Lunde [1, 3] has extended this to the case of accelerated motion in a canal of finite 

 depth. The first summand in the formula above is an added-mass term. If one takes 

 the case of a ship starting suddenly from rest at constant velocity c, this term vanishes 

 and the second summand approaches Michell's integral as t -> oo. Since towing-tank 

 tests approximate this situation, it would be of interest to have computations of the 

 transient part of this for a ship-like form. Lunde informs the author that such are in 

 the case of a ship starting suddenly from rest at constant velocity c, this term vanishes 

 progress. The author has obtained asymptotic expansions for large t. 



Other unsteady-state problems which would be of interest but which have not 

 been attacked as far as the author is aware are motion over a stepped bottom or ap- 

 proaching a beach. The steady-state problem of motion parallel to a beach or step has 

 also not been treated. Hansen [1] has considered the case of a pressure point moving 

 parallel to a beach, but, although this gives a qualitative notion of the form of the 

 waves, it doesn't relate them to the geometry of the ship or to the forces on it. 



Ship Waves. In concluison we should like to call the reader's attention to the 

 fact that we have seldom discussed the actual waves produced by a ship. There seems 

 to have been little investigation of the form of the free surface away from the ship as 

 it relates to the form of the ship. The studies of Hogner [2] and others on waves gen- 

 erated by moving pressure distributions give qualitative information about ship waves, 

 but not in relation to a specific hull geometry. 



REFERENCES 



Certain journals which occur frequently in the following list have been abbreviated 

 beyond easy recognition. The code follows: 



JSTG — Jahrbuch der Schiffbautechnischen Gesellschaft (Berlin). 

 PRSL — Proceedings of the Royal Society of London. Series A. 

 Trans. INA — Transactions of the Institution of Naval Architects (London). 

 Trans. NECIES — Transactions of the North-East Coast Institute of Engineers and 



Shipbuilders (Newcastle-upon-Tyne ) . 

 Trans. SNAME — Transactions of the Society of Naval Architects and Marine Engi- 

 neers (New York). 

 DTMB Rpt. — The David Taylor Model Basin (Washington, D. C), Report. 



Apukhtin, P. A., and Voitkunskii, Ya. I. 



2. Soprotivlenie vody dvizheniyu sudov (The resistance of water to the motion of ships). 

 Moscow-Leningrad, 1953. 



Birkhoff. Garrett; Korvin-Kroukovsky, B. V.; and Kotik, Jack. 



2. Theory of the wave resistance of ships. I. The significance of Michell's integral. 

 Trans. SNAME 62 (1954), 359-371. 



Birkhoff, Garrett; and Kotik, Jack. 



1. Theory of the wave resistance of ships. II. The calculation of Michell's integral. 

 Trans SNAME 62 (1954), 372-385, discussion, 385-396. 



2. Some transformations of Michell's integral. Publ. Nat. Tech. Univ. Athens, No. 10 

 (1954), 26 pp. 



Dorr, J. 



1. Zwei Integralgleichungen erster Art, die sich mit Hilfe Mathieuscher Funktionen 

 losen lassen. Z. angew. Math. Physik 3 (1952), 427-439. 

 Erdelyi, A. 



1. Asymptotic expansions. Dover, New York, 1956. 



130 



