LONG WAVE RUN-UP OVER SUBMERGED REEF AND BREAKWATER

LONG WAVE RUN-UP OVER SUBMERGED REEF AND BREAKWATER A THESIS SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAIʻI IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN CIVIL ENGINEERING August 2012 By Nathan T. Shimabuku Thesis Committee: Michelle H. Teng, Chairperson Janet M. Becker Marcelo Kobayashi H. Ronald Riggs 1

ACKNOWLEDGEMENT I would like to take this opportunity to thank my advisor Dr. Michelle Teng, who has mentored me through my thesis. Dr. Teng is responsible for igniting my initial interest in the field of fluid mechanics. I will be eternally grateful for her kindness and support. I am also grateful and appreciative to Dr. Janet Becker and Dr. Marcello Kobayashi for all the times when I needed help and encouragement. I would like to thank Dr. Riggs and Dr. Robertson for allowing me to travel to Japan to investigate tsunami damage and allowing me to participate in the joint Hawaii-Korea workshop on coastal engineering. I would also like to thank my undergraduate research assistants Nelson Fernandez and Isaiah Sato for their endless help and support. Thank you to Nadine Kawabata and Matthew Asada who has been extremely supportive as friends and colleagues. Lastly I would like to thank my parents Everett and Carol Shimabuku who have supported me unconditionally. This study was funded by National Science Foundation NEES-R Grant No. 0619083. 2

ABSTRACT In this thesis study, joint numerical and experimental studies were carried out to examine the effects that a submerged breakwater s length, distance from the beach and height have on the wave runup. The numerical program was based on a staggered grid leapfrog method on the shallow water equations which is used to simulate runup. Propagation over the reef is simulated with a predictor corrector Boussinesq scheme. We experimentally observed that a trend based on the wave s wavelength, length of the breakwater and lagoon spacing. We see that maximum and minimum runup reduction is observed at half integer and integer values of ζ and γ depending if the wave breaks. We define ζ and γ as the breakwater s lagoon spacing or the breakwater s length in terms of wavelength of the initial wave divided by the breakwater s height in terms of still water depth. We observed that the maximum runup reduction for a non breaking wave and minimum runup reduction for breaking waves were observed at half integer and integer values of γ. We also observed that if a breakwater is present the runup generated by waves can be reduced up to 40% but may be increased by 5%, with the exception of one measurement and that if wave breaking does occur the runup can be reduced by 80% and that it can be increased by 40%. 3

Contents ABSTRACT...3 1 INTRODUCTION...8 1.1 Tsunamis and Breakwaters...8 1.2 Past Work...8 1.3 Objectives of the Present Study... 11 2 THEORY... 12 2.1 Shallow Water Equation... 12 2.2 Boussinesq Equation... 14 3 NUMERICAL SCHEMES... 21 3.1 Numerical Scheme for Long Wave Propagation... 21 3.2 Numerical Scheme for Long Wave Propagation over a Sloping Beach:... 23 3.3 Boundary Conditions and Initial Conditions of the Boussinesq Equation:... 24 3.4 Boundary Conditions and Initial Conditions of the SWE:... 25 3.5 Boussinesq Propagation... 27 3.6 Boussinesq propagation over a step... 33 3.7 NSWE propagation... 34 3.8 NSWE against Synolakis s theoretical results... 48 3.9 Stability... 51 4 NUMERICAL RESULTS... 53 5 EXPERIMENTAL SETUP... 59 5.1 Wave Flume and Wave Generation... 59 5.2 Artificial Beach... 60 5.3 Measurement of Wave Height and Run-Up... 61 6 EXPERIMENTAL RESULTS... 63 6.1 Non Breaking Waves... 63 6.2 Breaking Waves... 74 7 CONCLUSION... 87 8 RERERENCES... 89 4

Figure 2.1 Definition sketch for shallow water equations... 12 Figure 2.2 Definition sketch Boussinesq equations... 15 Figure 3.1 Initial condition for the numerical simulation of the Boussinesq equations for an amplitude of 0.15... 25 Figure 3.2 A illustrative method of the numerical shallow water equation s boundary condition26 Figure 3.3 The initial condition for the numerical shallow water equations given in red and the Boussinesq equation s initial condition green.... 27 Figure 3.4 Boussinesq numerical scheme s propagation of a solitary wave whose α = 0.2... 28 Figure 3.5 Amplitude lost vs. CFL for α = 0.05... 29 Figure 3.6 Amplitude lost vs. CFL for α = 0.1... 30 Figure 3.7Amplitude lost vs. CFL for α = 0.15... 31 Figure 3.8 Amplitude lost vs. CFL for α = 0.05... 32 Figure 3.9 Wave fission comparison with Mei and Madsen... 34 Figure 3.10 Wave propagation over a constant depth... 35 Figure 3.11 Runup vs. CFL for α = 0.05 and a β = 20... 36 Figure 3.12 Runup vs. CFL for α = 0.1 and a β = 20... 37 Figure 3.13 Runup vs. CFL for α = 0.15 and a β = 20... 38 Figure 3.14 Runup vs. CFL for α = 0.2 and a β = 20... 39 Figure 3.15 Runup vs. CFL for α = 0.05 and a β = 15... 40 Figure 3.16 Runup vs. CFL for α = 0.1 and a β = 15... 41 Figure 3.17 Runup vs. CFL for α = 0.15 and a β = 15... 42 Figure 3.18 Runup vs. CFL for α = 0.2 and a β = 15... 43 Figure 3.19 Runup vs. CFL for α = 0.05 and a β = 10... 44 Figure 3.20 Runup vs. CFL for α = 0.1 and a β = 10... 45 Figure 3.21 Runup vs. CFL for α = 0.15 and a β = 10... 46 Figure 3.22 Runup vs. CFL for α = 0.2 and a β = 10... 47 Figure 3.23 Numerical and theoretical comparison for 20 degree sloping beach, where f(x) corresponds to Synolakis formula and 20 Degree.txt corresponds to the numerical model of the shallow water equations... 48 Figure 3.24 Numerical and theoretical comparison for 15 degree sloping beach, where f(x) corresponds to Synolakis formula and 15 Degree.txt corresponds to the numerical model of the shallow water equations... 49 Figure 3.25 Numerical and theoretical comparison for 10 degree sloping beach, where f(x) corresponds to Synolakis formula and 10 Degree.txt corresponds to the numerical model of the shallow water equations... 50 Figure 4.1 Numerical results of a 0.75 high breakwater on a 20 degree sloping beach for runup against lagoon spacing for various breakwater lengths... 53 Figure 4.2 Numerical results of a 0.75 high breakwater on a 20 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 54 Figure 4.3 Numerical results of a 0.75 high breakwater on a 10 degree sloping beach for runup against lagoon spacing for various breakwater lengths... 55 Figure 4.4 Numerical results of a 0.75 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 56 Figure 4.5 Numerical results of a 0.75 high breakwater on a 5 degree sloping beach for runup against lagoon spacing for various breakwater lengths... 57 5

Figure 4.6 Numerical results of a 0.75 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 58 Figure 5.1 Experimental Set up... 59 Figure 5.2 Artificial beach set up... 60 Figure 5.3Unistrut apparatus/ base... 60 Figure 5.4 Wave gauge output... 61 Figure 5.5 Analytical and Experimental comparison of a hand generated solitary wave... 62 Figure 6.1 Experimental results of a 0.5 high breakwater on a 20 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 63 Figure 6.2 Experimental results of a 0.5 high breakwater on a 20 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 64 Figure 6.3 Breakwater length lines for a 0.5 high breakwater on a 20 degree sloping beach.. 65 Figure 6.4 Breakwater length trends for a 0.5 high breakwater on a 20 degree sloping beach without a lagoon... 66 Figure 6.5 Breakwater length trends for a 0.5 high breakwater on a 20 degree sloping beach for various lagoon spaces... 67 Figure 6.6 Experimental results of a 0.75 high breakwater on a 20 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 68 Figure 6.7 Experimental results of a 0.75 high breakwater on a 20 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 69 Figure 6.8 Experimental results of a 0.5 high breakwater on a 10 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 70 Figure 6.9 Experimental results of a 0.5 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 71 Figure 6.10 Experimental results of a 0.75 high breakwater on a 10 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 72 Figure 6.11 Experimental results of a 0.75 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 73 Figure 6.12 Experimental results of a 0.5 high breakwater on a 20 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 75 Figure 6.13 Breaking (X) and non breaking (+) observations of figure 6.12... 76 Figure 6.14 Experimental results of a 0.5 high breakwater on a 20 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 77 Figure 6.15 Breaking (+) and non breaking (X) observations of figure 6.14... 78 Figure 6.16 Experimental results of a 0.75 high breakwater on a 20 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 79 Figure 6.17 Breaking (+) and non breaking (X) observations of figure 6.16... 80 Figure 6.18 Experimental results of a 0.5 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 81 Figure 6.19 Breaking (+) and non breaking (X) observations of figure 6.18... 82 Figure 6.20 Experimental results of a 0.75 high breakwater on a 10 degree sloping beach for runup against lagoon spaces for various breakwater lengths... 83 Figure 6.21 Breaking (+) and non breaking (X) observations of figure 6.20... 84 Figure 6.22 Experimental results of a 0.75 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces... 85 Figure 6.23 Breaking (X) and non breaking (+) observations of figure 6.22... 86 6

Table 3.1 Results for a α = 0.05... 29 Table 3.2 Results for α=0.1... 30 Table 3.3 Results for α=0.15... 31 Table 3.4 Amplitude lost vs. CFL for α = 0.2... 32 Table 3.5 Summary of the Boussinesq numerical simulation... 33 Table 3.6 Runup of α = 0.05 and a β = 20... 36 Table 3.7 Runup of α = 0.1 and a β = 20... 37 Table 3.8 Runup of α = 0.15 and a β = 20... 38 Table 3.9 Runup of α = 0.2 and a β = 20... 39 Table 3.10 Runup of α = 0.05 and a β = 15... 40 Table 3.11 Runup of α = 0.1 and a β = 15... 41 Table 3.12 Runup of α = 0.15 and a β = 15... 42 Table 3.13 Runup of α = 0.2 and a β = 15... 43 Table 3.14 Runup of α = 0.05 and a β = 10... 44 Table 3.15 Runup of α = 0.1 and a β = 10... 45 Table 3.16 Runup of α = 0.15 and a β = 10... 46 Table 3.17 Runup of α = 0.2 and a β = 10... 47 Table 3.18 Set of non breaking alphas... 51 Table 3.19 Numerical diffusion comparison... 52 7

1 INTRODUCTION 1.1 Tsunamis and Breakwaters Tsunamis or long waves have become a growing topic of interest to engineers in recent years. In this thesis we will numerically and experimentally investigate the effect that a reef or a submerged breakwater has on the final run up of tsunamis onto coastal land. A tsunami is defined as a series of waves generated by any large disturbance of seawater. From this definition we see that tsunamis are generated by earthquakes, volcanic eruptions, landsides or meteors. The term tsunami comes from the Japanese word for harbor wave, due to the fact that when a tsunami enters a harbor or bay it can cause large standing waves that last for hours. Tsunamis gain their massive destructive power through the initial rise of the sea surface which generates a large potential energy. This energy is transferred to kinetic energy in the form of wave propagation. Tsunamis usually do not break, and most of their destructive force is caused by debris and strong currents generated by a rapid local rise of sea level. The term long wave is defined as a wave whose wavelength is much greater than the water depth. Tsunamis fall into this category because a tsunami s wavelength is about 200 km and the maximum ocean depth is about 8 10 km; tsunamis clearly fit a long wave description. Breakwaters, as the name suggests are natural or manmade structures used to dissipate waves. They are implemented to mitigate coastal erosion. They can be structures that are below the water, above the water and floating. Engineers and researchers have a great interest in learning whether submerged reefs and breakwaters can help to reduce the tsunami run-up on land. 1.2 Past Work 8

Madsen and Mei (1969) were among the first to study a long wave propagating over a submerged topographical change such as a shelf or a step. Results from their 1969 paper found that a solitary wave of 0.12, here we define the average water depth as, amplitude traveling over a upward step of 0.5 would evolve into two waves, with the ratio of the new wave amplitude to the initial wave of 1.56. This phenomenon of one wave separating into several waves is known as wave fission. Madsen and Mei s (1969) numerical study was based on the nonlinear and dispersive Boussinesq model. A similar Boussinesq model is applied in the present study to simulate solitary waves propagating over submerged reef and breakwaters. For wave run-up on beaches, one of the earlier important studies was done by Synolakis in 1987 who derived the following formula to predict the maximum run-up of a solitary wave on a smooth plane beach: Here he defined R as the run-up, h as the mean water level, β as the angle of the beach and a as the initial wave amplitude. He also verified with his experimental data that the solution (1.1) is valid for non-breaking wave run-up. The analytical solution (1.1) was derived by Synolakis (1987) based on the nonlinear and non-dispersive shallow water equations using a hodograph transformation. During wave run-up on a beach, the nonlinear effect is dominant; therefore, the dispersive effect may be neglected. 1.1 The following limitations due to singularties have been cited in his paper for the non-breaking condition: β 1.2 β 1.3 9

Equation 1.2 was derived in a previous paper by Gjevik and Pedersen (1981) and equation 1.3 was obtained by Synolakis (1987). Another important study on wave run-up was carried out by Liu, Cho, Briggs, Synolakis and Kanoglu in 1995. They developed a finite difference scheme with a staggered grid and an explicit leapfrog time forward method to solve the shallow wave equations and simulated solitary wave run-up on a conical island. They verified that their numerical prediction was in good agreement with their laboratory data. The same equation and numerical scheme developed by Liu et al (1995) were adopted in the present study to simulate wave run-up on beaches. Further discussion on this topic occurs in chapter 4. In recent years, researchers have carried out many excellent studies on the effect of submerged reef and breakwaters on long wave run-up. For example, Kunkel et al (2006) predicted tsunami run-up after the tsunami wave passes over a submerged reef through numerical simulations. They found that the taller the reef, the more reduction in the final wave run-up. They also studied the effect of a lagoon on tsunami propagation and run-up reduction. However, the authors did not investigate or specify whether the reduction is due to wave breaking or wave fission. Mohandie (2008) carried out a joint numerical and experimental study to investigate the effectiveness of beach vegetation and a submerged reef as potential natural barriers for reducing long wave run-up. He found that for a submerged reef to be effective, the height of the reef needs to be taller than half of the water depth. Mohandie s (2008) numerical simulation showed that there exists an optimal reef length that can reduce the run-up the most. Specifically, it was found that when the reef length is about twice the wavelength, the reef is the most effective in reducing the run-up. In addition, Mohandie s (2008) experimental results showed that when waves break over the reef, the final run-up will be reduced much more than non-breaking waves over the reef. Irtem et al (2011) performed a series of wave tank experiments to study solitary wave run-up after propagating over a single submerged breakwater. It was observed that with a submerged breakwater, the run-up can be reduced by 20-40% especially if the breakwater is porous. Other excellent earlier work included those by Hanson and Kraus (1989,1990,1991), among others. While the aforementioned existing studies are very useful and valuable, each of these studies usually focused on only one or two factors. There are still other factors that have not been 10

fully investigated. For example, the length of the reef in Mohandie s (2008) study was relatively long. Shorter reefs were not studied. Also, the effect of multiple breakwaters with gaps in between was not examined. 1.3 Objectives of the Present Study The present study is an extension of the previous existing studies. Our main goal is to seek the conditions or configurations of reef and breakwater that are most effective in reducing long wave run-up on coastal land. Specifically, the objectives of the present joint numerical and experimental study are: (1) investigate the effects of reef length and height on run-up reduction (2) investigate the effect of reef height on run-up reduction; (3) examine the effect of lagoon spacings on run-up reduction; (4) study how wave fission and wave breaking affect the final wave run-up. 11

2 THEORY 2.1 Shallow Water Equation The shallow water equations (SWE) are used to approximate the propagation of a long wave. The shallow water equations are derived from the Navier-Stokes equation 2.1 and the continuity equation 2.2: 2.1 Where we define v as the velocity vector that is comprised of the velocity components (u, v, and w), ρ as the fluid s density, p as the fluid s pressure and μ as the dynamic viscosity of the fluid. 2.2 η z x h Figure 2.1 Definition sketch for shallow water equations 12

The kinematic boundary condition at the surface is obtained by writing the free surface, F, in terms of η, the position of the free surface and the vertical position z in 2.3 and taking the material derivative of 2.3 resulting in 2.4, the kinematic boundary condition: 2.3 Similarly taking the surface at the seabed and taking the material derivative of 2.5 results in the kinematic seabed boundary conditions 2.6 where, 2.4 2.5 The dynamic boundary condition is formulated from Euler s integral. We set the pressure equal to the atmospheric pressure, at the surface, η. We also assume that the horizontal length is much greater than the vertical, thus we can assume that the pressure is hydrostatic and set. Thus we obtain a function form of pressure given by 2.7: 2.6 The SWE are obtained by integrating equations 2.1 and 2.2 over the vertical axis, z, and applying Leibniz rule. The surface terms are evaluated with the kinematic boundary conditions 2.4 and 2.6. The pressure term in 2.1 is replaced by 2.7, thus the one dimensional momentum SWE 2.8 and the continuity SWE 2.9 are formulated. 2.7 13 2.8

2.9 Where we define: and 2.10 2.2 Boussinesq Equation The Boussinesq equations are derived using potential theory, which follows from the Laplace equation, where is the velocity potential: The following boundary conditions, that include the seabed, and the dynamic and kinematic free surface conditions, where is the position of the free surface and the average water depth yeilds: 2.11 2.12 2.13 2.14 14

z x Figure 2.2 Definition sketch Boussinesq equations We scale our variables with, the wavelength, the wave s relative velocity, the average water depth and as the relative amplitude of the wave. We also note that the unprimed variables are the non dimensional variables: 2.15 2.16 2.17 We define an aspect ratio, a velocity scale and a time scale: 2.18 15

2.19 2.20 2.21 We scale using 2.15-.18 such that: 2.22 We find a dominant scaling for the velocity potential as: 2.23 Thus we define the scaling parameters ε, the dispersive effect, δ, the nonlinear effect and the Ursell number: 2.24 2.25 2.26 Plugging in our scaled values to our original BVP, 2.11-.14, our scaled BVP becomes: 16

2.27 2.28 2.29 2.30 We see that Laplace s equation 2.27 and the sea bed boundary condition 2.28 do not contain δ thus we recast the velocity potential ϕ as a perturbation expansion in the form: 2.31 Where we rewrite Laplace s equation, 2.27 as: 2.32 Collecting the zeroth order terms from equation 2.27, we see that: 17

2.33 The solution to equation 2.33 using variation of parameters is: 2.34 Where we see A=0, from our seabed boundary condition, 3.28. Similarly we can obtain the first and second order terms and find that: 2.35 2.36 2.37 Thus combining the terms 2.35-2.37 we see that the velocity potential: 2.38 Also we see that we can rewrite the velocity as: 2.39 18

2.40 We now consider horizontal velocity reference frame relative to the bottom: 2.41 We substitute 2.41 into 2.39 and 2.40 to rewrite the velocities in terms of the relative bottom velocity: 2.42 We may rewrite the dynamic boundary condition, 2.29 and kinematic boundary condition, 2.30 in terms of the horizontal and vertical velocity: 2.43 2.44 Substituting and rewriting the dynamic, 2.44 and kinematic, 2.45 boundary conditions in terms of our reference velocity, 2.41: 2.45 2.46 19

2.47 Now we rewrite our reference bottom velocity as a vertically averaged horizontal velocity: 2.48 We rewrite the dynamic and kinematic boundary condition as: 2.49 2.50 Finally writing the dynamic and kinematic boundary conditions in terms of we arrive at the dimensional Boussinesq equations: 2.51 2.52 The following derivation was obtained from lecture notes of Dr. Joseph Hammack. 20

3 NUMERICAL SCHEMES 3.1 Numerical Scheme for Long Wave Propagation We use the Boussinesq propagation model to simulate the solitary wave s propagation and interaction with the submerged breakwater. Here we will discuss the numerical implementation that is carried out for the Boussinesq equations: 3.1 3.2 To solve the coupled equations above we implemented a predictor-corrector method (Teng 1990, Mohandie 2008) on both equations. For equation 3.1 we implement a forward difference on the time derivative and a central difference on the spatial derivative, thus allowing us to solve for the predicted value of η. We define G as the approximation of the spatial derivative: 3.3 3.4 For equation 3.2, we collect the time differentiated terms and approximate them by a forward difference in time to obtain the predicted value for u. We see that this yields an implicit scheme requiring a tri-diagonal matrix algorithm to solve this set of equations. 21

3.5 As before we define F as the terms not under time differentiation, and apply a central difference approximation to both u and η. We also average non differentiated term on the nonlinear part of equation 3.2 to yield: 3.6 3.7 We now apply the same treatment to find the corrected terms which are in the same form as the predicted step: We define spatial terms G and F by correcting them: 3.8 3.9 22

3.10 This concludes our discussion for the Boussinesq numerical scheme. 3.2 Numerical Scheme for Long Wave Propagation over a Sloping Beach: To simulate the propagation and the interaction with the sloping beach we use the nonlinear shallow water equations. Here we will discuss the numerical implementation that is carried out over the non linear shallow water equations. We implement the discretization as proposed in Liu et al (1995), where we implement a staggered explicit finite difference leap-frog scheme on the linear terms of the nonlinear shallow water equations (2.8 and 2.9). 3.11 For the nonlinear terms, we apply the upwind scheme as shown: 3.12 where we define 3.13 Combining the linear and non linear equations yield: 23 3.14

3.15 3.16 This concludes our discussion of the non linear shallow water equations. 3.3 Boundary Conditions and Initial Conditions of the Boussinesq Equation: For the case of wave propagation and interaction with the breakwater, we have chosen a large computational domain with at least 100 water depths such that the boundaries have no interaction with the solution. 3.17 The initial conditions are provided by a solitary wave which is a solution to the Boussinesq equation (Teng and Wu 1992, Teng 1997): 3.18 3.19 3.20 24

Figure 3.1 Initial condition for the numerical simulation of the Boussinesq equations for an amplitude of 0.15 3.4 Boundary Conditions and Initial Conditions of the SWE: The boundary conditions are modified to include run-up using a moving boundary scheme developed by Liu et al (1995). This is illustrated below in figure 3.2. We define the shoreline as the grid point between the wet cell and the dry cell. If the water level at the wet cell is greater than the height of the dry cell then the dry cell becomes a wet cell and the shore line moves, however if the water height is less than the height of the dry cell then the shoreline remains at the cell. 25

Figure 3.2 A illustrative method of the numerical shallow water equation s boundary condition The initial condition for the non linear shallow water equations follows from the wave profile generated from the Boussinesq part of the code. As generated below: 26

Figure 3.3 The initial condition for the numerical shallow water equations given in red and the Boussinesq equation s initial condition green. Here we see the green line as the initial wave input to Boussinesq numerical model and the red as initial wave for the SWE numerical model to further illustrate the sequence of inputs for the numerical model. 3.5 Boussinesq Propagation To test the accuracy and convergence of the Boussinesq equations we tested several CFL combinations over a length of about 90 water depths. Shown below are the results for our 4 amplitudes and its error or loss of amplitude in %. Shown below is a sample of a solitary wave with a 0.2 amplitude propagating over a constant depth of 1: 27

Figure 3.4 Boussinesq numerical scheme s propagation of a solitary wave whose α = 0.2 In the figure 3.4 we see that the initial wave is represented by the green solitary wave and the propagated wave is red solitary wave. Here we can clearly see that the amplitude is reduced, which is caused by an artificial dissipation due to the central difference disctritization used on equation 3.7. Located below are the results that were obtained from the Boussinesq numerical simulation for various amplitudes, where we determine the amplitude reduction, and its CFL number. We also graph the reduction against the CFL number as accuracy test. 28

Amplitude Reduction 0.05 Amplitude Test Reduction dx dt CFL Amplitude % 0.5 0.005 0.01 0.048711713 2.576574 0.4 0.005 0.0125 0.04915264 1.69472 0.3 0.005 0.016667 0.049426343 1.147314 0.2 0.005 0.025 0.04963096 0.73808 0.1 0.005 0.05 0.049535315 0.92937 0.09 0.005 0.055556 0.048994042 2.011916 0.08 0.005 0.0625 0.04942428 1.15144 0.07 0.005 0.071429 0.048848696 2.302608 0.06 0.005 0.083333 0.047980443 4.039114 0.05 0.005 0.1 0.04583642 8.32716 Table 3.1 Results for a α = 0.05 Convergence and Accuracy Of 0.05 Amplitude 10 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 CFL Figure 3.5 Amplitude lost vs. CFL for α = 0.05 29

Amplitude Reduction 0.1 Amplitude Test Reduction dx dt CFL Amplitude % 0.5 0.005 0.01 0.09438289 5.61711 0.4 0.005 0.0125 0.09616513 3.83487 0.3 0.005 0.016667 0.09748598 2.51402 0.2 0.005 0.025 0.098568276 1.431724 0.1 0.005 0.05 0.09879608 1.20392 0.09 0.005 0.055556 0.0976608 2.3392 0.08 0.005 0.0625 0.098614536 1.385464 0.07 0.005 0.071429 0.097407 2.593 0.06 0.005 0.083333 0.095562495 4.437505 0.05 0.005 0.1 0.09098442 9.01558 Table 3.2 Results for α=0.1 Convergence and Accuracy of 0.1 Amplitude 10 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 CFL Figure 3.6 Amplitude lost vs. CFL for α = 0.1 30

Amplitude Reduction 0.15 Amplitude Test dx dt CFL Amplitude Reduction 0.5 0.005 0.01 0.13847539 7.683073333 0.4 0.005 0.0125 0.14210953 5.260313333 0.3 0.005 0.016667 0.14483266 3.444893333 0.2 0.005 0.025 0.1470714 1.9524 0.1 0.005 0.05 0.1477683 1.4878 0.09 0.005 0.055556 0.14601937 2.653753333 0.08 0.005 0.0625 0.14753309 1.644606667 0.07 0.005 0.071429 0.1456724 2.885066667 0.06 0.005 0.083333 0.14277141 4.81906 0.05 0.005 0.1 0.13559447 9.603686667 Table 3.3 Results for α=0.15 Convergence and Accuracy of 0.15 Amplitude 12 10 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 CFL Figure 3.7Amplitude lost vs. CFL for α = 0.15 31

Amplitude Reduction 0.2 Amplitude Test Reduction dx dt CFL Amplitude % 0.5 0.005 0.01 0.18200746 8.99627 0.4 0.005 0.0125 0.18751782 6.24109 0.3 0.005 0.016667 0.19199507 4.002465 0.2 0.005 0.025 0.19544706 2.27647 0.1 0.005 0.05 0.19660959 1.695205 0.09 0.005 0.055556 0.19418797 2.906015 0.08 0.005 0.0625 0.19632322 1.83839 0.07 0.005 0.071429 0.19374 3.13 0.06 0.005 0.083333 0.18979314 5.10343 0.05 0.005 0.1 0.17993276 10.03362 Table 3.4 Amplitude lost vs. CFL for α = 0.2 Convergence and Accuracy of 0.2 Amplitude 12 10 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 CFL Figure 3.8 Amplitude lost vs. CFL for α = 0.05 32

From these results it is clear that there is an upper and lower bound for the CFL, namely we see that a dx of 0.1 and a dt of 0.005 obtains the best results which is a CFL of 0.05, however this is not the case for the amplitude of 0.05, where we found that a dx of 0.2, CFL of 0.025, works a little better. Located below is a summary of the results found from this study: Amplitude CFL Simulated Error (%) 0.05 0.025 0.04963096 0.73808 0.1 0.05 0.09879608 1.20392 0.15 0.05 0.1477683 1.4878 0.2 0.05 0.19660959 1.695205 Table 3.5 Summary of the Boussinesq numerical simulation We conclude that this agrees with the theory and that our error is acceptable. 3.6 Boussinesq propagation over a step To further validate the accuracy of our scheme we observed the effects and phenomena that occur when a solitary wave propagates over a step. Here we initially compare our model s results with Madsen and Mei (1969), who studied the numerical and experimental transformation of a solitary wave propagating over a step. They determined that a solitary wave would increase in amplitude as it propagated over the step. As it undergoes this transformation the wave also separates into two waves, this process is known a wave fission and can be observed in the figure below. 33

Figure 3.9 Wave fission comparison with Mei and Madsen Madsen and Mei also observed a value for the ratio between the maximum wave amplitude after the transformation and the initial wave amplitude. They found that for an initial wave amplitude of 0.12 the observed ratio was 1.56; from our results we obtained a value of 1.74 which is an error of 11.6% of Madsen and Mei as seen in figure 3.9. 3.7 NSWE propagation To validate the non linear shallow water equations we will test it s propagation over a constant depth: 34

Figure 3.10 Wave propagation over a constant depth As we know for linear nondispersive waves, its shape will not change over a constant depth. For our numerical results, we observe the effects of wave propagation over a constant depth; we have a 1.6% growth in the wave amplitude. Located below are the results that were obtained from the non linear shallow water equation numerical simulation for various amplitudes, where we determine the runup, and its CFL number. We also graph the reduction against the CFL number as accuracy test where we define the error as: 3.21 35

dx dt CFL runup Error % 0.01 0.005 0.5 0.12795-15.3253 0.02 0.005 0.25 0.11854-6.84399 0.03 0.005 0.166667 - - 0.04 0.005 0.125 - - 0.05 0.005 0.1 0.113308-2.12771 0.06 0.005 0.083333 0.110586 0.325251 0.07 0.005 0.071429 0.104473 5.834921 0.08 0.005 0.0625 0.113663-2.44772 0.09 0.005 0.055556 0.118583-6.883 0.1 0.005 0.05 0.126143-13.6969 0.2 0.005 0.025 0.122442-10.361 0.3 0.005 0.016667-0.00683 106.159 0.4 0.005 0.0125 0.116681-5.16815 0.5 0.005 0.01 0.11942-7.63683 Table 3.6 Runup of α = 0.05 and a β = 20 Figure 3.11 Runup vs. CFL for α = 0.05 and a β = 20 36

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 0.281223-6.57232 0.03 0.005 0.166667 0.279715-6.00074 0.04 0.005 0.125 0.272628-3.31519 0.05 0.005 0.1 0.276491-4.7789 0.06 0.005 0.083333 0.274065-3.85944 0.07 0.005 0.071429 0.274141-3.8883 0.08 0.005 0.0625 0.272577-3.29586 0.09 0.005 0.055556 0.271592-2.9226 0.1 0.005 0.05 0.27111-2.73981 0.2 0.005 0.025 0.262191 0.640256 0.3 0.005 0.016667 0.260139 1.417874 0.4 0.005 0.0125 0.244009 7.530555 0.5 0.005 0.01 0.236148 10.5096 Table 3.7 Runup of α = 0.1 and a β = 20 Figure 3.12 Runup vs. CFL for α = 0.1 and a β = 20 37

dx dt cal runup Error % 0.01 0.005 0.5 0.452651-3.33385 0.02 0.005 0.25 0.453974-3.63597 0.03 0.005 0.166667 0.440684-0.60201 0.04 0.005 0.125 0.437334 0.162877 0.05 0.005 0.1 0.433761 0.978475 0.06 0.005 0.083333 0.43037 1.752442 0.07 0.005 0.071429 0.426989 2.524396 0.08 0.005 0.0625 0.423748 3.26423 0.09 0.005 0.055556 0.42103 3.884834 0.1 0.005 0.05 0.417746 4.634541 0.2 0.005 0.025 0.394027 10.0491 0.3 0.005 0.016667-0.00559 101.2772 0.4 0.005 0.0125 0.363 17.1322 0.5 0.005 0.01 0.353277 19.35185 Table 3.8 Runup of α = 0.15 and a β = 20 Figure 3.13 Runup vs. CFL for α = 0.15 and a β = 20 38

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 0.38597 38.5022 0.03 0.005 0.166667 0.608632 3.024877 0.04 0.005 0.125 0.601983 4.084304 0.05 0.005 0.1 0.590256 5.95279 0.06 0.005 0.083333 0.585509 6.709208 0.07 0.005 0.071429 0.587055 6.462815 0.08 0.005 0.0625 0.571574 8.929428 0.09 0.005 0.055556 0.57368 8.593941 0.1 0.005 0.05 0.568319 9.448029 0.2 0.005 0.025 0.522191 16.79779 0.3 0.005 0.016667 20.31853-3137.41 0.4 0.005 0.0125 0.468629 25.33196 0.5 0.005 0.01 - - Table 3.9 Runup of α = 0.2 and a β = 20 Figure 3.14 Runup vs. CFL for α = 0.2 and a β = 20 39

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 0.135557 4.832965 0.03 0.005 0.166667 - - 0.04 0.005 0.125 - - 0.05 0.005 0.1 0.121952-5.68872 0.06 0.005 0.083333 - - 0.07 0.005 0.071429 - - 0.08 0.005 0.0625 - - 0.09 0.005 0.055556 0.132585 2.534391 0.1 0.005 0.05 - - 0.2 0.005 0.025 0.133969 3.604928 0.3 0.005 0.016667 0.130923 1.248863 0.4 0.005 0.0125 0.126551-2.13219 0.5 0.005 0.01 0.127077-1.72562 Table 3.10 Runup of α = 0.05 and a β = 15 Figure 3.15 Runup vs. CFL for α = 0.05 and a β = 15 40

dx dt CFL runup Error % 0.01 0.005 0.5 0.291407 5.248173 0.02 0.005 0.25 - - 0.03 0.005 0.166667 0.303576 1.291571 0.04 0.005 0.125 0.301579 1.940786 0.05 0.005 0.1 0.299629 2.57485 0.06 0.005 0.083333 0.297968 3.114941 0.07 0.005 0.071429 0.296768 3.505306 0.08 0.005 0.0625 0.296393 3.627151 0.09 0.005 0.055556 - - 0.1 0.005 0.05 0.291517 5.212562 0.2 0.005 0.025 0.276268 10.17075 0.3 0.005 0.016667 0.268152 12.80977 0.4 0.005 0.0125 0.257205 16.36903 0.5 0.005 0.01 0.250185 18.65185 Table 3.11 Runup of α = 0.1 and a β = 15 Figure 3.16 Runup vs. CFL for α = 0.1 and a β = 15 41

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 - - 0.03 0.005 0.166667 0.48255 5.481889 0.04 0.005 0.125 0.467064 8.515179 0.05 0.005 0.1 0.461524 9.600288 0.06 0.005 0.083333 0.462748 9.360597 0.07 0.005 0.071429 - - 0.08 0.005 0.0625-199.479 39172.29 0.09 0.005 0.055556 0.44835 12.18078 0.1 0.005 0.05 0.448009 12.24754 0.2 0.005 0.025 0.407982 20.08767 0.3 0.005 0.016667 0.389304 23.74631 0.4 0.005 0.0125 0.374939 26.55986 0.5 0.005 0.01 0.356589 30.15411 Table 3.12 Runup of α = 0.15 and a β = 15 Figure 3.17 Runup vs. CFL for α = 0.15 and a β = 15 42

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 - - 0.03 0.005 0.166667 - - 0.04 0.005 0.125 - - 0.05 0.005 0.1 - - 0.06 0.005 0.083333-0.01313 98.20488 0.07 0.005 0.071429 0.597735 18.28393 0.08 0.005 0.0625 0.589831 19.36443 0.09 0.005 0.055556 0.591396 19.15053 0.1 0.005 0.05 0.587975 19.61823 0.2 0.005 0.025 - - 0.3 0.005 0.016667 0.499661 31.69162 0.4 0.005 0.0125 - - 0.5 0.005 0.01 0.461044 36.97088 Table 3.13 Runup of α = 0.2 and a β = 15 Figure 3.18 Runup vs. CFL for α = 0.2 and a β = 15 43

dx dt CFL runup Error % 0.01 0.005 0.5 0.166158-4.23518 0.02 0.005 0.25 0.164037-2.90426 0.03 0.005 0.166667 0.157719 1.059006 0.04 0.005 0.125 0.157046 1.48132 0.05 0.005 0.1 0.156269 1.968495 0.06 0.005 0.083333 0.155553 2.417697 0.07 0.005 0.071429 0.150037 5.878101 0.08 0.005 0.0625 0.148962 6.552657 0.09 0.005 0.055556 - - 0.1 0.005 0.05 0.152607 4.265952 0.2 0.005 0.025 0.146931 7.826722 0.3 0.005 0.016667 0.142342 10.70513 0.4 0.005 0.0125 0.141227 11.40457 0.5 0.005 0.01 0.136 14.68403 Table 3.14 Runup of α = 0.05 and a β = 10 Figure 3.19 Runup vs. CFL for α = 0.05 and a β = 10 44

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 - - 0.03 0.005 0.166667-0.02141 105.6474 0.04 0.005 0.125 0.341129 10.02161 0.05 0.005 0.1 0.328044 13.47294 0.06 0.005 0.083333 0.332076 12.40948 0.07 0.005 0.071429 0.320924 15.35096 0.08 0.005 0.0625 0.325428 14.16303 0.09 0.005 0.055556 0.314583 17.02353 0.1 0.005 0.05 0.318395 16.01798 0.2 0.005 0.025 0.301727 20.41445 0.3 0.005 0.016667 0.275028 27.45694 0.4 0.005 0.0125 0.265303 30.02203 0.5 0.005 0.01 0.254888 32.76913 Table 3.15 Runup of α = 0.1 and a β = 10 Figure 3.20 Runup vs. CFL for α = 0.1 and a β = 10 45

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 - - 0.03 0.005 0.166667 - - 0.04 0.005 0.125 - - 0.05 0.005 0.1 - - 0.06 0.005 0.083333 - - 0.07 0.005 0.071429 - - 0.08 0.005 0.0625 - - 0.09 0.005 0.055556 - - 0.1 0.005 0.05 - - 0.2 0.005 0.025 0.41447 34.14313 0.3 0.005 0.016667 0.388166 38.32279 0.4 0.005 0.0125 0.369595 41.27347 0.5 0.005 0.01 0.354636 43.65046 Table 3.16 Runup of α = 0.15 and a β = 10 Figure 3.21 Runup vs. CFL for α = 0.15 and a β = 10 46

dx dt CFL runup Error % 0.01 0.005 0.5 - - 0.02 0.005 0.25 - - 0.03 0.005 0.166667 - - 0.04 0.005 0.125 - - 0.05 0.005 0.1 - - 0.06 0.005 0.083333-0.01195 98.67509 0.07 0.005 0.071429-0.00978 98.91533 0.08 0.005 0.0625 - - 0.09 0.005 0.055556 - - 0.1 0.005 0.05 - - 0.2 0.005 0.025 0.52471 41.80965 0.3 0.005 0.016667 0.490086 45.64935 0.4 0.005 0.0125 - - 0.5 0.005 0.01 - - Table 3.17 Runup of α = 0.2 and a β = 10 Figure 3.22 Runup vs. CFL for α = 0.2 and a β = 10 47

3.8 NSWE against Synolakis s theoretical results Here we present our best runup against Synolakis s theoretical result for 20, 15 and 10 degrees. Figure 3.23 Numerical and theoretical comparison for 20 degree sloping beach, where f(x) corresponds to Synolakis formula and 20 Degree.txt corresponds to the numerical model of the shallow water equations 48

Figure 3.24 Numerical and theoretical comparison for 15 degree sloping beach, where f(x) corresponds to Synolakis formula and 15 Degree.txt corresponds to the numerical model of the shallow water equations 49

Figure 3.25 Numerical and theoretical comparison for 10 degree sloping beach, where f(x) corresponds to Synolakis formula and 10 Degree.txt corresponds to the numerical model of the shallow water equations We see that for a 20 degree beach our numerical simulation matches Synolakis s prediction relatively well, however for 15 and 10 degree slopes our accuracy tends to diverge from Synolaki s result as alpha increases. This can be explained by the following formulas of wave breaking: β 3.22 β Thus we have a upper and lower bound for wave breaking on the beach: 50 3.23

slope minimum bound maximum bound 20 0.1558 0.2662 15 0.1108 0.1894 10 0.0696 0.1189 Table 3.18 Set of non breaking alphas We see that this is true as with all of our numerical simulations tend to predict runup for the upper bound amplitude very well, less than 5% error, but tend to deviate from Synolakis s prediction after that upper bound is met, this can be attributed to the upwind scheme used on the nonlinear terms which create an artificial dissipation. 3.9 Stability Here we discuss the implementation of adding a diffusion term at the end of the Boussinesq numerical scheme in the form of Where we let k = 0.005 or 0.01, we use this method to improve smoothness for the input of the shallow water equations. We see in the figure below that this reduces the amplitude by less that 1% and that this improves the stability of the SWE part of the code. 3.24 51

Table 3.19 Numerical diffusion comparison We see that the blue line represents the input wave and that the red represent the simulation without diffusion and the green represent the solution with diffusion. 52

4 NUMERICAL RESULTS We present our numerical results below for our lagoon spacing and the runup reduction for a 20 degree sloping beach and a 0.5 water depth high breakwater. We have let the color scale represent the breakwater length and have scaled both the lagoon spacing and the breakwater length by the wavelength of the initial wave. Figure 4.1 Numerical results of a 0.75 high breakwater on a 20 degree sloping beach for runup against lagoon spacing for various breakwater lengths We notice that numerically, lagoon spacing s below 0.4 provides the greatest runup reduction and more importantly runup cannot be increased in this range. We also note that in this area the greatest runup reduction was achieved at 0.3 LS/L0. We believe that this may occur 53

because the wave does not interact with the edge of the breakwater as it transitions from the Boussinesq model to the SWE model. We also present our numerical simulations of a 20 degree sloping beach by plotting the breakwater s length against runup reduction below and have left the color scale to represent the lagoon s spacing to illustrate the effect of breakwater length against runup. Figure 4.2 Numerical results of a 0.75 high breakwater on a 20 degree sloping beach for runup against breakwater lengths for various lagoon spaces Here we notice that a shorter lagoon spacing decrease runup and that longer spaces increase runup. We also observe two peaks at around 0.55 and 1.25 LB/L0 that had the showed most the most reduction. 54

We present our numerical results below for our lagoon spacing and the runup reduction for a 10 degree sloping beach. We have let the color scale represent the breakwater length and have scaled both the lagoon spacing and the breakwater length by the wavelength of the initial wave. Figure 4.3 Numerical results of a 0.75 high breakwater on a 10 degree sloping beach for runup against lagoon spacing for various breakwater lengths We notice that a lagoon space below 0.55 provides the greatest runup reduction and more importantly runup cannot be increased in this range. We suspect that this is also due to the transition of programs. 55

We also present our numerical simulations of a 10 degree sloping beach by plotting the breakwater s length against runup reduction below and have left the color scale to represent the lagoon s spacing to illustrate the effect of breakwater length against runup. Figure 4.4 Numerical results of a 0.75 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces Again we notice a shorter lagoon spacing decreases runup. We also observe three peaks at around 0.55, 0.85 and 1.15 LB/L0 that had the showed most the most reduction for the lagoon spacing below 0.55. We present our numerical results below for our lagoon spacing and the runup reduction for a 5 degree sloping beach. We have let the color scale represent the breakwater length and 56

have scaled both the lagoon spacing and the breakwater length by the wavelength of the initial wave. Figure 4.5 Numerical results of a 0.75 high breakwater on a 5 degree sloping beach for runup against lagoon spacing for various breakwater lengths We expected to see a similar trend such as the 10 and 20 degree results, however we feel that the range of no runup was not reached. We do notice that shorter breakwaters seem to have a stronger effect on runup and we see two peaks at about 0.3 and 0.6 that had the strongest effect on runup due to the lagoon spacing. We also notice that the cases where a lagoon was not present had the weakest results. 57

We also present our numerical simulations of a 5 degree sloping beach by plotting the breakwater s length against runup reduction below and have left the color scale to represent the lagoon s spacing to illustrate the effect of breakwater length against runup. Figure 4.6 Numerical results of a 0.75 high breakwater on a 10 degree sloping beach for runup against breakwater lengths for various lagoon spaces We observe a stronger trend then our 10 and 20 degree results, which show that a 0.3 lagoon spacing is the most effective for all breakwater lengths below 0.6. We also notice that a breakwater length of 0.45 seems to increase runup. 58

5 EXPERIMENTAL SETUP 5.1 Wave Flume and Wave Generation We examined effects of both the wave profile and its runup reduction using the experimental setup shown here: Figure 5.1 Experimental Set up where we define, as the wave amplitude, as the initial constant water depth, WG as the wave gauges, d as the height of the breakwater, L as the length of the breakwater, S as the spacing of the breakwater from the base of the beach, β as the angle of the beach and R as the run-up. The experiment was conducted in a wave flume in the hydraulics lab. The wave flume is 6 in wide, 1.25 ft deep and about 35 ft long with transparent sidewalls. Wave generation is created manually by pushing a vertical plate at one end of the flume. To control the consistency, we allowed only one person in the group to generate all the waves. As we see in the next section that the generated waves match a solitary wave rather well. 59

5.2 Artificial Beach The artificial beach is simulated by a plexi-glass board and a stilt used to prop up the board as shown below: Figure 5.2 Artificial beach set up The brace for the plexi glass board was constructed out of uni struts and rest upon the wave flume s frame; it is constructed such that the angle of the beach may be varied depending on the position and configuration of the uni strut apparatus. Figure 5.3Unistrut apparatus/ base 60

Amplitude (cm) 5.3 Measurement of Wave Height and Run-Up Two resistance-type of wave gauges were used to measure the wave height and record the free surface profile as a function of time at the fixed gage locations, thus resulting in a plot like the following: 2.5 Wave gauge output 2 1.5 1 wave gauge 1 wave gauge 2 0.5 0 0 2 4 6 8 10 12-0.5 time Figure 5.4 Wave gauge output Here we determine the inital wave height and how the wave profile changes near the beach, so that we may compare it to the numerical simulations, which are discussed in chapter 7. We compare a sample output and the solitary wave whose solution is of the form: 61

Figure 5.5 Analytical and Experimental comparison of a hand generated solitary wave From here we obtain a value of amplitude, wave parameter k and the peak position xc using gnu plot s fit, where we have plugged in the solution to the solitary wave equation and left, the amplitude, a, the wave parameter k and the non dimensional amplitude, α as parameters that need to fit the data. a = 0.161306 +/- 0.003709 (2.299%) k = 2.71058 +/- 0.07544 (2.783%) xc = 1.66397 +/- 0.007012 (0.4214%) Run-up was measured by manually marking the maximum excursion that the wave could reach on the artificial beach, then using a tape measure or ruler to measure the maximum distance D that the wave traveled on the beach. This distance was later converted to the run-up by R = D sin. To control the consistency of this measurement we designated one of our group members to measure all the run up, for the entire experiment. Since the run-up was measured manually, it is important to estimate and understand the potential error associated with the manual measurement. In the experiments, the typical maximum travel distance D was about 10 inches. Due to the error in human response time, marking and ruler measurement, the error in this distance is estimated to be 0.5 inches, which leads to a relative error of 5%. 62

6 EXPERIMENTAL RESULTS 6.1 Non Breaking Waves We present our experimental results below for our lagoon spacing and the runup reduction for a 0.5 breakwater on a 20 degree sloping beach. We have let the color scale represent the breakwater length and have scaled both the lagoon spacing and the breakwater length by the wavelength of the initial wave. Figure 6.1 Experimental results of a 0.5 high breakwater on a 20 degree sloping beach for runup against lagoon spaces for various breakwater lengths 63

We notice that a lagoon spacing of 0.25 and 0.4 wavelengths had the strongest effect on runup. We present our experimental data of a 0.5 high breakwater on a 20 degree sloping beach by plotting the breakwater s length against runup reduction below and have left the color scale to represent the lagoon s spacing to illustrate the effect of breakwater length against runup. Figure 6.2 Experimental results of a 0.5 high breakwater on a 20 degree sloping beach for runup against breakwater lengths for various lagoon spaces 64

We observe two peaks at around 0.5 and 1 LB/L0 that had the showed most the most reduction. We also notice that for similar lagoon spacings a trend line behavior is observed as shown below: Figure 6.3 Breakwater length lines for a 0.5 high breakwater on a 20 degree sloping beach The breakwater length lines are not trend lines but an illustrative way of grouping similar lagoon spacings that show the decay as the breakwater s length is increased. We notice that the data peaks at half integer and integer γ, where I define γ as the ratio between the breakwater length in terms of wavelength and breakwater height in terms of the still water level. 65

6.1 Shown below is a plot of runup reduction against γ. Figure 6.4 Breakwater length trends for a 0.5 high breakwater on a 20 degree sloping beach without a lagoon Located below is a figure comparing the different families of breakwater length lines at one wavelength breakwater. We see that the 0.5 wavelength lagoon spacing is the most effective at reducing wave runup, we also notice that all the breakwater length lines follow a similar trend. 66