Waves in the plane, punctured by excision of a small disk with radius much smaller than the wavelength, can be modified by being forced to vanish on the boundary of the disk. Such waves exhibit a logarithmically thin ‘pinprick’, and logarithmically weak oscillations persisting far away. As the radius vanishes, these modifications become asymptotically invisible. Examples are punctured plane waves, and a punctured unit disk; in the latter case, the pinprick causes a logarithmic shift in the eigenvalues. It is conjectured that the plane can be densely covered with asymptotically invisible pinpricks, and that there are analogous phenomena in higher dimensions. The curious phenomenon of pinpricks is not hard to understand, and would be worth presenting in graduate courses on waves.

中文翻译：

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波函数中的对数针刺

平面中的波，通过切除半径远小于波长的小圆盘而被刺穿，可以通过被迫在圆盘边界上消失来改变。这种波表现出对数级的细“针刺”，并且对数级的微弱振荡持续存在很远的地方。随着半径消失，这些修改逐渐变得不可见。例子有穿孔平面波和穿孔单位圆盘；在后一种情况下，针刺会导致特征值发生对数变化。据推测，平面上可能密布着渐近看不见的针孔，并且在更高维度中也存在类似的现象。这种奇怪的针刺现象并不难理解，值得在关于波浪的研究生课程中介绍。