It is shown how the solution for velocity potential may be determined when the effect of surface tension is included in the linearized theory of Ursell-type edge waves over a plane-sloping beach. The problem is examined without making a hydrostatic assumption. Explicit solutions for edge capillary-gravity waves are given and the dispersion equation is obtained. The influence of capillarity on gravity waves is discussed.

1.
G. G.
Stokes
, “
Report on recent researches in hydrodynamics
,” presented at the
16th meeting of the British Association for the Advancement of Science
,
1846
;
G. G.
Stokes
,
Mathematical and Physical Papers of G. G. Stokes
(
Cambridge University Press
, Cambridge,
1880
), Vol.
1
, p.
157
.
2.
F.
Ursell
, “
Edge waves on a sloping beach
,”
Proc. R. Soc. London, Ser. A
214
,
79
(
1952
).
3.
H. P.
Greenspan
, “
A note on edge waves in a stratified fluid
,”
Stud. Appl. Math.
49
,
381
(
1970
).
4.
B.
Saint-Guily
, “
Ondes de frontière dans un basin tournant dont le fond est incliné
,”
C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers
266
,
1291
(
1968
).
5.
S. V.
Muzylev
and
A. B.
Odulo
, “
Waves in a rotating stratified fluid on a sloping beach
,”
Dokl. Akad. Nauk SSSR
250
,
331
(
1980
).
6.
B.
Saint-Guily
, “
Ondes de frontière dans une mer stratifiée dont le fond est incliné
,”
Annales Hydrographiques
757
,
7
(
1982
).
7.
S. G.
Llewellin Smith
, “
Stratified rotating edge waves
,”
J. Fluid Mech.
498
,
161
(
2004
).
8.
D. A.
Allwood
, “
Capillary-gravity waves over a sloping beach
,”
Proc. Cambridge Philos. Soc.
74
,
539
(
1973
).
9.
P. S.
Williams
,
U. T.
Ehrenmark
, and
G. L.
Body
, “
Bounded capillary-gravity waves obliquely incident on a plane beach
,”
Wave Motion
39
,
143
(
2004
).
10.
L. D.
Landau
and
E. M.
Lifshitz
,
Fluid Mechanics
, 2nd ed. (
Pergamon
, New York,
1987
).
11.
J.
Lighthill
,
Waves in Fluids
(
Cambridge University Press
, Cambridge,
1996
).
12.
A. S.
Peters
, “
Water waves over sloping beaches and the solution of a mixed boundary value problem for Δϕk2ϕ=0 in a sector
,”
Commun. Pure Appl. Math.
5
,
87
(
1952
).
13.
M.
Roseau
, “
Short waves parallel to the shore over a sloping beach
,”
Commun. Pure Appl. Math.
11
,
433
(
1958
).
14.
P. S.
Williams
, “
Waves on a sloping beach
,”
Proc. Cambridge Philos. Soc.
57
,
160
(
1961
).
You do not currently have access to this content.