# A Non-Hydrostatic Model for Simulating Weakly Dispersive Landslide-Generated Waves

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## Abstract

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## 1. Introduction

## 2. Governing Equations and the NH-1L Scheme

- Calculate ${h}^{n+1}$ from Equation (8); then the free surface is ${\eta}^{n+1}={h}^{n+1}-{d}^{n+1}$.
- Calculate $\widehat{u}$ using Equation (9), where the non-hydrostatic pressure term is neglected.
- Solve the linear system of Equation (16) to obtain ${q}^{n+1}$.
- Calculate ${w}_{\mathrm{bot}}^{n+1}$ using Equation (12).
- Calculate the corrected values of ${u}^{n+1}$ using Equation (19).
- Calculate ${w}_{\mathrm{top}}^{n+1}$ using Equation (10).

## 3. Model Validation

#### 3.1. Waves Generated by Landslide with Constant Speed

#### 3.2. Waves Generated by Landslide with Acceleration

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Wave speed normalized by $\sqrt{g{d}_{0}}$ as a function of $k{d}_{0}$. (

**b**) Relative error of the wave speed.

**Figure 3.**Staggered grid partition with locations of the unknown variables in the one-layer non-hydrostatic model.

**Figure 4.**Comparison of the wave-field contours obtained from three different wave models: (

**a**,

**d**) NH-1L model, (

**b**,

**e**) LWD model, (

**c**,

**f**) LFD model, for (

**a**–

**c**) $\mu =0.1$, (

**d**–

**f**) $\mu =0.2$. The dashed line represents the characteristic curves with wave celerity $\pm \sqrt{g{d}_{0}}$.

**Figure 5.**The free surface profiles at time $t\sqrt{g/{L}_{s}}=12$ resulting from the NH-1L simulation are compared with the LSWE, LWD, and LFD models for two dispersion parameters (

**a**) $\mu =0.1$. (

**b**) $\mu =0.2$.

**Figure 6.**Comparison of wave-field contours obtained from three different wave models: (

**a**,

**d**) NH-1L model. (

**b**,

**e**) LWD model. (

**c**,

**f**) LFD model. (

**a**–

**c**) $\mu =0.3$. (

**d**–

**f**) $\mu =0.4$. The dashed lines represent the characteristic curves with wave celerity $\pm \sqrt{g{d}_{0}}$.

**Figure 7.**The free surface profiles at time $t\sqrt{g/{L}_{s}}=12$ resulting from the NH-1L simulation are compared with the LWD and LFD models for two dispersion parameters (

**a**) $\mu =0.3$. (

**b**) $\mu =0.4$.

**Figure 8.**Free surface profiles at time $t\sqrt{g/{L}_{s}}=12$ resulting from the NH-1L simulation are compared with the LWD and LFD models for the dispersion parameter $\mu =0.5$.

**Figure 9.**Comparison of wave-field contours obtained by different models with the $\mu =0.5$: (

**a**) NH-1L model. (

**b**) LWD model. (

**c**) LFD model.

**Figure 10.**Wave profiles at subsequent times from run 21; results of the NH-1L model are compared with the LWD and experimental data.

**Table 1.**Dimensionless parameters used in the landslide experiment of Whittaker et al. [34].

Run | $\mathit{\lambda}$ | $\mathit{Fr}$ | $\mathit{\mu}$ |
---|---|---|---|

21 | 0.153 | 0.125 | 0.35 |

23 | 0.153 | 0.375 | 0.35 |

12 | 0.102 | 0.500 | 0.70 |

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**MDPI and ACS Style**

Tarwidi, D.; Pudjaprasetya, S.R.; Tjandra, S.S.
A Non-Hydrostatic Model for Simulating Weakly Dispersive Landslide-Generated Waves. *Water* **2023**, *15*, 652.
https://doi.org/10.3390/w15040652

**AMA Style**

Tarwidi D, Pudjaprasetya SR, Tjandra SS.
A Non-Hydrostatic Model for Simulating Weakly Dispersive Landslide-Generated Waves. *Water*. 2023; 15(4):652.
https://doi.org/10.3390/w15040652

**Chicago/Turabian Style**

Tarwidi, Dede, Sri Redjeki Pudjaprasetya, and Sugih Sudharma Tjandra.
2023. "A Non-Hydrostatic Model for Simulating Weakly Dispersive Landslide-Generated Waves" *Water* 15, no. 4: 652.
https://doi.org/10.3390/w15040652