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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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A fresh look on old analytical solutions for water waves on a constant slope; pp. 422–429

(Full article in PDF format) doi: 10.3176/proc.2015.3S.13


Authors

Shanshan Xu, Frédéric Dias

Abstract

The first studies of water waves climbing a beach of constant slope were restricted to standing wave solutions. They all used potential flow theory. Various methods can be found in the literature. Hanson (The theory of ship waves. Proc. Roy. Soc. Lond. A Mat., 1926, 111, 491–529) and Stoker (Water Waves. Interscience, New York, 1957) gave the standing wave solution for slope angles of π /2n with n an integer. Stoker (Water Waves. Interscience, New York, 1957), Peters (Water waves over sloping beaches and the solution of a mixed boundary value problem for Δ2ø – k2ø = 0 in a sector. Commun. Pur. Appl. Math., 1952, 5, 87–108), and Isaacson (Water waves over a sloping bottom. Commun. Pur. Appl. Math., 1950, 3, 11–31) provided solutions for beaches of arbitrary slope angles. Due to the lack of numerical tools at the time, results were completely based on the theory of functions of complex variables, which is sometimes tedious and not easy to compare with modern numerical evaluation of analytical solutions. Here, we present four old solutions of standing waves and then evaluate the analytical solutions numerically to visualize results and perform comparisons. The run-up of waves in arbitrary water depth is also discussed.

Keywords

potential flow theory, standing wave, run-up, sloping beach.

References

1. Stoker , J. J. Water Waves. Interscience , New York , 1957.

2. Peters , A. S. Water waves over sloping beaches and the solution of a mixed boundary value problem for Δ2ø – k2ø = 0 in a sector. Commun. Pur. Appl. Math. , 1952 , 5 , 87–108.
http://dx.doi.org/10.1002/cpa.3160050103

3. Isaacson , E. Water waves over a sloping bottom. Commun. Pur. Appl. Math. , 1950 , 3 , 11–31.
http://dx.doi.org/10.1002/cpa.3160030103

4. Hanson , E. T. The theory of ship waves. Proc. Roy. Soc. Lond. A Mat. , 1926 , 111 , 491–529.

5. Synolakis , C. E. The runup of solitary waves. J. Fluid Mech. , 1987 , 185 , 523–545.
http://dx.doi.org/10.1017/S002211208700329X

6. Carrier , G. F. , Wu , T. T. , and Yeh , H. Tsunami run-up and draw-down on a plane beach. J. Fluid Mech. , 2003 , 475 , 79–99.
http://dx.doi.org/10.1017/S0022112002002653

7. Stefanakis , T. S. , Xu , S. S. , Dutykh , D. , and Dias , F. Run-up amplification of transient long waves. Q. Appl. Math. , 2015 , 73 , 177–199.
http://dx.doi.org/10.1090/S0033-569X-2015-01377-0

8. Madsen , P. A. and Fuhrman , D. R. Run-up of tsunamis and long waves in terms of surf-similarity. Coast. Eng. , 2008 , 55 , 209–223.
http://dx.doi.org/10.1016/j.coastaleng.2007.09.007

9. Keller , J. B. and Keller , H. B. Water Wave Run-up on a Beach. Technical Report. Department of the Navy , Washington , DC , 1964.

 
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Current Issue: Vol. 68, Issue 3, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December