# Effects of Slide Shape on Impulse Waves Generated by a Subaerial Solid Slide

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. Numerical Model for Landslide Generated Waves

#### 2.2. Model Validation

## 3. Numerical Experiments

#### 3.1. Design of Numerical Experiments

#### 3.2. Results and Discussions

#### 3.2.1. Maximum Amplitude and Its Location

#### 3.2.2. Wave Period

#### 3.2.3. Impact Energy Conversion

#### 3.2.4. Evolution of Leading Waves

## 4. Concluding Remarks

- Maximum wave amplitude, ${A}_{m}$, is inversely proportional to the slide shape parameter, $\sigma $. Location of the maximum amplitude, ${x}_{m}$, also depends on $\sigma $ (Section 3.2.1).
- The first crest always travels at the typical long wave speed while the second crest propagates at a speed about 15 to 25% slower depending on the value of $\sigma $ (Section 3.2.4).
- In the far field, the second crest becomes larger than the first crest if $\sigma $ is smaller (Section 3.2.4).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic (not drawn to scale) of the numerical wave tank: a solid slide moving downhill on a sloping beach connected to a region of constant depth ${h}_{0}$. $\beta $ is the beach slope, $\varphi $ denotes the slide front angle [68], and $\theta $ shows the included angle formed by the slide front face and the still free surface. Origin of the coordinate system is set at the initial shoreline. (

**b**) The computational domain and a sample numerical mesh. No-slip condition is required on all boundaries, except at the top boundary open to free atmosphere.

**Figure 2.**Model validation for subaerial landslide generated waves: snapshots of free-surface profile at select time instants (Top to bottom: $t=0.6,1.0,1.5$ s). Dot: model predictions. Circle: measurements reported by Heinrich [51].

**Figure 3.**Model validation for subaerial landslide generated waves: records of free-surface elevation at fixed locations (Top to bottom: $x=4,8,12$ m). Dot: model predictions. Circle: measurements reported by Heinrich [51].

**Figure 4.**Reproduction of submarine landslide generated waves: snapshots of the free surface profile at two different time instants (Left: $t=0.5$ s; Right: $t=1.0$ s). Dot: model predictions. Circle: measurements reported by Heinrich [51].

**Figure 5.**Reproduction of submarine landslide generated waves: records of free surface elevation at fixed locations (Top to bottom: $x=4,8,12$ m). Dot: model predictions. Circle: measurements reported by Heinrich [51].

**Figure 6.**Sketch of triangular prism shaped model solid slides used in the numerical experiments. The two-digit code indicates the case number. The filled isosceles right triangle shows the base of the subaerial slide used in the laboratory experiments by Heinrich [51]. ${\mathcal{L}}_{s}$: slide length. ${\mathcal{S}}_{s}$: maximum slide thickness. The slide shape parameter $\sigma $ measures the distance from the front vertex to the position of maximum slide thickness. Parameters of the model slides are given in Table 1.

**Figure 7.**Effects of slide shape on maximum wave amplitude. (

**Left**) amplitude against slide shape parameter $\sigma $ (■: maximum amplitude ${A}_{m}$; □: amplitude measured at $x=1$ m; —: fitted regression line for ■; ---: linear fit for □). (

**Right**) location of the maximum amplitude, ${x}_{m}$. In both panels, triangles represent the results by the reference solid slide used in the laboratory testing.

**Figure 8.**Wave periods observed at $x=1$ m under slides with various slide shape parameter $\sigma $. ■: results of model slides shown in Figure 6. ▲ shows the baseline result by the solid slide used in the physical modeling. Dashed lines indicate the $\pm 6\%$ range.

**Figure 9.**Energy conversion ratio, ${e}_{0}$, verses slide shape parameter, $\sigma $. ■: model predictions. ▲: result by the reference solid slide. Curve: exponential fitting by ${e}_{0}={c}_{1}\mathrm{exp}\left({c}_{2}\sigma \right)$.

**Figure 10.**Evolution of wave profiles: records of free-surface elevation at select locations (Top to bottom: $x=1,3,5,7$ m).

**Left**: results for Case 00 ($\sigma =0$).

**Right**: Case 07 ($\sigma =0.7$ m).

**Figure 11.**Records of free-surface elevation at $x=1$ m. (

**Top**) Results for Case 00 (solid line); Case 01 (dash-dotted line); Case 02 (dashed line); and Case 03 (dotted line). (

**Bottom**) Results for Case 04 (solid line); Case 05 (dash-dotted line); Case 06 (dashed line); and Case 07 (dotted line).

**Figure 12.**Records of free-surface elevation at $x=7$ m. (

**Top**) Results for Case 00 (solid line); Case 01 (dash-dotted line); Case 02 (dashed line); and Case 03 (dotted line). (

**Bottom**) Results for Case 04 (solid line); Case 05 (dash-dotted line); Case 06 (dashed line); and Case 07 (dotted line).

**Figure 13.**Comparison of the first crest’s amplitude (${A}_{m,1}$) with the second crest’s amplitude (${A}_{m,2}$) recorded at $x=7$ m under various slide shape parameter, $\sigma $. ■: model predictions. ▲ represents the result by the reference solid slide used in the laboratory experiment. Line: linear regression fit.

**Table 1.**Values of slide shape parameter $\sigma $ and slide front angle $\varphi $ for all triangular prism shaped model solid slides. Case REF represents the subaerial slide used in the physical modeling by Heinrich [51]. All slides have the same volume (${\mathcal{V}}_{s}$), slide length (${\mathcal{L}}_{s}$), and maximum slide thickness (${\mathcal{S}}_{s}$).

Case # | $\mathit{\sigma}$ (m) | $\mathit{\varphi}$ (degree) |
---|---|---|

00 | 0.0 | 90.0${}^{\circ}$ |

01 | 0.1 | 74.2${}^{\circ}$ |

02 | 0.2 | 60.5${}^{\circ}$ |

03 | 0.3 | 50.0${}^{\circ}$ |

REF | 0.35 | 45.0${}^{\circ}$ |

04 | 0.4 | 41.5${}^{\circ}$ |

05 | 0.5 | 35.3${}^{\circ}$ |

06 | 0.6 | 30.5${}^{\circ}$ |

07 | 0.7 | 26.8${}^{\circ}$ |

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Huang, C.-S.; Chan, I.-C. Effects of Slide Shape on Impulse Waves Generated by a Subaerial Solid Slide. *Water* **2022**, *14*, 2643.
https://doi.org/10.3390/w14172643

**AMA Style**

Huang C-S, Chan I-C. Effects of Slide Shape on Impulse Waves Generated by a Subaerial Solid Slide. *Water*. 2022; 14(17):2643.
https://doi.org/10.3390/w14172643

**Chicago/Turabian Style**

Huang, Chiung-Shu, and I-Chi Chan. 2022. "Effects of Slide Shape on Impulse Waves Generated by a Subaerial Solid Slide" *Water* 14, no. 17: 2643.
https://doi.org/10.3390/w14172643