scholarly journals A mathematical formulation for estimating maximum run-up height of 2018 Palu tsunami

2020 ◽  
Author(s):  
Ikha Magdalena ◽  
Antonio Hugo Respati Dewabrata ◽  
Alvedian Mauditra Aulia Matin ◽  
Adeline Clarissa ◽  
Muhammad Alif Aqsha
Keyword(s):  
Shallow Water ◽  
Cross Section ◽  
Incident Wave ◽  
Shallow Water Equation ◽  
Mathematical Formulation ◽  
Equation System ◽  
Type Transformation ◽  
Linear Equation System ◽  
Nonlinear Shallow Water Equations ◽  
Run Up

Abstract. Run-up is defined as sea wave up-rush on a beach. Run-up height is affected by many factors, including the shape of the bay. As an archipelagic country, Indonesia consists of thousands of islands with bays of diverse profiles, including Palu Bay, which is a well-known example of a bay with a drastically-increasing wave run-up height. In the case of the 2018 Palu tsunami, scientists found that the incident wave was amplified by the shape of the bay. The amplifying wave played a large role in the significant increase of run-up height. The run-up in question caused severe inundation, which led to a high number of casualties and damages. Therefore a mathematical model will be constructed to investigate the wave run-up. The bay's geometry will be approximated using three linearly-inclined channel types: one of parabolic cross-section, one of triangular cross-section, and a plane beach. We use the generalized nonlinear shallow water equations, which is then solved analytically using a hodograph-type transformation. As a result, the nonlinear shallow water equation system can be reduced to a one-dimensional linear equation system. Assuming the incident wave is sinusoidal, we can obtain a simple formula for calculating maximum run-up height on the shoreline.

A Class of Exact Solution of (3+1)-Dimensional Generalized Shallow Water Equation System

2015 ◽  
Vol 16 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Jian-Guo Liu ◽  
Zhi-Fang Zeng ◽  
Yan He ◽  
Guo-Ping Ai
Keyword(s):  
Shallow Water ◽  
Nonlinear Models ◽  
Shallow Water Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Equation System ◽  
Wave Solutions ◽  
Spurious Oscillation ◽  
Localized Excitations ◽  
Undetermined Function

AbstractShallow water wave equation has increasing use in many applications for its success in eliminating spurious oscillation, and has been widely studied. In this paper, we investigate (3+1)-dimensional generalized shallow water equation system. Based on the $(G'/G)$-expansion method and the variable separation method, we choose $\xi (x,y,z,t) = f(y + cz) + ax + h(t)$ and suppose that ${a_i}(i = 1,2, \ldots,m)$ is an undetermined function about $x,y,z,t$ instead of a constant in eq. (3), which are different from those in previous literatures. With the aid of symbolic computation, we obtain a family of exact solutions of the (3+1)-dimensional generalized shallow water equation system in forms of the hyperbolic functions and the trigonometric functions. When the parameters take special values, in addition to traveling wave solutions, we also get the nontraveling wave solutions by using our method; these obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves. The $(G'/G)$-expansion method is a very general and powerful tool that will lead to further insights and improvements of the nonlinear models.

Nonlinear landslide tsunami run-up

2011 ◽  
Vol 691 ◽  
pp. 440-460 ◽  
Author(s):  
M. Sinan Özeren ◽  
Nazmi Postacioglu
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Shallow Water Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Submarine Landslides ◽  
Partial Derivatives ◽  
Nonlinear Shallow Water Equations ◽  
Run Up

AbstractInhomogeneous nonlinear shallow-water equations are studied using the Carrier–Greenspan approach and the resulting equations are solved analytically. The Carrier–Greenspan transformations are commonly used hodograph transformations that transform the nonlinear shallow-water equations into a set of linear equations in which partial derivatives with respect to two auxiliary variables appear. Yet, when the resulting initial-value problem is treated analytically through the use of Green’s functions, the partial derivatives of the Green’s functions have non-integrable singularities. This has forced researchers to numerically differentiate the convolutions of the Green’s functions. In this work we remedy this problem by differentiating the initial condition rather than the Green’s function itself; we also perform a change of variables that renders the entire problem more easily treatable. This particular Green’s function approach is especially useful to treat sources that are extended in time; we therefore apply it to model the run-down and run-up of the tsunami waves triggered by submarine landslides. Another advantage of the method presented is that the parametrization of the landslide using sources is done within the integral algorithm that is used for the rest of the problem instead of treating the landslide-generated wave as a separate incident wave. The method proves to be more accurate than the techniques based on Bessel function expansions if the sources are very localized.

scholarly journals Run-up of nonlinear monochromatic wave on a plane beach in presence of a tide

2019 ◽  
Vol 59 (4) ◽  
pp. 529-532
Author(s):  
I. I. Didenkulova ◽  
E. N. Pelinovsky
Keyword(s):  
Exact Solution ◽  
Shallow Water ◽  
Nonlinear Problem ◽  
Incident Wave ◽  
Wave Amplitude ◽  
Monochromatic Wave ◽  
Shallow Water Theory ◽  
Long Wave ◽  
Run Up

The nonlinear problem of long wave run-up on a plane beach in a presence of a tide is solved within the shallow water theory using the Carrier-Greenspan approach. The exact solution of the nonlinear problem for wave run-up height is found as a function of the incident wave amplitude. Influence of tide on characteristics of wave run-up on a beach is studied.

Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

2014 ◽  
Vol 748 ◽  
pp. 416-432 ◽  
Author(s):  
Alexei Rybkin ◽  
Efim Pelinovsky ◽  
Ira Didenkulova
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Cross Section ◽  
Exact Analytical Solution ◽  
Auxiliary Function ◽  
Arbitrary Cross Section ◽  
Channel Cross Section ◽  
Linear Wave ◽  
Nonlinear Shallow Water Theory ◽  
Run Up

AbstractWe present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.

scholarly journals Identifikasi Pendekatan Shallow Water Equation Dalam Simulasi 2D Gelombang Tsunami di Pantai Keburuhan Purworejo

2020 ◽  
Vol 18 (1) ◽  
pp. 127-152
Author(s):  
Benny Hartanto ◽  
Ningrum Astriawati
Keyword(s):  
Shallow Water ◽  
Shallow Water Equation ◽  
Run Up ◽  
Tsunami Model

Kabupaten Purworejo merupakan salah satu dari lima daerah yang terkena dampak run-up tsunami Jawa 17 Juli 2006. Berdasarkan hasil Rapid Survey oleh BPDP dan BPPT, sepanjang Pantai Keburuhan merupakan lokasi terjadinya run-up tsunami di Kabupaten Purworejo di koordinat 109.912 LS -7.85 BT sebesar 1,7 meter. Tujuan dari penelitian ini adalah memperkirakan waktu tempuh tsunami, distribusi tinggi gelombang tsunami dan daerah jangkauan tsunami akibat dampak gempa tsunami Jawa 17 Juli 2006 di Pantai Keburuhan, Purworejo. Metode yang digunakan adalah metode deskriptif analitis dengan pendekatan kuantitatif. Data yang digunakan pada penelitian ini adalah titik tinggi, batimetri, parameter gempa, peramalan pasang surut wilayah perairan Pantai Keburuhan, data citra Geo Eye 1, dan kelerengan pantai. Pemodelan tsunami menggunakan perangkat lunak COMCOT v1.7 dengan kejadian gempa Jawa 17 Juli 2006. Berdasarkan pengolahan data, diketahui bahwa kecepatan gelombang maksimal sebesar 3.8788 m/s. Pada menit ke- 40, amplitudo awal gelombang tsunami sebesar 1.644  meter telah mencapai Pantai Keburuhan. Daerah jangkauan tsunami terluas dan jarak jangkauan maksimum terjauh  terjadi di Pantai Keburuhan adalah 1,23 km2 dan 1,4 km. Berdasarkan hasil verifikasi dengan nilai RSR sebesar 0,26. Hasil validasi simulasi tsunami menggunakan COMCOT v1.7 diketahui bahwa tinggi run-up tsunami model sudah cukup sesuai dengan data run-up yang terjadi saat kejadian, dengan nilai RSR sebesar 0,29 dan CF sebesar 1,63.

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