
On one hand, Dean (1979) based on empirical estimates of fluid drag forces acting on vertical, rigid cylinders, the model for the damping of incident wave height (
$ {H}_{0} $ ) by coastal plants:in which
where
$ {K}_{v} $ (–) is the scaled wave height,$ L $ (m) is the length of vegetation area,$ \alpha $ ($ =\alpha {'}L $ ) (–) is the scaled damping factor, and$ x $ (–) is the scaled distance through the vegetation field.On the other hands, Kobayashi et al. (1993) linearized the horizontal drag force as a function of fluid particle velocity. The local wave height was assumed to decay exponentially with propagation through a vegetation bed according to the following form:
where
$ k{'} $ (m^{–1}) is an exponential damping factor, indicating a slighter decrease in a lower value.$ k $ ($ =k{'}L $ ) (–) is the scaled exponential damping factor.Based on reliable calibration methods, these two expressions are linked. Using the Taylor expansion, when the scaled distance
$ x $ equals half, the following equations are derived:and
where
$ {R}_{1}\left(x\right) $ and$ {R}_{2}\left(x\right) $ are the residual terms. To analyze the importance of each term in Eq. (9), these terms are represented:In these above Eqs (11)–(14),
$ \alpha $ is larger than zero due to the fact of wave attenuation. Since$ x $ is in the range of zero to unit, Eq. (12) can obtain its largest value when$ x $ equals zero, and in this case,Similarly, Eq. (13) has the largest value when
$ x $ equals zero or unit:And Eq. (14) can obtain the largest value in the case of
$ x=0 $ :To evaluate the relative magnitudes of the different terms of Eq. (9), Fig. 1 presents the factors. The result demonstrates that the first two terms can play the most significant roles.
Comparably, the importance of each term in Eq. (10) is analyzed and the following expressions are obtained:
Hence, Fig. 2 shows the comparison between these equations as a function of the exponential damping factor. Based on experiences, the value of
$ k $ is always in the range of zero to two (see for instance, Table 1 in Section 4.2). Under this circumstance, it is obvious that the first two terms are the key ingredients to Eq. (10) and the lower the value of$ k $ , which means the slower the wave attenuate, the more important the first term can be.Figure 2. Comparison between the factors in Eq. (10) as a function of the exponential damping factor
$ k $ .Reference Type of
vegetationCase Plant
height
/mPlant
diameter
/mPlant
densitystem
/m^{2}Incident
wave
period/sIncident
wave
height/mCollected
$ {C}_{D} $/–Calibrated
$ k $/–Wu et al. (2011) birch dowels 12436301 0.63 0.009 4 350 1.2 0.085 2.55 0.47 12435001 0.48 0.009 4 350 1.2 0.084 1.71 0.33 12636301 0.63 0.009 4 623 1.2 0.083 2.74 0.72 Wu and Cox (2015) plastic strips 5a 0.12 0.005 2 100 1.6 0.016 2.23 0.43 5b 0.12 0.005 2 100 1.6 0.024 2.12 0.55 5c 0.12 0.005 2 100 1.6 0.033 1.95 0.66 5d 0.12 0.005 2 100 1.6 0.041 1.62 0.76 Wu and Cox (2016) plastic strips Case2 0.12 0.005 1 618 0.6 0.018 7 3.74 0.43 Case5 0.12 0.005 1 618 1.2 0.031 3 2.21 0.38 Yao et al. (2018) PVC pipes wave0712 0.2 0.02 139 1.2 0.07 2.68 0.58 Table 1. Values of parameters from references and the calibrated
$ k $ .Therefore, consider only the first two terms, instead of distancedependent terms, in Eqs (9)–(10), the proportionality results in:
Then, a new expression for the drag coefficient is derived based on Eq. (2):
Therefore, by measuring local wave height
$ H\left(x\right) $ , the exponential damping rate ($ k $ ) can be calibrated easily, for instance, by Microsoft EXCEL, instead of more professional numerical tools, and$ {C}_{D} $ can be obtained by Eq. (23).To look upon Eq. (23) further, a derivative of this equation with respect to
$ k $ is taken and the result is obtained:It is obvious that Eq. (24) is larger than zero when
$ k $ is unequal to 2 since all these parameters are positive. Then$ {C}_{D} $ is a monotone function of the exponential damping factor. In other word, the more considerably the wave attenuate, the larger the drag coefficient. Besides, the length of the vegetation field is introduced for the bulk drag coefficient comparing to traditional Eq. (2), and it has an inverse correlation with$ {C}_{D} $ which is reasonable: A wider vegetated area drives a larger (exponential) damping factor and these parameters compensate with each other.Furthermore, the force
$ F $ can be expressed as:
A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation
doi: 10.1007/s1313101900000

Key words:
 wave attenuation by vegetation /
 naturebased coast /
 drag coefficient /
 empirical validation
Abstract: Vegetation in wetlands is a largescale naturebased resource that can provide multiple benefits to human beings and the environment, such as wave attenuation in coastal zones. Traditionally, there are two main calibration approaches to calculating the attenuation of wave driven by vegetation. The first method is a straightforward one based on the exponential attenuation of wave height in the direction of wave transmission, which, however, overlooks the crucial drag coefficient (
Citation:  Zhilin Zhang, Bensheng Huang, Hongxiang Ji, Xin Tian, Jing Qiu, Chao Tan, Xiangju Cheng. A rapid assessment method for calculating the drag coefficient in wave attenuation by vegetation[J]. Acta Oceanologica Sinica. doi: 10.1007/s1313101900000 