Surface waves have been introduced to explore the directional motion (DM) mechanism of a droplet on an obliquely vibrated substrate (Guo et al., Phys. Fluids, 2020), where the surface wave was decomposed for analysis by using a proper orthogonal decomposition (POD) method. In this work, the elaborate responses of the POD modes are explored given varying excitation frequency ($f$) and amplitude. It is found the first two POD modes evolve quite differently as $f$ changes. The first (swaying) POD mode always follows the excitation, while the second (spreading) POD mode transforms from co-frequency (linear) to double-frequency (non-linear) response as $f$ decreases, which lets the second POD mode have a frequency of $\sim$ $2f_0$ constantly when $f_0<f<2f_0$ ($f_0$ and $2f_0$ are the lowest natural frequencies of swaying and spreading vibration, respectively). That makes the POD modes contribute differently on the DM in different regimes, and the first POD mode may not contribute more to the DM. On the other hand, the DM speed is found to grow gradually when $f$ decreases or the vibration acceleration amplitude increases, which is attributed to the increasing vibration nonlinearity derived from more and more kinetic energy input. This work improves the understanding of the DM from the perspective of surface waves.

引入了表面波以探索在倾斜振动的基板上液滴的定向运动（DM）机制（Guo等人，Phys.Fluids，2020），其中通过使用适当的正交分解将表面波分解以进行分析（ POD）方法。在这项工作中，探索了在不同激励频率下POD模式的精细响应（$F$）和幅度。发现前两种POD模式随着$F$变化。第一个（摇摆）POD模式始终跟随激励，而第二个（扩展）POD模式则从同频（线性）响应转换为双频（非线性）响应，如下所示：$F$ 减小，从而使第二个POD模式的频率为 $〜$ $2F_0$ 不断地 $F_0<F<2F_0$ （$F_0$ 和 $2F_0$分别是摆动和传播振动的最低固有频率。这使得POD模式以不同的方式对DM做出不同的贡献，并且第一POD模式可能不会对DM做出更大的贡献。另一方面，发现DM速度在$F$减小或振动加速度振幅增加，这是由于越来越多的动能输入导致振动非线性增加。这项工作从表面波的角度提高了对DM的理解。