Optimization of parabolic bars for maximum Bragg resonant reflection of long waves

This paper presents an analytical solution for the problem of the long wave reflection by a series of artificial bars with parabolic configuration in terms of the associated Legendre functions. It is shown that both the reflection and transmission coefficients depend solely upon the number of bars, the dimensionless bar height, the dimensionless bar width and the dimensionless bar distance. Particularly, under the Bragg resonance condition, i.e., the distance between two adjacent bars is about half of the wavelength of the normal incident waves, the analytical solution for the peak Bragg resonant reflection is obtained, which reveals that the peak Bragg resonance depends upon the number of bars, the dimensionless bar height and the dimensionless bar width. Based on this solution, the optimization of the parabolic bars is made to obtain the maximum Bragg resonance and a group of optimal curves, which may be very useful in the design of Bragg breakwaters with parabolic bars.