@article{yu_2014, title={Approximations of the Dispersion Relationship of Water Waves}, volume={140}, ISSN={["1943-7889"]}, DOI={10.1061/(asce)em.1943-7889.0000620}, abstractNote={ABSTRACTA simple explicit formula is presented here to approximate the dispersion relationship of linear water waves, which allows direct and accurate calculation of wavenumber k for a given frequency ω. This formula is valid for all water depths, having an accuracy of the maximum error being 0.33% compared with the true solution (which is typically obtained by iterations or other root-finding numerical algorithms). It can easily be realized on a simple hand calculator (e.g., approved calculators for the Fundamentals of Engineering and the Principles and Practice of Engineering exams), and does not need any iteration, appealing in particular for practical engineering and pedagogical purposes. Another explicit formula, with still higher accuracy but slightly less simple form, is also discussed.}, number={1}, journal={JOURNAL OF ENGINEERING MECHANICS}, author={Yu, Jie}, year={2014}, month={Jan}, pages={233–235} }
 @article{yu_zheng_2012, title={Exact solutions for wave propagation over a patch of large bottom corrugations}, volume={713}, ISSN={["0022-1120"]}, DOI={10.1017/jfm.2012.460}, abstractNote={Abstract}, journal={JOURNAL OF FLUID MECHANICS}, author={Yu, Jie and Zheng, Guangfu}, year={2012}, month={Dec}, pages={362–375} }
 @article{yu_2010, title={Effects of Finite Water Depth on Natural Frequencies of Suspended Water Tanks}, volume={125}, ISSN={["0022-2526"]}, DOI={10.1111/j.1467-9590.2010.00492.x}, abstractNote={Linear sloshing problems (inviscid irrotational flows) in suspended tanks are revisited, with an intention to address some issues in the previous study based on shallow water wave theory. Time‐periodic solutions are considered, which describe the synchronized oscillation of the water and tank, reached after the initial transient dies out. The solutions are developed for arbitrary water depths, and separate explicitly the propagating and evanescent wave components of the fluid motion, illuminating clearly the physics and converging rapidly. At the limit of infinite string length, these solutions describe the sloshing motions in tanks that are free to oscillate horizontally on a frictionless plane. Various effects on the lowest sloshing mode are discussed, emphasizing the physical interpretations and examining the limitations of the shallow water approximations. Comparisons with existing laboratory experiments are made, showing agreements with the analysis.}, number={4}, journal={STUDIES IN APPLIED MATHEMATICS}, author={Yu, J.}, year={2010}, month={Nov}, pages={373–391} }
 @article{yu_howard_2010, title={On higher order Bragg resonance of water waves by bottom corrugations}, volume={659}, ISSN={["0022-1120"]}, DOI={10.1017/s0022112010002582}, abstractNote={The exact theory of linearized water waves in a channel of indefinite length with bottom corrugations of finite amplitude (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209–234) is extended to study the higher order Bragg resonances of water waves occurring when the corrugation wavelength is close to an integer multiple of half a water wavelength. The resonance tongues (ranges of water-wave frequencies) are given for these higher order cases. Within a resonance tongue, the wave amplitude exhibits slow exponential modulation over the corrugations, and slow sinusoidal modulation occurs outside it. The spatial rate of wave amplitude modulation is analysed, showing its quantitative dependence on the corrugation height, water-wave frequency and water depth. The effects of these higher order Bragg resonances are illustrated using the normal modes of a rectangular tank.}, journal={JOURNAL OF FLUID MECHANICS}, author={Yu, Jie and Howard, Louis N.}, year={2010}, month={Sep}, pages={484–504} }
 @article{howard_yu_2007, title={Normal modes of a rectangular tank with corrugated bottom}, volume={593}, ISSN={["0022-1120"]}, DOI={10.1017/S0022112007008695}, abstractNote={We study some effects of regular bottom corrugations on water waves in a long rectangular tank with vertical endwalls and open top. In particular, we consider motions which are normal modes of oscillation in such a tank. Attention is focused on the modes whose internodal spacing, in the absence of corrugations, would be near the wavelength of the corrugations. In these cases, the perturbation of the eigenfunctions (though not of their frequencies) can be significant, e.g. the amplitude of the eigenfunction can be greater by a factor of ten or more near one end of the tank than at the other end. This is due to a cooperative effect of the corrugations, called Bragg resonance. We first study these effects using an asymptotic theory, which assumes that the bottom corrugations are of small amplitude and that the motions are slowly varying everywhere. We then present an exact theory, utilizing continued fractions. This allows us to deal with the rapidly varying components of the flow. The exact theory confirms the essential correctness of the asymptotic results for the slowly varying aspects of the motions. The rapidly varying parts (evanescent waves) are, however, needed to satisfy accurately the true boundary conditions, hence of importance to the flow near the endwalls.}, journal={JOURNAL OF FLUID MECHANICS}, author={Howard, Louis N. and Yu, Jie}, year={2007}, month={Dec}, pages={209–234} }