Discussion Discussion of “A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics” ☆ Chondros M., Memos C. 1. The rationale behind utilizing variable coefficients in Boussinesq equations is not valid The discussers stated that the assumption of space and time depen- dent coefficients (α 1 , β 1 , α 2 ) is not valid for wave signals containing more than two components. (See Fig. 28.) We do not subscribe to the previous statement and actually this is the improvement upon the original model that this research offers. In dealing with regular wave propagation where the wave period is uniquely known everywhere, the wave number k and its non- dimensional counterpart κ = kd can be easily and accurately deter- mined over the whole domain. Thus the simulated monochromatic wave has the exact wave celerity (first order analysis), the exact second harmonic amplitude (second order analysis), improved third harmonic amplitude (third order analysis) and the exact linear shoaling gradient in any depth. This proposed improvement to Boussinesq-type models is a significant advancement over existing models, since it produces ex- actly the above characteristics in any depth for the case of regular waves. Coming to irregular waves the technique presented in Section 2.4 is used in the computations. In summary, an instantaneous local “wave” celerity is derived by associating it with the deformation rate of the free surface (Eq. (50)). Then κ can be deduced at any point through the local dispersion relationship, and the said coefficients are evaluated over the whole computational domain. This approach is quite general and applicable to regular waves too. The special case of regular waves propagating over a submerged bar, that the discussers bring up, can be treated by the general approach mentioned previously. It is noted that for this special case the simple technique applicable to regular waves was also applied with very satisfactory results. It follows from the above that the introduction and treatment of co- efficients variable in time and water depth are based on sound physical ground, and as one would expect produce results of equal or superior accuracy than constant coefficients, as verified by the relevant results and comparisons. Regarding the energy transfer functions it should be noted at the outset that the proposed variable coefficients reproduce the corre- sponding values of Stokes wave up to the second order for every κ and thus the expected energy transfer matches exactly that of Stokes wave over the entire range of depths for the specified order of nonlinearity. However, following a reviewer's suggestion, we opted to provide sam- ple results, mainly for the sake of comparison with other models. In order to do this the same methodology of evaluating the transfer func- tions should be used in the compared models. This requirement leads to some distortion of the proposed approach in determining the above coefficients of the modified model, in the following sense. The bichromatic wave although regarded as a simple case of a wave group, it is actually an irregular wave train, as regards the evaluation of the var- iable coefficients under consideration. Thus the proposed technique to calculate its wave number k p can only be reached through the computa- tional algorithm at each node and time step as presented in Section 2.4, since no analytical expression would be available. Therefore, instead of applying this latter treatment that would lead to perfect to the 2nd order agreement in each individual case as mentioned above, the con- ventional approach was followed in order to obtain some information on energy transfer and make comparisons. This approach is based on three conditions for k p described by the relations: k p ¼ k n þ k m k p ¼ k m k p ¼ k n þ k m ð Þ=2; where k n , k m are the wave numbers of the two individual regular waves. All three cases as above were studied but the most significant first one is dealt with in the paper. The validity range of the proposed model under the assumptions presented in the paper was found to cor- respond to κ ≈1.94 extending further the respective limiting depth of the original model that occurs for κ ≈1.76. It is obvious from the above that since the proposed coefficients up to second order are identical to those of Stokes wave for every κ, the expect- ed energy transfer functions should be exact in the whole range of water depths. By using the above well-known expressions for k p , somehow un- fairly to our model, results were obtained that offer insight in its behavior regarding energy transfer in wave propagation into shallow waters. 2. The technique of using wave surface elevation to obtain wave number is inaccurate Liu and Fang claimed that the technique of using wave surface eleva- tion to obtain wave number is inaccurate. It has to be made clear that in the case of monochromatic wave prop- agation the error of calculating the wave number is designed to be 0% everywhere, as noted in the first paragraph of section 1 above. The fig- ure of 2.89% error given in Table 2 for Stokes 1st order wave was pro- duced by applying the general technique for estimating the wave Coastal Engineering 95 (2015) 181–182 ☆ We thank the discussers for their interest in our work pertaining to the derivation and applicability of a nonlinear Boussinesq-type wave model. Also, for the opportunity they provided to us for giving extra clarifications on some issues presented densely in the paper due to lack of space. Their disagreements are summarized in the following three sec- tions, where our corresponding rebuttal can be found. http://dx.doi.org/10.1016/j.coastaleng.2014.11.006 0378-3839/© 2014 Published by Elsevier B.V. Contents lists available at ScienceDirect Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng