An inverse problem for the wave equation in plane-stratified media
An inverse problem for the wave equation in plane-stratified media
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平面分层介质中波动方程的反问题
DOI：
10.18910/5408
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发表时间：
2005
影响因子：
0.4
通讯作者：
S. Nagayasu
中科院分区：
文献类型：
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作者：
S. Nagayasu
假设半空间中存在具有奇异性的介质。我们通过在半空间边界附近的某个点处引起人工冲击并通过观察边界上的波的行为来研究介质的奇异性。例如，Rakesh [2]和Wang [4]研究了波动方程和弹性方程的这些问题。在这些论文中，他们使用了“线性化”方法，即假设介质的奇异性等很小（Sacks-Symes [3]）。我们讨论的情况下，奇点可能是大的。我们专门讨论这种情况并讨论以下问题：假设两种介质，介质1和介质2，位于半空间中，并且界面壁平行于半空间的边界（见图1）。我们假设介质1中的波的速度和边界反射的方式是已知的，但是介质1的宽度、介质2中的波的速度以及界面和透射条件是未知的。在这种情况下，我们试图通过使用已知的数据或在边界附近可以观察到的数据来识别这些未知的东西。现在，我们引入符号并将上面的问题公式化。假设n ≥ 2。设x′ =（x1，. . .，xn−1），且x′′ =（x2，. . .，xn），对于坐标x =（x1，. . .，xn）在R.变量x1扮演时间的角色，x“扮演物理空间的角色。设h > 0且Ω1：= {x′′ ∈ Rn−1：0 h}。我们设置Dxj：=（1/i）（λ/λ xj）。设ak为正真实的数，
Assume that there exist media which have singularities in a half-space. We investigate the singularities of the media by causing an artificial shock at a certain point near the boundary of the half-space and by observing the behavior of waves on the boundary. These problems for wave equations and elastic equations were studied by Rakesh [2] and Wang [4], for example. In these paper, they use the “linearization” method, that is, they assume the smallness of the singularities of the media and so on (Sacks-Symes [3]). We discuss the case when the singularities may be large. We specialize the situation and discuss the following problem: Assume that two media, Medium 1 and Medium 2, are laying in the half-space, and the interface wall is parallel to the boundary of a half-space (see Figure 1). We assume that the speed of waves in Medium 1 and the way of the reflection by the boundary are known, but the width of Medium 1, the speed of waves in Medium 2, and interface and transmission conditions are unknown. In this situation, we try to identify these unknown things by using the known data or the data which can be observed near the boundary. Now, we introduce the notation and formulate the problem above. Suppose n ≥ 2. Let us write x′ = (x1, . . . , xn−1), and x′′ = (x2, . . . , xn) for the coordinate x = (x1, . . . , xn) in R. The variable x1 plays the role of the time and x′′ the physical space. Let h > 0 and Ω1 := {x′′ ∈ Rn−1 : 0 h}. We set Dxj := (1/i)(∂/∂xj). Let ak be positive real number and