# The Maximum Height and Attenuation of Impulse Waves Generated by Subaerial Landslides

Authors: Baoliang Wang; Lingkan Yao; Haixin Zhao; Cong Zhang
Journal: Shock and Vibration
 (2018)

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## Abstract

High-speed landslides that flow into reservoirs can cause impulsive water waves. To study the characteristics of the maximum impulse wave’s height and its attenuation, 25 sets of flume experiments were conducted using orthogonal theory and 6 main influencing factors were considered. Taking the impulse wave heights as the evaluation criteria and analyzing the 6 influencing factors at 5 different levels, the characteristics of the maximum impulse wave’s height and its attenuations were obtained. Then, statistical relationships between the maximum wave height and the controlling factors were proposed. Then, by combining the continuity equation and the hydrodynamic open channel transient flow movement equation, the process of landslide wave height attenuation was studied, and it was found that the attenuation of the wave is consistent with exponential attenuation. Then, combined with the data obtained from the orthogonal experiments, an attenuation equation for the surge was derived. Finally, the proposed equation was validated by applying it to the landslides that took place along the shore of the Zipingpu reservoir, which were triggered by the Wenchuan earthquake, and the results indicate that the calculated results are very close to the observed results.


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## Body

## 1. Introduction

Landslides that flow into reservoirs can cause catastrophic impulse waves, threatening the safety of ships, dams, lives, and property. Several catastrophic events that resulted in a large loss of human life and property have highlighted the necessity of understanding this complex phenomenon, including the Vajont disaster, which is one of the most catastrophic phenomena ever documented involving subaerial landslide generated waves; it occurred in Italy, on 9 October, 1963. In this event, a 300 × 106 m3 landslide flowed into the Vajont artificial reservoir and formed an 80 m high wave, which overtopped the dam, destroyed the city of Longarone, and killed 1909 people [1]. Another example of a submarine landslide related disaster is the well-documented tsunami generated by an underwater landslide triggered by a 7.0 magnitude earthquake on July 17, 1998, close to the Sissano Lagoon, Sang Dawn province, Papua New Guinea [2]. The tsunami hit an area ~30 km wide, and although the affected area was small, it caused a 15 m high run-up that killed more than 2,100 people. Many similar events have also been reported in other articles [3–6].At present, the research on landslide waves can be divided into three categories: mathematical theories, numerical simulations, and physical models.In terms of theoretical analysis and numerical simulation, Kennard [7] derived an expression for surface waves caused by the movement of an object in water by solving the velocity potential function of the Laplace equation. Noda [8] considered the extreme solution for landslides for the two extreme conditions of horizontal and vertical movement and deduced the analytical solution of the maximum wave height produced by these two conditions using linear approximation and infinite approximation. On the basis of Noda’s work, Pan [9] proposed an equation for maximum wave by combining the results of a unidirectional flow analysis with the characteristics of the reflection and superposition of the surge. It is also found that transient waves generated on the shore may remain trapped close to the coast due to refraction [10, 11]. Based on the Navier–Stokes equations, a two-dimensional hydrodynamic model was used to simulate the formation and propagation of waves. Pelinovsky [12] and Quecedo et al. [13] proposed a solution for the full Navier–Stokes equations, the results of which are in good agreement with real data. Ataie-Ashtiani and Malek-Mohammadi [14] simulated the surge of a rigid landslide and a flexible landslide entering the water along an inclined sliding surface using an SPH numerical simulation. Fritz et al. [15] gave a detailed analysis of landslide related surge characteristics from a two-dimensional perspective. Di Risio and Sammarco [16] proposed a new analytical solution for the leading wave based on linear theory, the result of which indicates that the wave height and period are affected by the impulse pressure and the velocity field generated during the underwater phase. By simplifying the hydrodynamic momentum equation and considering the fluid acceleration, Mulligan and Take [17] developed an equation that does not measure or estimate the length and time scale of the landslide. This equation yields the upper limit of the surge amplitude and provides a theoretical means to predict the near-field maximum water surface elevation. Yin et al. [18] employed a fluid-solid coupling model based on a general moving object collision model and the renormalization of the group turbulent model. Bosa and Petti [19] established one-dimensional and two-dimensional models for landslide surges based on the nonlinear shallow water model.In terms of physical experiments, the triggering mechanism, that is, the landslide, can be divided into subaerial, partially submerged, and completely submerged according to the landslide’s initial position [20]. Fritz et al. [21] studied the wave formation process for sliders of different sizes, shapes, and densities at varying degrees of depth along the slope. He found that the maximum wave height is related to the impact velocity, the thickness of the landslide, and the initial water depth. Noda [8] used the solid slider model to simulate the initial surge characteristics of a slider entering the water under the two extreme conditions of the horizontal and vertical movement, assuming that the initial wave height is a function of the Froude number. Heller and Hager [22] concluded through model experiments that the surge height is mainly related to the unit volume and the Froude number. Walder et al. [23] used two-dimensional physical model experiments to study the near-field characteristics of solid sliders. Their results indicate that the parameters of the near-field surge characteristics are determined by the dimensionless landslide volume per unit width, the dimensionless underwater movement time, and the dimensionless vertical impact speed. Some researchers have also studied granular landslides into lakes [24–26]. These studies suggest that the wave behavior is highly dependent on the water depth relative to the size of the landslide and that density plays a minor role in wave formation. Through three-dimensional model experiments, Frtiz et al. [27] concluded that the wave height is related to the dimensionless landslide volume, the dimensionless vertical impact speed, the dimensionless landslide thickness, and the initial water depth. In a subsequent study, by considering the energy exchange between the slider and the water, Frtiz et al. [15] concluded that the Froude number is the main factor controlling the maximum surge height. Liu et al. [28] and Enet and Grilli [29] independently conducted experiments using tanks with widths 4 times and 5.4 times larger than the landslide, respectively. These findings are very useful for near-field research but cannot be used for far-field propagation analysis because the measurements are quickly contaminated by waves reflected off of the tank’s walls. Based on mixture theory, Zitti et al. [30] proposed a simple model to estimate the momentum transfer between the granular flow and the water phases. Several researchers have studied impulse waves caused by submerged landslides [10, 28, 31, 32]. The most important of these results are those of Watts [32]; that is, the formal coupling of the landslide motion to the generated waves and most of the landslide’s energy are transferred to the water during an underwater landslide [33, 34]. The importance of the nondimensional time characteristic of a landslide’s underwater motion, ts∗=tsg/d, which can express the role of the landslide’s mechanism in the process of wave generation in a simple but meaningful way, was stressed by Walder et al. [33] and Watts [31]. Through 288 sets of three-dimensional physical model experiments, Heller and Spinneken [35] also considered the underwater movement time and summarized the empirical equation for the relationship between the underwater movement time and the maximum wave height. Heller and Spinneken [35] and Heller et al. [36] directly compared 2D and 3D block models on planar hill slopes. Ataie-Ashtiani and Malek-Mohammadi [14] analyzed the wave period, height, and energy of the surge and developed an empirical equation for the period and height of the surge wave. Recently, the wave generation problem caused by a coastal tsunamigenic source, that is, a coastal mountain landslide, has been solved using experiments [37–39] in the case of both a straight coast and a circular island, and it was found that the location of the maximum run-up along the coast occurs in the near distance (about 2-3 times the width of the landslide) rather than very close to the landslide [10].Most previous studies have considered the shape of the slider, the speed, the water depth, and other factors, but there is no specific analysis of the influence of each factor. This paper focuses on the maximum wave height of the landslide and its attenuation characteristics.


## 2. Model Experiments

### 2.1. Experimental Setup

Figure1 shows an illustration of the flume apparatus used in this study. The flume is 9 m long, 0.5 m wide, and 0.7 m high. The two sidewalls are transparent tempered glass, so the wave propagation in the flume can be observed. A digital camera was placed near the point where the concrete block impacts the water to record the form of the landslide surges. Seven wave height gauges were arranged along the central axis of the water flume in the direction of the surge motion, and a water level hydrograph at each point was set to record after the water wave was formed. To convey solid blocks, an inclined plane with an adjustable slope (30–90°) was designed. The slope’s length was 2.5 m, the slope is made of stainless steel marked with a scale, and the plane’s surface was lubricated to reduce friction.Figure 1
Schematic of the flume device.


### 2.2. Experimental Measurements

The blocks used were solid blocks with zero porosity, and the landslide model was designed to simulate real geometry and bulk density characteristics. The back of the landslide block has a hanger, and the hanger is attached to a string. The landslide body was placed on the surface of the inclined plane. The string was cut before sliding starts, and the landslide was free to slide on the surface under the force of gravity. The trajectory of the block was recorded by the digital camera. According to the time series pictures of the different displacements, the displacement (L) of the landslide body at time (t) was obtained. The maximum wave height was obtained from the digital camera, and the time of the underwater landslide motion (ts) was read frame by frame from industrial camera footage.


### 2.3. Experimental Design

Many physical variables can affect the maximum wave height, and the wave height can be represented as a function of all of the physical parameters involved as follows:(1)A=fs,w,l,vs,d,h,wf,ts,α,ρs,ρw,g,μ,v,r,p,where l, w, and s are the length, width, and thickness of the modeled landslide, respectively; d is the still water depth; h is the drop height of the landslide; wf is the width of the flume; ts is the time of the landslide’s underwater motion; α is the angle of the inclined plane; ρs and ρw are the landslide and water densities, respectively; g is the gravitational acceleration; μ is the water viscosity; v is the landslide impact velocity; r is the distance from the drop point; and p is the porosity of the landslide. With the simplifications introduced by Panizzo et al. [40], the main parameters selected for this study are the sliding impact velocity (v), the slide width (w), the slide thickness (s), the slide length (l), the slide’s slope (α), the still water depth (d), the drop height of the mass’s center (h), and the landslide’s underwater motion time (ts).These factors can be combined into five dimensionless factors as follows:Slide Froude number:(2)F=vgd.Dimensionless slide volume:(3)V=Vswd2.Blockage ratio:(4)B=wwf.Nondimensional time of the landslide underwater motion:(5)Ts=tsgd.Slide impact angle:(6)α0=α360×2πin  radians.The experimental sets were designed using the orthogonal method, which is the most useful design method in multifactor experiments. The theory of orthogonal design is a branch in the field of mathematics and is deeply studied by several great number theorists, such as Radom and Hurwitz. The encyclopedic work of Geramita and Seberry [41] is a good reference for the orthogonal method. A classic result of this method is attributed to Radom, since he determined the dimensions in which orthogonal designs exist; more information on this method and physical discussion of how this technique was applied in this study is provided in the Appendix. The experiment sets of this study were performed at 5 levels for each factor using an L2556 orthogonal table. The experimental scheme was compiled according to the orthogonal table, and 25 sets of experiments were carried out, as is shown in Table 1.Table 1
Factors and levels of the orthogonal experiment.
Level
parameter
d (m)
w (m)
s (m)
l (m)
h (m)
α (°)
1
0.25
0.07
0.07
0.1
0.3
30
2
0.3
0.1
0.1
0.12
0.45
45
3
0.35
0.12
0.12
0.15
0.6
60
4
0.4
0.15
0.15
0.2
0.75
70
5
0.45
0.2
0.2
0.25
0.9
90


## 2.1. Experimental Setup

Figure1 shows an illustration of the flume apparatus used in this study. The flume is 9 m long, 0.5 m wide, and 0.7 m high. The two sidewalls are transparent tempered glass, so the wave propagation in the flume can be observed. A digital camera was placed near the point where the concrete block impacts the water to record the form of the landslide surges. Seven wave height gauges were arranged along the central axis of the water flume in the direction of the surge motion, and a water level hydrograph at each point was set to record after the water wave was formed. To convey solid blocks, an inclined plane with an adjustable slope (30–90°) was designed. The slope’s length was 2.5 m, the slope is made of stainless steel marked with a scale, and the plane’s surface was lubricated to reduce friction.Figure 1
Schematic of the flume device.


## 2.2. Experimental Measurements

The blocks used were solid blocks with zero porosity, and the landslide model was designed to simulate real geometry and bulk density characteristics. The back of the landslide block has a hanger, and the hanger is attached to a string. The landslide body was placed on the surface of the inclined plane. The string was cut before sliding starts, and the landslide was free to slide on the surface under the force of gravity. The trajectory of the block was recorded by the digital camera. According to the time series pictures of the different displacements, the displacement (L) of the landslide body at time (t) was obtained. The maximum wave height was obtained from the digital camera, and the time of the underwater landslide motion (ts) was read frame by frame from industrial camera footage.


## 2.3. Experimental Design

Many physical variables can affect the maximum wave height, and the wave height can be represented as a function of all of the physical parameters involved as follows:(1)A=fs,w,l,vs,d,h,wf,ts,α,ρs,ρw,g,μ,v,r,p,where l, w, and s are the length, width, and thickness of the modeled landslide, respectively; d is the still water depth; h is the drop height of the landslide; wf is the width of the flume; ts is the time of the landslide’s underwater motion; α is the angle of the inclined plane; ρs and ρw are the landslide and water densities, respectively; g is the gravitational acceleration; μ is the water viscosity; v is the landslide impact velocity; r is the distance from the drop point; and p is the porosity of the landslide. With the simplifications introduced by Panizzo et al. [40], the main parameters selected for this study are the sliding impact velocity (v), the slide width (w), the slide thickness (s), the slide length (l), the slide’s slope (α), the still water depth (d), the drop height of the mass’s center (h), and the landslide’s underwater motion time (ts).These factors can be combined into five dimensionless factors as follows:Slide Froude number:(2)F=vgd.Dimensionless slide volume:(3)V=Vswd2.Blockage ratio:(4)B=wwf.Nondimensional time of the landslide underwater motion:(5)Ts=tsgd.Slide impact angle:(6)α0=α360×2πin  radians.The experimental sets were designed using the orthogonal method, which is the most useful design method in multifactor experiments. The theory of orthogonal design is a branch in the field of mathematics and is deeply studied by several great number theorists, such as Radom and Hurwitz. The encyclopedic work of Geramita and Seberry [41] is a good reference for the orthogonal method. A classic result of this method is attributed to Radom, since he determined the dimensions in which orthogonal designs exist; more information on this method and physical discussion of how this technique was applied in this study is provided in the Appendix. The experiment sets of this study were performed at 5 levels for each factor using an L2556 orthogonal table. The experimental scheme was compiled according to the orthogonal table, and 25 sets of experiments were carried out, as is shown in Table 1.Table 1
Factors and levels of the orthogonal experiment.
Level
parameter
d (m)
w (m)
s (m)
l (m)
h (m)
α (°)
1
0.25
0.07
0.07
0.1
0.3
30
2
0.3
0.1
0.1
0.12
0.45
45
3
0.35
0.12
0.12
0.15
0.6
60
4
0.4
0.15
0.15
0.2
0.75
70
5
0.45
0.2
0.2
0.25
0.9
90


## 3. Experimental Results

### 3.1. Velocity of Slide

The block sliding process was recorded by a digital camera, which recorded 25 frames per second with a frame interval of 0.04 seconds. Thus, the location of the mass’s center at different times could be obtained. The block’s movement can be simplified as the movement of the slider on the inclined plane with a friction coefficient(μ). Figure 2 shows the force analysis of the slide block.Figure 2
Forces acting on the sliding block.The acceleration along the inclined plane can be calculated by(7)a=gsin⁡α-μcos⁡α.The velocity can be obtained by (8)vs=2as=2gh1-μcot⁡α,where g is the acceleration due to gravity (9.8 m/s2); α is the slope angle (°); h is the drop height of the mass’s center (m); and μ is the friction coefficient.Using (8), we analyzed 8 sets of experimental videos and obtained 8 friction coefficients, with an average value of 0.460 and a variance of 1.77 × 10−5. Figure 3 shows a comparison between the calculated and measured slide velocities derived from three experiments. The measured velocities are very close to those calculated using (2); therefore, we can obtain the slide velocity using (8).Figure 3
Comparison between the measured evaluated slide velocities and those obtained from (8).


### 3.2. Analysis of the Importance of the Influencing Factors

In orthogonal design, due to the different experimental conditions, we cannot compare the two specific sets of experimental data. However, if the experimental results are properly combined, they can be compared. The impulse wave heights obtained under the various experimental conditions are listed in Table2. In the following discussion, we analyze and calculate the data in the table.Table 2
Experiment schemes and results.
Experiment number
d (cm)
b (cm)
s (cm)
l (cm)
h (cm)
α (°)
Results (mm)
(
1
)
25
7
7
10
30
30
21
(
2
)
25
10
10
12
45
45
33
(
3
)
25
12
12
15
60
60
42
(
4
)
25
15
15
20
75
70
51
(
5
)
25
20
20
25
90
90
68
(
6
)
30
7
10
15
75
90
44
(
7
)
30
10
12
20
90
30
46
(
8
)
30
12
15
25
30
45
38
(
9
)
30
15
20
10
45
60
41
(
10
)
30
20
7
12
60
70
35
(
11
)
35
7
12
25
45
70
42
(
12
)
35
10
15
10
60
90
42
(
13
)
35
12
20
12
75
30
46
(
14
)
35
15
7
15
90
45
42
(
15
)
35
20
10
20
30
60
37
(
16
)
40
7
15
12
90
60
47
(
17
)
40
10
20
15
30
70
41
(
18
)
40
12
7
20
45
90
38
(
19
)
40
15
10
25
60
30
42
(
20
)
40
20
12
10
75
60
41
(
21
)
45
7
20
20
60
45
51
(
22
)
45
10
7
25
75
60
41
(
23
)
45
12
10
10
90
70
42
(
24
)
45
15
12
12
30
90
34
(
25
)
45
20
15
15
45
30
46


#### 3.2.1. Influence of the Different Values of the Experimental Indexes for the Same Factors

We usedj to represent the jth level of factor d, where j=1,2,3,4,5. The d1 levels (25 cm) of the d factors were calculated, and the experimental indexes in experimental schemes 1, 2, 3, 4, and 5 were calculated for d1. For these conditions, the cumulative height of the surge is(9)d1=I1=21+33+42+51+68=215.The average value for experimental index A1 is as follows:(10)d1¯=I1¯=2155=43.The average values of the experimental indexes for d2, d3, d4, and d5 are 40.8, 41.8, 41.8, and 42.8, respectively.In the 5 experiments usingd1 (experiments (1)–(5)), b, s, l, h, and α were evaluated at 5 different levels, and each appeared 1 time. Similarly, for the 5 experiments using d2 (experiments (6)–(10)), b, s, l, h, and α were evaluated at 5 different levels, and each appeared 1 time. By this analogy, in the d1, d2, d3, d4, and d5 conditions of the 5 experiments, although other conditions (b,s,l,h,α) have changed, these changes are “balanced,” which makes d1 (Ii), d2 (IIi), d3 (IIIi), d4 (IVi), and d5 (V) comparable. The difference between the four factors reflects the effect of the four values of the d factor (still water depth) on the wave height. Through data comparison of different levels, it can be seen that d1>d5>d4>d3>d2. d1 has a maximum value of 215, and d2 has a minimum value of 204, so when the d factor is at level 1, the wave height is maximized. The significance of this is that when the d factor is 25 cm, the wave height value can be increased by an average of 2.2 mm compared to the minimum value (2 level; 30 cm). Similarly, we can calculate the average value ofb,s,l,h, andα for different values. The results of the experimental data are listed in Table 3. Ii, IIi, IIIi, IVi, and Vi in Table 3 represent the sum of the experimental indicators of factor i at the first, second, third, and fourth levels.Table 3
Orthogonal analysis of experimental data.
d
b
s
l
h
α
I
i
215
205
177
187
171
201
IIi
204
203
198
195
200
164
IIIi
209
206
205
215
212
249
IVi
209
210
224
223
223
211
Vi
214
227
247
231
245
226
k
1
=
I
1
/
5
43
41
35.4
37.4
34.2
40.2
k
2
=
I
I
1
/
5
40.8
40.6
39.6
39
40
41
k
3
=
I
I
I
1
/
5
41.8
41.2
41
43
42.4
41.5
k
4
=
I
V
1
/
5
41.8
42
44.8
44.6
44.6
42.2
k
5
=
V
1
/
5
42.8
45.4
49.4
46.2
49
45.2
R
i
2.2
4.8
14
8.8
14.8
5
Note. Ri represents the range of values under various factor conditions.


#### 3.2.2. Range Analysis

The range is defined as the difference between the best and worst level of the factor. Range analysis is mainly used to clarify the significance levels of the different influencing factors on the wave height, and those most significant factors can be determined based on the results of the range analysis. It is important to measure the fluctuation of the data. The range of each factor in Table4 is calculated as Ri, the range of the d factor is R1=I1¯-II1¯=43-40.8=2.2. Similarly, the range of theb,s,l,h, andα factor is 4.8, 14, 8.8, 14.8, and 5, respectively. Figure 4 shows the influence of the various factors on the wave height. The abscissa is expressed by the different levels of the factors, and the ordinate is the wave height. Comparison of the graphs can clearly show the wave height at the most effective value of the factors. As can be seen from Figure 4, of the factors affecting the wave height, the range of the drop height is the largest, and it produces a large difference in the wave height; therefore, the drop height is the main influencing factor. Through range comparison and intuitive graphical analysis, it can be concluded that the sliding slope is the main factor affecting the wave height, followed by the drop height, the slider thickness, the slider length, the slider width, and the still water depth.Table 4
Relationship betweenHr/h0 and x/h0 for each experiment.
H
m
a
x
/
d
H
r
/
d
=
k
e
-
m
(
x
/
d
)
k
m
R
2
T1
0.084
0.081
0.050
0.913
T2
0.119
0.120
0.049
0.955
T3
0.155
0.146
0.045
0.940
T4
0.181
0.166
0.043
0.915
T5
0.260
0.238
0.042
0.950
T6
0.140
0.126
0.049
0.934
T7
0.153
0.135
0.052
0.882
T8
0.127
0.127
0.058
0.960
T9
0.130
0.118
0.052
0.870
T10
0.117
0.103
0.049
0.851
T11
0.113
0.111
0.040
0.950
T12
0.120
0.109
0.054
0.933
T13
0.131
0.128
0.050
0.938
T14
0.120
0.125
0.056
0.948
T15
0.106
0.093
0.039
0.968
T16
0.118
0.109
0.043
0.909
T17
0.103
0.100
0.047
0.908
T18
0.095
0.100
0.056
0.909
T19
0.105
0.101
0.062
0.952
T20
0.103
0.110
0.048
0.865
T21
0.113
0.113
0.064
0.888
T22
0.091
0.080
0.043
0.884
T23
0.093
0.087
0.049
0.869
T24
0.076
0.065
0.055
0.900
T25
0.102
0.103
0.054
0.858Figure 4
Relationship between the wave height and all of the influencing factors.


### 3.3. Prediction of the Maximum Wave Height

#### 3.3.1. Definition of the Maximum Wave Height

Figure5 shows the four distinct phases defined by Huber and Hager [43]. Phase 1 is when the landslide begins moving, accelerates, and then falls into the water. Phase 2 is the underwater movement of the landslide, which is also referred to as the generation area or the splash zone in the case of a subaerial landslide. During phase 2, the energy of the landslide is transferred to the water [33, 34]. Phase 3 refers to the propagation of the water waves. Phase 4 is when the impulse wave reaches the water body boundaries and runs up or over the top of the dams, which eventually poses a great threat to human lives and property.Figure 5
Main phases of landslides-generated impulse waves.Figure6 depicts a typical waveform signal record from the experiments. As is defined by Panizzo et al. [1] and Walder et al. [33], a standard zero-crossing analysis was carried out to define the maximum wave amplitude (aM). The wave amplitude reported here is the elevation of the water surface relative to the datum (still water) within the zero-crossing wave. The maximum impulse wave height is the difference between the absolute maximum and the absolute minimum values within the maximum zero-crossing wave. The related wave period (Tmax), which is important to wave propagation, and the first wave of the generated transient train (a1(+), a1(-), and T1) were also characterized. The video images were analyzed frame by frame, and, then, the captured images were printed to measure the maximum height of the wave. The error of the measured maximum wave height is caused by the image resolution (0.4 mm per pixel) and blurring of the image (from a limited exposure time). We estimate that the error is typically within the range of 1 to 2 mm. The wavelet analysis method was used to compute the zero cross section of the wave records. Based on the work of Panizzo et al. [44], wavelet analysis was used to make a part of the wave field propagate away from the impact point, thus reducing the effect of the reflected wave on the recorded wave signal.Figure 6
Sketch of an impulse wave and the primary wave parameters.


#### 3.3.2. Equation for the Maximum Wave Height

The data recorded by the wave gauges was used to derive the prediction equations for the impulse wave’s characteristics. Based on dimensional analysis, the relationship between the wave height and the four dimensionless parameters is as follows:(11)Hmaxd=fvgd,wd,Vsbd2,TsAw∗,α,where the nondimensional parameter Aw∗=bs/d2 summarizes the landslide geometry. We can obtain an empirical equation through multiple linear regression analysis, and the relationship between the maximum wave height and the five dimensionless parameters can be expressed by (11) as follows:(12)Hmaxd=0.145F0.672B0.09V0.217TsAw∗-0.045α0.038.Figure7 compare the calculated and measured wave heights of each experiment. It can be seen from Figure 7 that the empirical equation is basically consistent with the experimental data for all of the different experimental conditions.Figure 7
Comparison between measured values and predicted values.In actual landslides,Ts is a dependent variable and must be calculated. Panizzo et al. [1] developed an empirical equation for estimating Ts:(13)Ts=0.43V-0.27F-0.66sin⁡α-1.32.Equation (13) correlates well with our experimental measurements and is therefore useful in determining Ts.


### 3.4. Equation for Wave Height Attenuation

The attenuation calculation of the landslide surge is based on the basic continuity equation and the motion equation for unsteady flow in an open channel (Li [45]):(14)T∂h∂t+u∂h∂x+A∂u∂x+bu-q=0,g∂h∂x+∂u∂t+u∂u∂x+gj-i=0,where T(h,x) is the water surface width; A(h,x) is the section area; u(x,t) is the flow velocity; h(x,t) is the water depth; j is the energy slope; i is the bed slope; and q is the increased flow per unit length. We use the hydrodynamic perturbation method to expand the continuity equation and the motion equation into polynomials. If we assume that the increased flow per unit length (q) is zero and that the friction slope (j) in the fluid is proportional to the local velocity, then (14) can be simplified as follows:(15)T∂h∂t+uT∂h∂x+A∂u∂x+bu=0,g∂h∂x+∂u∂t+u∂u∂x+gku-i=0.In order to solve (15), we can use the hydrodynamic perturbation method to linearize the equation. We assume that h¯ is the representative water depth, δ is the wave height, and ε=δ/h¯. In the perturbation method, the solutions of u and h can often be expressed as follows:(16)ux,t=u0x,t+εu1x,t+ε2u2x,t+⋯,hx,t=h0x,t+εh1x,t+ε2h2x,t+⋯,where the values of u0,u1,…,h0,h1,… need to be determined and sometimes need other forms of expansion as noted by Kevorkian and Cole [46]. By substituting (16) into (15), we obtained the following equations:(17)εu1+⋯∂∂xεu1+⋯+∂∂tεu1+⋯+g∂∂xh0+εh1+⋯+gεku1+⋯-i=0,A0+⋯∂∂xεu1+⋯+εu1+⋯T0+⋯∂∂xh0+⋯+T0+⋯∂∂th0+εh1+⋯+εu1+⋯b0+⋯=0.By simplifying (17) to the idempotent ε, we can obtain the following equations:(18)εT0∂h1∂t+u1dA0dx+A0∂u1∂x+T1∂h0∂t+⋯=0,g∂h0∂x-i+ε∂u1∂t+g∂h1∂x+gku1+⋯=0.Since (18) are assumed to be valid for any small value of ε, they are effective for each derivative of ε, so the equations for the zero order are as follows:(19)T0∂h0∂t=0,g∂h0∂x-i=0,where us represents the flow velocity (εu1) and hs represents the displacement (εh1). Thus, the equations for the first order are as follows:(20)T0∂hs∂t+usdA0dx+A0∂us∂x=0,∂us∂t+g∂hs∂x+gkus=0.Using (20) to eliminate u, we obtain the following:(21)T0∂2hs∂t2+gkT0∂hs∂t=A0g∂2hs∂x2+gdA0dx∂hs∂x.For rectangular or approximately rectangular channels, dA0/dx=0, so Equation (21) can be written as follows:(22)T0∂2hs∂t2+gkT0∂hs∂t=A0g∂2hs∂x2.After the landslide occurs, the surface of the water rises, then falls rapidly, and soon develops an oscillation with a small height. It can be assumed that the descent process obeys an exponential distribution. Therefore, we assumehs(x,t)=f(x)e-at, and (22) can be changed to(23)T0a2fxe-at-gkT0afxe-at=A0gf′′xe-at.If k1=T0a2-gkT0a/A0g, then f′′(x)=k1f(x), and we obtain the following equation:(24)fx=C1ek1x+C2ek2x.Applying the boundary condition f(0)=hj, limx→∞f(x)=0, we get C1=0, C2=hj; therefore,(25)hsx,t=hje-k1x-at=ke-mx.Next, we use the experimental data to verify and fit the equation. According to the data recorded by each wave instrument, the free surface height of the water body at different times can be obtained. Figure8 shows the variation in wave height with time for different positions in the same experiment.Figure 8
Variation in height of the surge with time for different points in the same experiment.The study of the factors affecting the attenuation of the landslide induced surge was carried out based on the wave height data of the 6 wave instruments. At different water depths, the relative distances of the six wave instruments along the cross section of the flume are shown in Figure9.Figure 9
Maximum wave heights versus the relative propagation distance.H
r
/
d can be roughly fitted by an exponential power function(26)Hrd=ke-mx/d.Hr is the height of the waves propagating along the flume. Table 4 lists the parameters for each experiment with R2≈1 for almost all cases.We find that, in each experiment, the values ofk and Hmax/d are very close; thus, the equation for the propagation of waves along the propagating path can be expressed as follows:(27)Hrd=Hmaxde-0.05x/d.Equation (27) is still able to effectively describe the attenuation of the impulse wave’s height, with a maximum error value of ≤15%.


## 3.1. Velocity of Slide

The block sliding process was recorded by a digital camera, which recorded 25 frames per second with a frame interval of 0.04 seconds. Thus, the location of the mass’s center at different times could be obtained. The block’s movement can be simplified as the movement of the slider on the inclined plane with a friction coefficient(μ). Figure 2 shows the force analysis of the slide block.Figure 2
Forces acting on the sliding block.The acceleration along the inclined plane can be calculated by(7)a=gsin⁡α-μcos⁡α.The velocity can be obtained by (8)vs=2as=2gh1-μcot⁡α,where g is the acceleration due to gravity (9.8 m/s2); α is the slope angle (°); h is the drop height of the mass’s center (m); and μ is the friction coefficient.Using (8), we analyzed 8 sets of experimental videos and obtained 8 friction coefficients, with an average value of 0.460 and a variance of 1.77 × 10−5. Figure 3 shows a comparison between the calculated and measured slide velocities derived from three experiments. The measured velocities are very close to those calculated using (2); therefore, we can obtain the slide velocity using (8).Figure 3
Comparison between the measured evaluated slide velocities and those obtained from (8).


## 3.2. Analysis of the Importance of the Influencing Factors

In orthogonal design, due to the different experimental conditions, we cannot compare the two specific sets of experimental data. However, if the experimental results are properly combined, they can be compared. The impulse wave heights obtained under the various experimental conditions are listed in Table2. In the following discussion, we analyze and calculate the data in the table.Table 2
Experiment schemes and results.
Experiment number
d (cm)
b (cm)
s (cm)
l (cm)
h (cm)
α (°)
Results (mm)
(
1
)
25
7
7
10
30
30
21
(
2
)
25
10
10
12
45
45
33
(
3
)
25
12
12
15
60
60
42
(
4
)
25
15
15
20
75
70
51
(
5
)
25
20
20
25
90
90
68
(
6
)
30
7
10
15
75
90
44
(
7
)
30
10
12
20
90
30
46
(
8
)
30
12
15
25
30
45
38
(
9
)
30
15
20
10
45
60
41
(
10
)
30
20
7
12
60
70
35
(
11
)
35
7
12
25
45
70
42
(
12
)
35
10
15
10
60
90
42
(
13
)
35
12
20
12
75
30
46
(
14
)
35
15
7
15
90
45
42
(
15
)
35
20
10
20
30
60
37
(
16
)
40
7
15
12
90
60
47
(
17
)
40
10
20
15
30
70
41
(
18
)
40
12
7
20
45
90
38
(
19
)
40
15
10
25
60
30
42
(
20
)
40
20
12
10
75
60
41
(
21
)
45
7
20
20
60
45
51
(
22
)
45
10
7
25
75
60
41
(
23
)
45
12
10
10
90
70
42
(
24
)
45
15
12
12
30
90
34
(
25
)
45
20
15
15
45
30
46


### 3.2.1. Influence of the Different Values of the Experimental Indexes for the Same Factors

We usedj to represent the jth level of factor d, where j=1,2,3,4,5. The d1 levels (25 cm) of the d factors were calculated, and the experimental indexes in experimental schemes 1, 2, 3, 4, and 5 were calculated for d1. For these conditions, the cumulative height of the surge is(9)d1=I1=21+33+42+51+68=215.The average value for experimental index A1 is as follows:(10)d1¯=I1¯=2155=43.The average values of the experimental indexes for d2, d3, d4, and d5 are 40.8, 41.8, 41.8, and 42.8, respectively.In the 5 experiments usingd1 (experiments (1)–(5)), b, s, l, h, and α were evaluated at 5 different levels, and each appeared 1 time. Similarly, for the 5 experiments using d2 (experiments (6)–(10)), b, s, l, h, and α were evaluated at 5 different levels, and each appeared 1 time. By this analogy, in the d1, d2, d3, d4, and d5 conditions of the 5 experiments, although other conditions (b,s,l,h,α) have changed, these changes are “balanced,” which makes d1 (Ii), d2 (IIi), d3 (IIIi), d4 (IVi), and d5 (V) comparable. The difference between the four factors reflects the effect of the four values of the d factor (still water depth) on the wave height. Through data comparison of different levels, it can be seen that d1>d5>d4>d3>d2. d1 has a maximum value of 215, and d2 has a minimum value of 204, so when the d factor is at level 1, the wave height is maximized. The significance of this is that when the d factor is 25 cm, the wave height value can be increased by an average of 2.2 mm compared to the minimum value (2 level; 30 cm). Similarly, we can calculate the average value ofb,s,l,h, andα for different values. The results of the experimental data are listed in Table 3. Ii, IIi, IIIi, IVi, and Vi in Table 3 represent the sum of the experimental indicators of factor i at the first, second, third, and fourth levels.Table 3
Orthogonal analysis of experimental data.
d
b
s
l
h
α
I
i
215
205
177
187
171
201
IIi
204
203
198
195
200
164
IIIi
209
206
205
215
212
249
IVi
209
210
224
223
223
211
Vi
214
227
247
231
245
226
k
1
=
I
1
/
5
43
41
35.4
37.4
34.2
40.2
k
2
=
I
I
1
/
5
40.8
40.6
39.6
39
40
41
k
3
=
I
I
I
1
/
5
41.8
41.2
41
43
42.4
41.5
k
4
=
I
V
1
/
5
41.8
42
44.8
44.6
44.6
42.2
k
5
=
V
1
/
5
42.8
45.4
49.4
46.2
49
45.2
R
i
2.2
4.8
14
8.8
14.8
5
Note. Ri represents the range of values under various factor conditions.


### 3.2.2. Range Analysis

The range is defined as the difference between the best and worst level of the factor. Range analysis is mainly used to clarify the significance levels of the different influencing factors on the wave height, and those most significant factors can be determined based on the results of the range analysis. It is important to measure the fluctuation of the data. The range of each factor in Table4 is calculated as Ri, the range of the d factor is R1=I1¯-II1¯=43-40.8=2.2. Similarly, the range of theb,s,l,h, andα factor is 4.8, 14, 8.8, 14.8, and 5, respectively. Figure 4 shows the influence of the various factors on the wave height. The abscissa is expressed by the different levels of the factors, and the ordinate is the wave height. Comparison of the graphs can clearly show the wave height at the most effective value of the factors. As can be seen from Figure 4, of the factors affecting the wave height, the range of the drop height is the largest, and it produces a large difference in the wave height; therefore, the drop height is the main influencing factor. Through range comparison and intuitive graphical analysis, it can be concluded that the sliding slope is the main factor affecting the wave height, followed by the drop height, the slider thickness, the slider length, the slider width, and the still water depth.Table 4
Relationship betweenHr/h0 and x/h0 for each experiment.
H
m
a
x
/
d
H
r
/
d
=
k
e
-
m
(
x
/
d
)
k
m
R
2
T1
0.084
0.081
0.050
0.913
T2
0.119
0.120
0.049
0.955
T3
0.155
0.146
0.045
0.940
T4
0.181
0.166
0.043
0.915
T5
0.260
0.238
0.042
0.950
T6
0.140
0.126
0.049
0.934
T7
0.153
0.135
0.052
0.882
T8
0.127
0.127
0.058
0.960
T9
0.130
0.118
0.052
0.870
T10
0.117
0.103
0.049
0.851
T11
0.113
0.111
0.040
0.950
T12
0.120
0.109
0.054
0.933
T13
0.131
0.128
0.050
0.938
T14
0.120
0.125
0.056
0.948
T15
0.106
0.093
0.039
0.968
T16
0.118
0.109
0.043
0.909
T17
0.103
0.100
0.047
0.908
T18
0.095
0.100
0.056
0.909
T19
0.105
0.101
0.062
0.952
T20
0.103
0.110
0.048
0.865
T21
0.113
0.113
0.064
0.888
T22
0.091
0.080
0.043
0.884
T23
0.093
0.087
0.049
0.869
T24
0.076
0.065
0.055
0.900
T25
0.102
0.103
0.054
0.858Figure 4
Relationship between the wave height and all of the influencing factors.


## 3.2.1. Influence of the Different Values of the Experimental Indexes for the Same Factors

We usedj to represent the jth level of factor d, where j=1,2,3,4,5. The d1 levels (25 cm) of the d factors were calculated, and the experimental indexes in experimental schemes 1, 2, 3, 4, and 5 were calculated for d1. For these conditions, the cumulative height of the surge is(9)d1=I1=21+33+42+51+68=215.The average value for experimental index A1 is as follows:(10)d1¯=I1¯=2155=43.The average values of the experimental indexes for d2, d3, d4, and d5 are 40.8, 41.8, 41.8, and 42.8, respectively.In the 5 experiments usingd1 (experiments (1)–(5)), b, s, l, h, and α were evaluated at 5 different levels, and each appeared 1 time. Similarly, for the 5 experiments using d2 (experiments (6)–(10)), b, s, l, h, and α were evaluated at 5 different levels, and each appeared 1 time. By this analogy, in the d1, d2, d3, d4, and d5 conditions of the 5 experiments, although other conditions (b,s,l,h,α) have changed, these changes are “balanced,” which makes d1 (Ii), d2 (IIi), d3 (IIIi), d4 (IVi), and d5 (V) comparable. The difference between the four factors reflects the effect of the four values of the d factor (still water depth) on the wave height. Through data comparison of different levels, it can be seen that d1>d5>d4>d3>d2. d1 has a maximum value of 215, and d2 has a minimum value of 204, so when the d factor is at level 1, the wave height is maximized. The significance of this is that when the d factor is 25 cm, the wave height value can be increased by an average of 2.2 mm compared to the minimum value (2 level; 30 cm). Similarly, we can calculate the average value ofb,s,l,h, andα for different values. The results of the experimental data are listed in Table 3. Ii, IIi, IIIi, IVi, and Vi in Table 3 represent the sum of the experimental indicators of factor i at the first, second, third, and fourth levels.Table 3
Orthogonal analysis of experimental data.
d
b
s
l
h
α
I
i
215
205
177
187
171
201
IIi
204
203
198
195
200
164
IIIi
209
206
205
215
212
249
IVi
209
210
224
223
223
211
Vi
214
227
247
231
245
226
k
1
=
I
1
/
5
43
41
35.4
37.4
34.2
40.2
k
2
=
I
I
1
/
5
40.8
40.6
39.6
39
40
41
k
3
=
I
I
I
1
/
5
41.8
41.2
41
43
42.4
41.5
k
4
=
I
V
1
/
5
41.8
42
44.8
44.6
44.6
42.2
k
5
=
V
1
/
5
42.8
45.4
49.4
46.2
49
45.2
R
i
2.2
4.8
14
8.8
14.8
5
Note. Ri represents the range of values under various factor conditions.


## 3.2.2. Range Analysis

The range is defined as the difference between the best and worst level of the factor. Range analysis is mainly used to clarify the significance levels of the different influencing factors on the wave height, and those most significant factors can be determined based on the results of the range analysis. It is important to measure the fluctuation of the data. The range of each factor in Table4 is calculated as Ri, the range of the d factor is R1=I1¯-II1¯=43-40.8=2.2. Similarly, the range of theb,s,l,h, andα factor is 4.8, 14, 8.8, 14.8, and 5, respectively. Figure 4 shows the influence of the various factors on the wave height. The abscissa is expressed by the different levels of the factors, and the ordinate is the wave height. Comparison of the graphs can clearly show the wave height at the most effective value of the factors. As can be seen from Figure 4, of the factors affecting the wave height, the range of the drop height is the largest, and it produces a large difference in the wave height; therefore, the drop height is the main influencing factor. Through range comparison and intuitive graphical analysis, it can be concluded that the sliding slope is the main factor affecting the wave height, followed by the drop height, the slider thickness, the slider length, the slider width, and the still water depth.Table 4
Relationship betweenHr/h0 and x/h0 for each experiment.
H
m
a
x
/
d
H
r
/
d
=
k
e
-
m
(
x
/
d
)
k
m
R
2
T1
0.084
0.081
0.050
0.913
T2
0.119
0.120
0.049
0.955
T3
0.155
0.146
0.045
0.940
T4
0.181
0.166
0.043
0.915
T5
0.260
0.238
0.042
0.950
T6
0.140
0.126
0.049
0.934
T7
0.153
0.135
0.052
0.882
T8
0.127
0.127
0.058
0.960
T9
0.130
0.118
0.052
0.870
T10
0.117
0.103
0.049
0.851
T11
0.113
0.111
0.040
0.950
T12
0.120
0.109
0.054
0.933
T13
0.131
0.128
0.050
0.938
T14
0.120
0.125
0.056
0.948
T15
0.106
0.093
0.039
0.968
T16
0.118
0.109
0.043
0.909
T17
0.103
0.100
0.047
0.908
T18
0.095
0.100
0.056
0.909
T19
0.105
0.101
0.062
0.952
T20
0.103
0.110
0.048
0.865
T21
0.113
0.113
0.064
0.888
T22
0.091
0.080
0.043
0.884
T23
0.093
0.087
0.049
0.869
T24
0.076
0.065
0.055
0.900
T25
0.102
0.103
0.054
0.858Figure 4
Relationship between the wave height and all of the influencing factors.


## 3.3. Prediction of the Maximum Wave Height

### 3.3.1. Definition of the Maximum Wave Height

Figure5 shows the four distinct phases defined by Huber and Hager [43]. Phase 1 is when the landslide begins moving, accelerates, and then falls into the water. Phase 2 is the underwater movement of the landslide, which is also referred to as the generation area or the splash zone in the case of a subaerial landslide. During phase 2, the energy of the landslide is transferred to the water [33, 34]. Phase 3 refers to the propagation of the water waves. Phase 4 is when the impulse wave reaches the water body boundaries and runs up or over the top of the dams, which eventually poses a great threat to human lives and property.Figure 5
Main phases of landslides-generated impulse waves.Figure6 depicts a typical waveform signal record from the experiments. As is defined by Panizzo et al. [1] and Walder et al. [33], a standard zero-crossing analysis was carried out to define the maximum wave amplitude (aM). The wave amplitude reported here is the elevation of the water surface relative to the datum (still water) within the zero-crossing wave. The maximum impulse wave height is the difference between the absolute maximum and the absolute minimum values within the maximum zero-crossing wave. The related wave period (Tmax), which is important to wave propagation, and the first wave of the generated transient train (a1(+), a1(-), and T1) were also characterized. The video images were analyzed frame by frame, and, then, the captured images were printed to measure the maximum height of the wave. The error of the measured maximum wave height is caused by the image resolution (0.4 mm per pixel) and blurring of the image (from a limited exposure time). We estimate that the error is typically within the range of 1 to 2 mm. The wavelet analysis method was used to compute the zero cross section of the wave records. Based on the work of Panizzo et al. [44], wavelet analysis was used to make a part of the wave field propagate away from the impact point, thus reducing the effect of the reflected wave on the recorded wave signal.Figure 6
Sketch of an impulse wave and the primary wave parameters.


### 3.3.2. Equation for the Maximum Wave Height

The data recorded by the wave gauges was used to derive the prediction equations for the impulse wave’s characteristics. Based on dimensional analysis, the relationship between the wave height and the four dimensionless parameters is as follows:(11)Hmaxd=fvgd,wd,Vsbd2,TsAw∗,α,where the nondimensional parameter Aw∗=bs/d2 summarizes the landslide geometry. We can obtain an empirical equation through multiple linear regression analysis, and the relationship between the maximum wave height and the five dimensionless parameters can be expressed by (11) as follows:(12)Hmaxd=0.145F0.672B0.09V0.217TsAw∗-0.045α0.038.Figure7 compare the calculated and measured wave heights of each experiment. It can be seen from Figure 7 that the empirical equation is basically consistent with the experimental data for all of the different experimental conditions.Figure 7
Comparison between measured values and predicted values.In actual landslides,Ts is a dependent variable and must be calculated. Panizzo et al. [1] developed an empirical equation for estimating Ts:(13)Ts=0.43V-0.27F-0.66sin⁡α-1.32.Equation (13) correlates well with our experimental measurements and is therefore useful in determining Ts.


## 3.3.1. Definition of the Maximum Wave Height

Figure5 shows the four distinct phases defined by Huber and Hager [43]. Phase 1 is when the landslide begins moving, accelerates, and then falls into the water. Phase 2 is the underwater movement of the landslide, which is also referred to as the generation area or the splash zone in the case of a subaerial landslide. During phase 2, the energy of the landslide is transferred to the water [33, 34]. Phase 3 refers to the propagation of the water waves. Phase 4 is when the impulse wave reaches the water body boundaries and runs up or over the top of the dams, which eventually poses a great threat to human lives and property.Figure 5
Main phases of landslides-generated impulse waves.Figure6 depicts a typical waveform signal record from the experiments. As is defined by Panizzo et al. [1] and Walder et al. [33], a standard zero-crossing analysis was carried out to define the maximum wave amplitude (aM). The wave amplitude reported here is the elevation of the water surface relative to the datum (still water) within the zero-crossing wave. The maximum impulse wave height is the difference between the absolute maximum and the absolute minimum values within the maximum zero-crossing wave. The related wave period (Tmax), which is important to wave propagation, and the first wave of the generated transient train (a1(+), a1(-), and T1) were also characterized. The video images were analyzed frame by frame, and, then, the captured images were printed to measure the maximum height of the wave. The error of the measured maximum wave height is caused by the image resolution (0.4 mm per pixel) and blurring of the image (from a limited exposure time). We estimate that the error is typically within the range of 1 to 2 mm. The wavelet analysis method was used to compute the zero cross section of the wave records. Based on the work of Panizzo et al. [44], wavelet analysis was used to make a part of the wave field propagate away from the impact point, thus reducing the effect of the reflected wave on the recorded wave signal.Figure 6
Sketch of an impulse wave and the primary wave parameters.


## 3.3.2. Equation for the Maximum Wave Height

The data recorded by the wave gauges was used to derive the prediction equations for the impulse wave’s characteristics. Based on dimensional analysis, the relationship between the wave height and the four dimensionless parameters is as follows:(11)Hmaxd=fvgd,wd,Vsbd2,TsAw∗,α,where the nondimensional parameter Aw∗=bs/d2 summarizes the landslide geometry. We can obtain an empirical equation through multiple linear regression analysis, and the relationship between the maximum wave height and the five dimensionless parameters can be expressed by (11) as follows:(12)Hmaxd=0.145F0.672B0.09V0.217TsAw∗-0.045α0.038.Figure7 compare the calculated and measured wave heights of each experiment. It can be seen from Figure 7 that the empirical equation is basically consistent with the experimental data for all of the different experimental conditions.Figure 7
Comparison between measured values and predicted values.In actual landslides,Ts is a dependent variable and must be calculated. Panizzo et al. [1] developed an empirical equation for estimating Ts:(13)Ts=0.43V-0.27F-0.66sin⁡α-1.32.Equation (13) correlates well with our experimental measurements and is therefore useful in determining Ts.


## 3.4. Equation for Wave Height Attenuation

The attenuation calculation of the landslide surge is based on the basic continuity equation and the motion equation for unsteady flow in an open channel (Li [45]):(14)T∂h∂t+u∂h∂x+A∂u∂x+bu-q=0,g∂h∂x+∂u∂t+u∂u∂x+gj-i=0,where T(h,x) is the water surface width; A(h,x) is the section area; u(x,t) is the flow velocity; h(x,t) is the water depth; j is the energy slope; i is the bed slope; and q is the increased flow per unit length. We use the hydrodynamic perturbation method to expand the continuity equation and the motion equation into polynomials. If we assume that the increased flow per unit length (q) is zero and that the friction slope (j) in the fluid is proportional to the local velocity, then (14) can be simplified as follows:(15)T∂h∂t+uT∂h∂x+A∂u∂x+bu=0,g∂h∂x+∂u∂t+u∂u∂x+gku-i=0.In order to solve (15), we can use the hydrodynamic perturbation method to linearize the equation. We assume that h¯ is the representative water depth, δ is the wave height, and ε=δ/h¯. In the perturbation method, the solutions of u and h can often be expressed as follows:(16)ux,t=u0x,t+εu1x,t+ε2u2x,t+⋯,hx,t=h0x,t+εh1x,t+ε2h2x,t+⋯,where the values of u0,u1,…,h0,h1,… need to be determined and sometimes need other forms of expansion as noted by Kevorkian and Cole [46]. By substituting (16) into (15), we obtained the following equations:(17)εu1+⋯∂∂xεu1+⋯+∂∂tεu1+⋯+g∂∂xh0+εh1+⋯+gεku1+⋯-i=0,A0+⋯∂∂xεu1+⋯+εu1+⋯T0+⋯∂∂xh0+⋯+T0+⋯∂∂th0+εh1+⋯+εu1+⋯b0+⋯=0.By simplifying (17) to the idempotent ε, we can obtain the following equations:(18)εT0∂h1∂t+u1dA0dx+A0∂u1∂x+T1∂h0∂t+⋯=0,g∂h0∂x-i+ε∂u1∂t+g∂h1∂x+gku1+⋯=0.Since (18) are assumed to be valid for any small value of ε, they are effective for each derivative of ε, so the equations for the zero order are as follows:(19)T0∂h0∂t=0,g∂h0∂x-i=0,where us represents the flow velocity (εu1) and hs represents the displacement (εh1). Thus, the equations for the first order are as follows:(20)T0∂hs∂t+usdA0dx+A0∂us∂x=0,∂us∂t+g∂hs∂x+gkus=0.Using (20) to eliminate u, we obtain the following:(21)T0∂2hs∂t2+gkT0∂hs∂t=A0g∂2hs∂x2+gdA0dx∂hs∂x.For rectangular or approximately rectangular channels, dA0/dx=0, so Equation (21) can be written as follows:(22)T0∂2hs∂t2+gkT0∂hs∂t=A0g∂2hs∂x2.After the landslide occurs, the surface of the water rises, then falls rapidly, and soon develops an oscillation with a small height. It can be assumed that the descent process obeys an exponential distribution. Therefore, we assumehs(x,t)=f(x)e-at, and (22) can be changed to(23)T0a2fxe-at-gkT0afxe-at=A0gf′′xe-at.If k1=T0a2-gkT0a/A0g, then f′′(x)=k1f(x), and we obtain the following equation:(24)fx=C1ek1x+C2ek2x.Applying the boundary condition f(0)=hj, limx→∞f(x)=0, we get C1=0, C2=hj; therefore,(25)hsx,t=hje-k1x-at=ke-mx.Next, we use the experimental data to verify and fit the equation. According to the data recorded by each wave instrument, the free surface height of the water body at different times can be obtained. Figure8 shows the variation in wave height with time for different positions in the same experiment.Figure 8
Variation in height of the surge with time for different points in the same experiment.The study of the factors affecting the attenuation of the landslide induced surge was carried out based on the wave height data of the 6 wave instruments. At different water depths, the relative distances of the six wave instruments along the cross section of the flume are shown in Figure9.Figure 9
Maximum wave heights versus the relative propagation distance.H
r
/
d can be roughly fitted by an exponential power function(26)Hrd=ke-mx/d.Hr is the height of the waves propagating along the flume. Table 4 lists the parameters for each experiment with R2≈1 for almost all cases.We find that, in each experiment, the values ofk and Hmax/d are very close; thus, the equation for the propagation of waves along the propagating path can be expressed as follows:(27)Hrd=Hmaxde-0.05x/d.Equation (27) is still able to effectively describe the attenuation of the impulse wave’s height, with a maximum error value of ≤15%.


## 4. A Case Study

The Zipingpu reservoir is located on the upper reaches of the Ming River in Maxi Town, which is about 30 km from Wenchuan county, Sichuan province, China. The reservoir was completed in 2006, and its water storage capacity is 11 billion cubic meters. The height of the dam is 156 meters, and the full-pool level altitude is 817 m.The Zipingpu landslide was a typical high-velocity landslide that occurred 15.5 km away from the earthquake epicenter. It occurred at 14:29 on 12 May 2008, about half a minute after the Wenchuan earthquake. A 25 m high huge impulse wave was observed to spread rapidly downstream. This surge killed more than 70 fishermen and destroyed more than 10 fishing boats along the shore. The Zipingpu landslide area is an irregular quadrilateral plane 180 m × 100 m × 25 m with a total volume of 4.5 × 105 m3. The vertical distance from the landslide center to the water surface is 900 m, and the average sliding surface angle is 48°.Assuming that the landslide entered the reservoir as one large mass, we can use (28), as recommended by the American Society of Civil Engineers [42], to calculate the entry velocity of the landslide:(28)vs=2gh1-cot⁡αtan⁡φ-cLmgsin⁡α,where α is the sliding surface angle; c and φ are shear strength parameters of sliding surface; H is the vertical distance from the landslide’s center to the water’s surface; m is the mass of the slider; and L is the length of the slider and the sliding surface. The calculated velocity of the landslide is 66.31 m/s. Using this velocity, the calculated maximum impulse wave height is 26.6 m.We use the empirical equation proposed by previous researchers to calculate 26.6 m for the case study, after which the calculated values are compared with the results of this paper and the measured data. The results are presented in Table5.Table 5
Comparison of the presented forecasting equations.
Method of calculation
Formula
Wave height (m)
Observed value
-
25
Equation in this paper
-
26.6
Cui and Zhu [42]
0.108F1.669V0.045S0.36α0-1.636
23.1
Noda [8]
1.32F
24.3
Walder et al. [33]
1.32(τ/M)−0.68, with τ = 4.5L0.5
22.6
Fritz et al. [15]
0.25F1.4S0.8
19.6
Panizzo et al. [40]
0.102∑0.572F0.297R−0.44(sin⁡4)−0.286exp⁡(0.6cos⁡θ)
16.4As can be seen form Table5, the calculated result is very close to the result of the eye-witness observations and the calculations of other researchers. In addition, the proposed equation takes into account parameters such as the landslide’s shape and the slope of the sliding surface, both of which are important to the formation of the impulse wave. Thus, the equations proposed in this paper are more suitable for practical cases.


## 5. Conclusions

The characteristics of the maximum impulse wave’s height and its attenuation were experimentally investigated using the orthogonal method. The experimental results were employed to develop an empirical equation to predict the maximum wave height for different landslide sizes and volumes. The following conclusions were generated.(
1
) It is possible to reduce the number of experiments and shorten the experimental period by using the orthogonal design. The reasonable mathematical statistical analysis of the experimental data in the orthogonal table can reveal the regularity and internal relationships between the index and the factors and can be used to guide the practice.(
2
) The results of the orthogonal experiments indicate that the initial drop height and slider thickness are the main factors affecting the wave height, then the slider length, slider width, sliding slope, and still water depth.(
3
) The relationship between the wave height and the four dimensionless parameters can be expressed as follows:(29)Hmaxd=0.145F0.672B0.09V0.217TsAW∗-0.045α0.038.(
4
) The hydrodynamic perturbation method was used to expand the continuity and motion equations into polynomials; it is concluded that the wave decay process exhibits an exponential form, and, by separating the variables, the attenuation law of the landslide wave was obtained.



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