An exact analytical solution is proposed for the pressure oscillations within deep coastal aquifers under the action of tidal level time-variations attributable to train of solitary waves originating off-shore. The purpose of the study is to relate the characteristics of the response of the system to amplitude, steepness, and asymmetry of the soliciting waves, in order to assess its vulnerability to events like violent seaquakes and consequent tsunamis. The time needed by the forcing perturbations, approximated by consecutive triangular impulses, to attain their maximum is assumed to be always smaller than the aquifer diffusive time, in order to evaluate the consequences of the sudden raise of water level along the shoreline, typical of those quite extreme phenomena. 1. Introduction Common waves propagating on the sea surface are a direct consequence of the wind. When it starts to blow, any even weak irregularity in the flow pattern will produce corresponding local variations of the atmospheric pressure on the water table, triggering off the formation of series of ripples amplified and pushed away by the wind itself. That phenomenon of gradual growth and associated coalescence can lead, off-shore and under the action of particularly intense fluxes, to waves of considerable height, which in extreme cases can measure up to 8 meters in the Mediterranean sea and over 15 meters in the oceans. Due to their very slow attenuation, mainly attributable to the effect of the friction, those sea waves keep on propagating when the forcing mechanism stops, reaching zones that can be even very far from the origin. Totally different is the genesis of the solitary waves, also known as “tsunami” (a Japanese term meaning “port wave”), which are induced by impulsive phenomena like submarine earthquakes (e.g., [1, 2], see also Figure 1) or the fall in the sea of big rock and ice masses. Figure 1: Example of earth-tsunami genesis. The tsunami waves travel on the sea surface according to peculiar laws, with almost no decay, over distances of thousands of kilometres. Their rate of propagation (Lagrange velocity) is given by where indicates the local sea depth and is the specific gravity. For instance, a sea depth of 5？km would correspond to a tsunami velocity of about 800？km/h (comparable to the regime speed of an aircraft). Characterized by a limited amplitude when they propagate in open sea (about 1？m), tsunamis’ waves become gigantic water walls (10–30？m) while breaking on the coast, with predictable ruinous effects, which sometimes can be even catastrophic. Two eloquent
References
[1]
H. Kanamori, “Mechanism of tsunami earthquakes,” Physics of the Earth and Planetary Interiors, vol. 6, no. 5, pp. 346–359, 1972.
[2]
E. Bryant, “5. Earth-generated tsunami,” in Tsunami: The Underrated Hazard, pp. 129–138, Springer, New York, NY, USA, 2nd edition, 2008.
[3]
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, UK, 2nd edition, 1959.
[4]
S. B. Hooghoudt, “Contribution to the knowledge of some physical quantities of the soil. 6,” in Verslagen van Landbouwkundige Onderzoekingen, vol. 43 B, pp. 461–676, Algemene Landsdrukkerij, 's-Gravenhage, the Netherlands, 1937.
[5]
S. B. Hooghoudt, “Contribution to the knowledge of some physical quantities of the soil. 7,” in Verslagen van Landbouwkundige Onderzoekingen, vol. 46 B, pp. 515–707, Algemene Landsdrukkerij, 's-Gravenhage, the Netherlands, 1940.
[6]
J. H. Edelman, Over the berekening van grondwaterstomingen, [Ph.D. thesis], Delft University, Delft, The Netherlands, 1947.
[7]
I. A. Charni, “A rigorous derivation of Dupuit's formula for unconfined seepage with seepage surface,” Proceedings of the Academy of Sciences, USSR, vol. 6, p. 79, 1951.
[8]
R. E. Glover, “The pattern of fresh water flow in a coastal aquifer,” Journal of Geophysical Research, vol. 64, pp. 439–475, 1959.
[9]
R. E. Glover, “Studies of ground water movement,” Technical Memorandum 657, Bureau of Reclamation, Denver, Colo, USA, 1960.
[10]
R. E. Glover, “Ground water movement,” Engineering Monograph 31, Bureau of Reclamation, Denver, Colo, USA, 1966.
[11]
L. D. Dumm, “Drain-spacing formula,” Agricultural Engineering, vol. 35, pp. 726–730, 1954.
[12]
L. D. Dumm, “Transient flow concept in subsurface drainage: its validity and use’,” Transactions of the American Philosophical Society, vol. 7, pp. 142–146, 1964.
[13]
J. Van Schilfgarde, “Approximate solutions to drainage flow problems,” in Drainage of Agricultural Lands. Agronomy 7., J. N. Luthin, Ed., pp. 79–112, American Society of Agronomy, Madison, Wis, USA, 1957.
[14]
J. van Schilfgarde, “Design of tile drainage of falling water tables,” Journal of the Irrigation and Drainage Division, vol. 89, pp. 1–11, 1963.
[15]
H. K. Brooks, “The submarine spring off Crescent Beach, Florida,” Quarterly Journal of the Florida Academy of Sciences, vol. 24, pp. 122–134, 1961.
[16]
H. Hammad, “Depth and spacing of tile drain system,” Journal of the Irrigation and Drainage Division, vol. 88, pp. 15–34, 1962.
[17]
J. Bear, Dynamics of Fluids in Porous Media, Courier Dover Publications, New York, NY, USA, 1988.
[18]
J. A. Williams, R. N. Wada, and R. Wang, “Model studies of tidal effects on groundwater hydraulics,” Tech. Rep. 39, University of Hawaii Water Resources Research Center, Honolulu, Hawaii, USA, 1970.
[19]
P. Nielsen, R. Aseervatham, J. D. Fenton, and P. Perrochet, “Groundwater waves in aquifers of intermediate depths,” Advances in Water Resources, vol. 20, no. 1, pp. 37–43, 1997.
[20]
P. L. F. Liu and J. Wen, “Nonlinear diffusive surface waves in porous media,” Journal of Fluid Mechanics, vol. 347, pp. 119–139, 1997.
[21]
M. E. Serfes, “Determining the mean hydraulic gradient of ground water affected by tidal fluctuations,” Ground Water, vol. 29, no. 4, pp. 549–555, 1991.
[22]
K. F. Liu, “Tide-induced ground-water flow in deep confined aquifer,” Journal of Hydraulic Engineering, vol. 122, no. 2, pp. 104–110, 1996.
[23]
J. J. Jiao and Z. Tang, “An analytical solution of groundwater response to tidal fluctuation in a leaky confined aquifer,” Water Resources Research, vol. 35, no. 3, pp. 747–751, 1999.
[24]
C. Wang, L. Huang, and M. Lin, “Water well resonance induced by pre-earthquake signals,” Journal of Engineering Mechanics, vol. 136, no. 9, pp. 1167–1175, 2010.
[25]
G. A. El, R. H. J. Grimshaw, and N. F. Smyth, “Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory,” Physica D, vol. 237, no. 19, pp. 2423–2435, 2008.
