1. (doubly warped product case) the coefficients of the metric are $L^\infty$ and bounded from below by a positive constant.

2. (Warped product case) the coefficients of the metrics belong to a critical $L^p$ space where $p < \infty$ depends on the dimension of the compact fibers of the cylinder.

Eventually, we show (in the warped product case and for zero frequency) that these uniqueness results are sharp by giving simple counterexamples for a class of singular metrics whose coefficients do not belong to the critical $L^p$ space. All these counterexamples lead in fact to a region of space that is invisible to boundary measurements.

1.（双重变形的乘积情况）度量的系数为$ L ^ \ infty $，并从下方以正常数为界。

2.（翘曲的产品案例）度量的系数属于关键的$ L ^ p $空间，其中$ p <\ infty $取决于圆柱体的紧密纤维的尺寸。

最终，我们通过给出一类奇异度量的简单反例（其系数不属于关键的$ L ^ p $空间）证明了（在翘曲的乘积情况下，对于零频率）这些唯一性结果很明显。所有这些反例实际上导致了边界测量不可见的空间区域。