*3.5. Assessing the Derived CD by Reproducing Acting Force*

In order to test the derived period-averaged *CD*, we used the derived values to reproduce the total force from the velocity signals using Equation (1). The reproduced total force (*Frep*) is subsequently compared with the measured actual total force (Figure 9). The reproduced total force (*Frep'*) using *CD* = 1 is also included as reference. It is clear that *Frep* is in good agreement with the measured force over the shown two wave periods, although small differences exist between them. Notably, the measured maximum force is well captured in *Frep* near x = 0.5 π, which is important as the maximum force is critical not only for energy dissipation but also for assessing the stem strength to wave loading. As a comparison, the difference between *Frep'* and the measurement is large, which shows the validity of using period-averaged *CD* to reproduce the total force.

**Figure 9.** Comparison between reproduced total acting force and measured total force. The red solid line is the measured total force; The black dash line is quantified by using derived period-averaged *CD* in Equation (1) (i.e., *Frep*); The blue dash line is quantified by using *CD* = 1 in Equation (1) for reference (i.e., *Frep'*). The shown test case is wave0712, with 7 cm wave height and 1.2 s wave period.

Figure 10 compares the maximum *Frep* and maximum measured total force obtained at all three functional measuring locations in all test cases. In general, the reproduced maximum force is well in-line with the measurement, as most of the data points are fairly close to the 1:1 reference line. The *R*<sup>2</sup> value is 0.759 for data of all three functional measuring locations in all test cases. This result indicates the *CD* deriving procedure is valid, and the intrinsic errors associated with this procedure are acceptable.

**Figure 10.** Comparison between the maximum reproduced total acting force (*Frep*) and the measured maximum total force at three functional measuring locations in all the tested cases.

#### **4. Discussion**

#### *4.1. Advantages of the Current Measuring System and Alignment Algorithm*

Our results have shown that large spatial variations can exist in the wave particle velocity and *CD* (Figure 5 and Table 2). Thus, it is important to have synchronized force–velocity measurement at multiple locations by a number of force–velocity measuring systems. The selected standard force sensors are small enough to be installed at multiple locations in wave flumes. Additionally, these sensors are designed with built-in tapped holes, which facilitate testing various vegetation mimics, e.g., rigid cylinders, flexible stripes, and real vegetation stems.

Our results further show that the realignment process is important to derive both time-varying and period-averaged *CD* values (Figures 6 and 7). In our experiment, we aligned the instruments as good as possible (please see Figure 1d of the manuscript), but it is inevitable to have small misalignment to cause the delay. The main source causing the delay may be the inherent difference in instruments' speed of recording and receiving data, as the force and velocity measurements have their separate data acquisition systems. Thus, the automatic synchronizing algorithm is necessary and valuable in the current study. By using this algorithm, the time shift between two signals can be reduced to 0.003 s, which is only 1% of the original time shift before the realignment. The obtained time shifts are believed to be acceptable when comparing to normal wave periods (1–2 s) tested in our lab flume. The time shifts are merely 0.3% to the tested wave period.

The overall good performance between maximum measured force and reproduced force shows the reliability of this alignment algorithm. Importantly, this algorithm can automatically process the force and velocity data. No manual tuning is needed. Hence, it provides a generic solution to the alignment problems in deriving *CD*. Furthermore, this algorithm can run very efficiently, which is desirable when processing large data sets from multiple measuring locations. Lastly, this alignment algorithm is applied to process the velocity data from ADVs, but it is worth noticing that this algorithm is also applicable for other velocity measuring technologies, e.g., EMF (electromagnetic flow manufacture meter) and PIV (particle image velocimetry) [41–43].

#### *4.2. Current Limitations and Future Applications*

It is noted that the time shifts after the realignment are non-zero, but they are significantly reduced. To further reduce the time shifts, high frequency force and velocity measurements (e.g., >100 Hz) are required to obtain finite time steps for the realignment algorithm. However, it is perhaps not possible to completely eliminate the time shifts for all the tested cases, especially when multiple wave periods are included in the analysis, as the realignment procedure needs to account for time shifts at multiple peaks.

The velocity measurement in the current experiment was conducted by ADV measurement, which is a conventional method in flume experiments. The main limitation of the ADV measurement is that it is a point measurement. To obtain vertical velocity profile, it is required to manually adjust the ADV-measuring locations and repeat the same test conditions for each measuring location. This process is very time-consuming. Thus, we only conducted the velocity profile measurement for two cases, whereas for other cases, the velocity data is taken at the half water depth, which roughly equaled to the mean in-canopy velocity (Figure 3). The same practice is also done in previous study [37]. However, it is possible that the small deviation between the point velocity and depth-averaged in-canopy velocity can lead to errors in the derived *CD* values. This may partly explain the difference between the maximum measured force and the reproduced force. In order to improve the velocity measuring accuracy and reduce the labor involved, PIV system can be applied in future experiments. The PIV system can provide detailed velocity information of velocity field [41,42]. By applying such a system, it is also possible to obtain the relative velocity between water motion and the motion of flexible vegetation stems. Thus, the developed technics in the current study can be further applied in flexible

vegetation canopies, e.g., saltmarshes and seagrasses, which is interesting to both coastal engineers and ecologists.
