{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:34:56Z","timestamp":1774629296519,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,14]],"date-time":"2019-01-14T00:00:00Z","timestamp":1547424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1706226"],"award-info":[{"award-number":["U1706226"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Wave height and wave period are important oceanic environmental factors that are used to describe the randomness of a wave. Within the field of ocean engineering, the calculation of design wave height is of great significance. In this paper, a periodic maximum entropy distribution function with four undetermined parameters is derived by means of coordinate transformation and solving conditional variational problems. A double entropy joint distribution function of wave height and wave period is also derived. The function is derived from the maximum entropy wave height function and the maximum entropy periodic function, with the help of structures of the Copula function. The double entropy joint distribution function of wave height and wave period is not limited by weak nonlinearity, nor by normal stochastic process and narrow spectrum. Besides, it can fit the observed data more carefully and be more widely applicable to nonlinear waves in various cases, owing to the many undetermined parameters it contains. The engineering cases show that the recurrence level derived from the double entropy joint distribution function is higher than that from the extreme value distribution using the single variables of wave height or wave period. It is also higher than that from the traditional joint distribution function of wave height and wave period.<\/jats:p>","DOI":"10.3390\/e21010064","type":"journal-article","created":{"date-parts":[[2019,1,14]],"date-time":"2019-01-14T12:20:07Z","timestamp":1547468407000},"page":"64","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Double Entropy Joint Distribution Function and Its Application in Calculation of Design Wave Height"],"prefix":"10.3390","volume":"21","author":[{"given":"Guilin","family":"Liu","sequence":"first","affiliation":[{"name":"College of Engineering, Ocean University of China, Qingdao 266100, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5702-2169","authenticated-orcid":false,"given":"Baiyu","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Engineering, University of California Berkeley, Berkeley, CA 94720, USA"}]},{"given":"Song","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Management, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6817-374X","authenticated-orcid":false,"given":"Hanliang","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Management, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"}]},{"given":"Liping","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5402-2203","authenticated-orcid":false,"given":"Wei","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Forest Resources and Conservation, University of Florida, Gainesville, FL 32611, USA"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2688","DOI":"10.1029\/JC080i018p02688","article-title":"On the joint distribution of the periods and amplitudes of sea waves","volume":"80","year":"1975","journal-title":"J. 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