### 1. Introduction

### 2. Separation of Reflected Waves

*ω*, the incident wave elevation (

*ζ*) and the reflected wave elevation (

_{I}*ζ*) can be expressed using Eqs. (1) and (2), respectively.

_{R}*H*and

_{I}*H*denote the incident wave height and reflected wave height, respectively,

_{R}*∊*and

_{I}*∊*denote the phase difference of the incident wave and reflected wave, respectively, and

_{R}*k*denotes the wave number. The time series of waves measured at a particular position by the superposition method of the incident and reflected waves can be represented by Eq. (3): where

*e*(

_{n}*t*) refers to the measurement error for the non-linear interference or noise of waves included in the wave height data measured by the

*n*measurement position (

^{th}*n*wave gauge), and

^{th}*l*indicates the distance from the reference point to the

_{n}*n*measurement position.

^{th}*L*) from the wave maker to the first wave gauge (Fig. 1). If Eq. (3) was organized using a trigonometric theorem, it can be represented as Eq. (4), and the equation for

_{k}*X*is provided in the appendix.

*N*represents the number of wave gauges, and

*T*represents the measured time. With error (

_{m}*E*), which is the sum of measurement errors, if the differential value for

*X*

_{i}, a coefficient related to the wave heights and phases of incident and reflected waves, becomes 0, the error for each coefficient can be minimized. If four

*X*

_{1}–

*X*

_{4}were calculated using this method, the height and phase of the incident and reflected waves can be obtained. Eq. (7), which is a linear algebraic matrix equation, can be composed using Eq. (6), which can minimize the minimum/maximum of the error (

*E*): where

*C*expresses the right side of Eq. (4) as a trigonometric function, and

_{ij}*F*denotes the wave measured at each wave gauge. This equation is provided in detail in the appendix. By calculating Eq. (7), the coefficient

_{j}*X*

_{i}can be obtained. Finally, Eqs. (8)–(11) were applied to calculate the wave height and phase of the incident and reflected waves.

### 3. Experimental Equipment and Description

### 3.1 Two-dimensional Wave Tank

*h*) was 0.35 m, and the flat punching plate (Position A) was installed such that it was submerged 0.01 m (Gap) from the free surface. The ultrasonic wave gauges were named as wave gauge #1 (

*W*

_{1}), #2 (

*W*

_{2}), #3 (

*W*

_{3}), and #4 (

*W*

_{4}), respectively. The spacing between each wave gauge was all set uniformly to 0.3 m, and the spacing (

*L*) between the wave maker and wave gauge #1 (

_{k}*W*

_{1}) was set to be 2 m considering the evanescent wave mode. The length (

*a*) of the flat punching plate was 0.5 m, and in Case 1, the length (

*b*) and angle (

*θ*) of the sloping punching plate were 1 m and 18.6°, respectively. The experiment was conducted on three conditions for the length (

*b*) and angle (

*θ*) of the sloping punching plate. In the plan view of Fig. 3, the width of the wave tank (

*c*) was 0.3 m, and in Case 1, the projected length (

*b*′) of the sloping punching plate was 0.94 m. The signals measured by the ultrasonic wave gauges were amplified using an amplifier (AMP) and sent to a data acquisition device (DAQ). Then, the data were stored and analyzed on a computer.

### 3.2 Punching Plate Wave Absorber

*D*) and spacing (

*B*) were drilled onto the plates to maximize the wave absorption effect.

*D*and

*B*of all punching plates were identical, and the number of pores (circles) varied depending on the porosity of the punching plate. The punching plates were fabricated using an acrylic material having a thickness of 10 mm.

*P*) is given as a ratio of the perforated area to the total area of the plate.

*P*= 0) for the sloping punching plate (b).

### 3.3 Ultrasonic Wave Height Gauge

### 4. Results and Analysis of Reflected Wave Separation Experiment

*P*= 0.0980), which combined a flat punching plate with a porosity of 10% (

*P*= 0.1043) and a sloping punching plate with a porosity of 10% (

*P*= 0.0948), was installed as the wave absorber, and the time series data measured by the third wave gauge were compared. The wave period of the incident wave was 1.2 s and the wave height was 2 cm. The free surface elevation measured between approximately 6 and 9 s after the generation of an incident wave confirmed that the steady-state was shown irrespective of whether the wave absorber was installed or not. Therefore, it was determined that the reflected waves did not enter the measurement section of the mini wave tank completely. However, after approximately 9 s, when the incident wave was reflected from the end wall of the tank entered the measurement section after being re-reflected by the wave maker plate, the free surface elevation increased by approximately 100% in the case of no wave absorber. On the other hand, when the combined punching plate was installed, crest and trough values of the free surface elevations were well maintained in a section of 2 cm, i.e., the incident wave height. This meant that the incident wave was no longer reflected from the wave absorber and most of the energy was dissipated, thereby showing the wave absorption performance of the punching plate.

*R*denotes the reflection coefficient,

*k*denotes the wave number, and

*h*denotes the water depth. The incident waves were generated at 0.1 s intervals with wave height of 2 cm and wave period of 0.8–1.2 s. The experiment was performed using punching plates with five porosities (

*P*= 0–0.2087).

*P*= 0) is installed and when no punching plate is installed. Furthermore, the reflection coefficient generally increases from the short-wave to the long-wave. This shows that although the flat punching plate (0% porosity,

*P*= 0) has been installed, the incident waves are not attenuated by the flat plate alone. However, when the flat plate was porous, considerably superior wave absorption performance was observed, and particularly, good wave absorption performance was confirmed for long-waves. Furthermore, it was confirmed that the deviation of the reflection coefficient according to the wavelength was not very large. When the porosity of the flat punching plate was approximately 5% (

*P*= 0.0521), the reflection coefficient was less than or equal to 0.3, but when the porosity was approximately 10% (

*P*= 0.1043), the reflection coefficient was less than or equal to 0.1, i.e., it demonstrated the best wave absorption performance. However, for a large porosity of 20% (

*P*= 0.2087), the reflection coefficient increased rather slightly in the long-wave region (

*k*

*h*= 1.2). In brief, when the porosity was 10% (

*P*= 0.1043) on the flat punching plate, the mean reflection coefficient was approximately 0.07, i.e., the best reflection coefficient.

*P*= 0). Based on this, it was determined that the installation of the sloping plate results in a shoaling effect of incident waves, and the wave energy decreases due to the interaction with the flat plate. Similar to the case of the flat punching plate, it was confirmed that the best wave absorption performance (a low reflection coefficient) was observed for a porosity of 10% (

*P*= 0.0980) under the combined punching plate condition. This shows that the optimal porosity was generally approximately 10% irrespective of the incident wavelength. Cho (2013) also obtained similar results. Therefore, it can be expected that the energy loss rate of the incident wave is the largest when the porosity of the punching plate is 10%. As reported by Ko and Cho (2018), the load on the punching plates is expected to decrease if the porosity increases. This is because the pressure difference between the punching plates decreases. Furthermore, for the combined punching plate, the effect of installing the sloping punching plate is greater than the wave absorption performance produced by the porosity change.

*P*= 0). When the installation angle of the sloping plate was 90°, the result was very similar to that of the case where there was no wave absorber. Moreover, the reflection coefficient increased in the short-wave region. When the angle of the sloping plate was 39.3°, the reflection coefficient was approximately 0.3 to 0.4, showing a certain degree of wave absorption performance. However, when the slope angle was 18.6°, the best wave absorption performance was recorded. Therefore, for the given conditions of this experiment (incident wave period: 0.8–1.2 s, water depth: 0.35 m), it was appropriate to maintain the angle of the sloping plate at approximately 18.6° to achieve effective wave absorption performance. Ko and Cho (2018) reported that the optimal angle for the sloping punching plate ranged from 10°–20°. Therefore, we determined the angle that can produce an appropriate shoaling effect when an incident wave enters the punching plate area. In this experiment, no breaking wave occurred on the sloping punching plate. However, some breaking waves occurred on the flat punching plate because the waves that entered the flat plate were reflected at the end wall of the wave tank, resulting in superposed waves.

*P*= 0.098). When the incident wave height was 0.5 cm, the reflection coefficient increased in all wave frequencies. This is because the installation depth of the flat punching plate was 1 cm below the free surface, so when the incident wave height is small, the wave absorption effect of the flat punching plate cannot be expected. When the wave height was greater than 1 cm, the reflection coefficient was less than or equal to 0.1, which showed significantly effective wave absorption performance. Generally, similar wave absorption performance was shown irrespective of the wave height. Furthermore, the reflection coefficient generally increases as the incident wavelength increases (as

*k*

*h*decreases). This result is similar to the conclusion of a study conducted by Yuan et al. (2013), which states that the reflection rate increases as the wave frequency increases on a sloping punching plate.

### 5. Conclusion

*P*= 0.1043).

*P*= 0.0980) with the punching plate having an installation angle of 18.6°, the reflection coefficient was reduced by up to 95%, thereby showing the most effective wave absorption performance. This means that the combination of a flat punching plate and a sloping punching plate can effectively attenuate the incident wave energy. When the incident wave height was greater than the installation depth of the flat punching plate, the difference in wave absorption performance due to the wave height change was not that large, and all cases showed excellent performance.