Observations of highly localized oscillons with multiple crests and troughs

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Observations of highly localized oscillons with multiple crests and troughs

Xiaochen Li, Dali Xu, and Shijun Liao

Phys. Rev. E 90, 031001(R) – Published 25 September 2014

Abstract

Two types of stable, highly localized Faraday resonant standing waves with multiple crests and troughs are observed in an ethanol-water solution partly filled in a Hele-Shaw cell vertically oscillated with a single frequency. Systematical experiments are performed to investigate the properties of these oscillons. It is found that the wave height of these oscillons is independent of fluid depth from 1 to 5 cm. In particular, some experiments are performed to indicate the high localization of the oscillons, which suggests that these oscillons may be regarded as a combination of the two elementary oscillons discovered by Rajchenbach et al. [Phys. Rev. Lett. 107, 024502 (2011)], for instance, $(2, 3) = (1, 1) + (1, 2)$ , where $(m, n)$ denotes an oscillon with $m$ crests and $n$ troughs. So, our experiments also reveal an elegant “arithmetic” of these oscillons. These experimental phenomena are helpful to deepen and enrich our understanding about Faraday waves.

Received 3 April 2014
Revised 15 July 2014

DOI:https://doi.org/10.1103/PhysRevE.90.031001

Authors & Affiliations

Xiaochen Li and Dali Xu

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China

Shijun Liao^*

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China and Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia

^*sjliao@sjtu.edu.cn

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Issue

Vol. 90, Iss. 3 — September 2014

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Images

Figure 1
The oscillons (3,3) and (4,3). The new oscillons observed in $15 %$ ethanol-water solution with a fluid depth of 2 cm, an oscillation frequency of 18 Hz, and an acceleration amplitude of $20.503 m / s^{2}$ . (a) and (b) represent the typical images of the oscillons (3,3) and (4,3), respectively. The corresponding supplementary movies [13] are available online.
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Figure 2
Averaged wave height versus fluid depth and acceleration amplitude. (a) Wave height of the (2,2) oscillon versus fluid depth, in a $15 %$ ethanol-water solution at the frequency of 18 Hz and the acceleration amplitude of $20.503 m / s^{2}$ . The square symbols denote the mean values of the experimental results, with error bars for standard derivation. The dashed line represents the averaged value 10.45 mm. (b) Wave height of the (2,2) oscillon versus acceleration amplitude, in a $15 %$ ethanol-water solution at a frequency of 18 Hz and fluid depth of 2 cm. Square symbols denote the mean value of the experimental results, with error bars for standard derivation. The solid line represents the optimal approximation obtained by the minimized linear least squares. $a_{low}$ and $a_{up}$ denote the lower and up thresholds of acceleration amplitude for the existence of oscillons, respectively.
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Figure 3
Existence window of oscillons. Experiments were performed in a $15 %$ ethanol-water solution with a fluid depth of 2 cm. The solid line with circles denotes the lower threshold of acceleration amplitude, and the dashed line with triangles represents the upper one, respectively.
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Figure 4
Wave height and crest-to-crest distance of oscillons. Experiments were performed in a $15 %$ ethanol-water solution at a frequency of 18 Hz, an acceleration amplitude of $20.503 m / s^{2}$ , and a fluid depth of 2 cm. (a) The averaged wave height of different types of oscillons. The integers 1 to 6 in the horizontal axis denote the oscillons of (1,1), (1,2), (2,2), (2,3), (3,3), and (3,4), respectively. Square symbols denote the mean value of the experimental results, with error bars for standard derivations. The dashed line represents their averaged value. (b) The averaged crest-to-crest distance of different types of oscillons. The integers 1 to 4 in the horizontal axis denote the oscillons of (2,2), (2,3), (3,3), and (3,4), respectively. Oscillons are created five times under the same physical situation. Each time when one oscillon is created, the crest-to-crest distance is measured once.
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Figure 5
Combinations of the elementary oscillons.
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Figure 6
High localization of oscillons. Experiments were performed in $15 %$ ethanol-water solution at a frequency of 18 Hz, an acceleration amplitude of $20.503 m / s^{2}$ , and a fluid depth of 2 cm. The right oscillon (1,2) remains the same in (a) and (b). The left oscillon is first (1,2) in (a) but then becomes (2,2) in (b) by additional disturbance of the free surface.
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