We investigate theoretically the effects of capillarity and gravity on interfacial Marangoni waves (M-waves). Previous studies which neglected these effects have shown the decay rate of M-waves to vary monotonically. In this work, we disprove it by deriving a new dispersion relation (accounting for these effects) for the M-waves, and show that their decay rate passes through a minimum, rather than varying monotonically. This result, obtained numerically elsewhere, was explained using a hypothetical directional packaging process. Our explanation here is consistent with the bigger picture, which includes both capillary-gravity waves (CG-waves) and M-waves, which coexist at the contaminated interface of two fluids. It is known that the CG-waves undergo maximum dissipation at a critical value of interfacial elasticity, which provides the restoring force for the M-waves. Here, we complete the story; we show that the M-waves undergo minimum dissipation at a critical value of capillarity/gravity, which provide the restoring forces for the CG-waves. Previously, it has been hypothesized that the maximum decay rate for CG-waves is caused due to their resonance with M-waves. Here, we show this hypothesis to be true for a temporally-damped wave system, and that the resonance (in addition to causing the maximum decay rate for CG-waves) also causes the minimum decay rate for M-waves.

我们从理论上研究毛细作用和重力对界面Marangoni波（M波）的影响。先前忽略这些影响的研究表明，M波的衰减速率单调变化。在这项工作中，我们通过推导M波的新色散关系（考虑了这些影响）来证明这一点，并证明了M波的衰减率是通过最小值的，而不是单调变化的。使用假设的定向包装过程解释了在其他地方以数字方式获得的结果。我们在这里的解释与更大的图景是一致的，它包括毛细管重力波（CG波）和M波，它们共存于两种流体的污染界面处。众所周知，CG波在界面弹性的临界值处经历最大耗散，为M波提供恢复力。在这里，我们完成了故事；我们表明，在毛细作用/重力的临界值处，M波经历了最小的耗散，这为CG波提供了恢复力。以前，已经假设CG波的最大衰减率是由于它们与M波的共振引起的。在这里，我们证明了这个假设对于时间阻尼波系统是正确的，并且共振（除了导致CG波的最大衰减率之外）还导致了M波的最小衰减率。据推测，CG波的最大衰减率是由于它们与M波的共振引起的。在这里，我们证明了这个假设对于时间阻尼波系统是正确的，并且共振（除了导致CG波的最大衰减率之外）还导致了M波的最小衰减率。据推测，CG波的最大衰减率是由于它们与M波的共振引起的。在这里，我们证明了这个假设对于时间阻尼波系统是正确的，并且共振（除了导致CG波的最大衰减率之外）还导致了M波的最小衰减率。