为之间的平滑过渡，焦散线本身的值很大，实际上，焦散线附近有一个边界层：实际上，R.Buchal和J.B.Keller121和R.Buchal[L]已经使用了类似于我们的解，它们在焦散线附近的边界层中是有效的。然而，我们的程序不涉及在边界层分析中经常遇到的坐标拉伸或解的匹配。在我们的解中，涉及艾里函数的导数的项在焦散附近相对较小，但在焦散之外很突出。这一项没有出现在Buchal和Keller的解中，它使得一致展开成为可能。
Au+ kzu= 0, which are valid for large values of k, can be constructed by means of the rays of geometrical optics (see, for example, J. B. Keller [lo]). Such solutions consist of a sum of terms of the form u= eik+'z, where p and z depend on the independent variables, p is independent of k, and z is a formal series in positive integral powers of kl. A caustic is a locus where the rays of geometrical optics have an envelope; at a caustic the amplitude t has a singularity and the asymptotic expansion of geometrical optics given above is not vaIid. We give a formal asymptotic series which has a caustic, but which is nevertheless uniformly valid, ie, which satisfies the reduced wave equation to arbitrarily high order in kl in a region whose size is independent of k, and which contains the caustic. In the simplest case of a smooth convex caustic, the solution is represented as a linear combination of the Airy function and its derivative. The coefficients in the expansion are closely related to the transport coefficients (2) of geometrical optics, however the coefficients in our expansion are not singular at the caustic. At points away from the caustic, the Airy function can be replaced by its asymptotic expansion for large argument: on one side of the caustic we obtain the expansion given by geometrical optics, and on the other side of the caustic we obtain an exponentially damped solution. In the vicinity of the caustic, we obtain a smooth transition between oscillatory and exponentially damped behavior, with large values on the caustic itself, In effect, there is a boundary layer near the caustic: indeed solutions similar to ours, which are valid in a boundary layer near the caustic, have been used by R. Buchal and J. B. Keller 121 and R. Buchal [l]. However, our procedure does not involve stretching of coordinates or matching of solutions, which are frequently encountered in boundary layer analysis. The term in our solution which involves the derivative of the Airy function is relatively small near the caustic, but prominent away from the caustic. This term, which does not appear in the solution of Buchal and Keller, makes the uniform expansion possible.