Ripples arise at edges of petals of blooming Lilium casablanca flowers and at edges of torn plastic sheets. In both systems, ripples are a consequence of excess length along the edge of a sheet. Through the use of both time-lapse videos of blooming lilies and still images of torn plastic sheets, we find that ripples in both systems are well-described by the scaling relationship a ∝ √(w(L − w)), where a is amplitude, w is wavelength, and L is arc length. By approximating that the arc length is proportional to the wavelength, we recover a phenomenological relationship previously reported for self-similar ripple patterns, namely ⟨a⟩ ∝ ⟨w⟩. Our observations imply that a broad class of systems in which morphological changes are driven by excess length along an edge will produce ripples described by a ∝ √(w(L − w)).

中文翻译：

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盛开的百合花边缘的波纹和塑料片撕裂

在盛开的百合木百合花的花瓣边缘和撕裂的塑料片的边缘会出现波纹。在这两种系统中，波纹都是沿纸张边缘长度过长的结果。通过使用盛开的百合花的延时视频和撕裂的塑料片的静止图像，我们发现两个系统中的波纹都可以通过比例关系a ∝√（w（L-w））很好地描述，其中a为振幅，w是波长，L是弧长。通过近似弧长与波长成正比，我们恢复了以前报道的自相似波纹模式的现象学关系，即⟨a⟩∝ww⟩。我们的观察表明，由沿边缘的超长长度驱动形态变化的一大类系统会产生由∝√（w（L-w））描述的波纹。