68 



Only a short enumeration of the most Important facts is possible. 

 The wave pattern of a ship advancing with a constant speed v in shallow water 

 differs from the corresponding pattern in infinite depth by an increase in 

 the wave length X and by a change in the configuration of the waves, more en- 

 ergy being stored in the echo waves. 



The existence of a critical velocity is due to the fact that for 

 v > Vgh the transverse waves must disappear. 



The phenomena are complicated, but useful results have been reached 

 by a simple method devised by Schllchting^^ which yields an estimate of the 

 resistance in shallow water when a resistance curve for Infinite depth is 

 given. Schlichting Introduces the hypothesis that the resistance in deep and 

 shallow water is the same when the length of the free wave corresponding to 

 the ship speed is the same. Thus differences in height and configuration of 

 the wave pattern are neglected. Using well known formulas for the wave (ship) 

 velocity 



Vq = Vg2^(^®®P water) and v^ = j/g^^ tanh -^ [32] 



(shoal water) it is easy to calculate the shallow-water speed v, from the con- 

 dition X = const, when v and h are given. Experiments give considerable sup- 

 port to this rather bold idea, but the method fails at speeds above the 

 critical . 



A hydrodynamic solution for shallow-watfer resistance has been given 

 by Sretensky;®® it is valid approximately under the same assumptions as 

 Mlchell's formula (see Appendix 2). Sretensky's integral has been used to 

 demonstrate that Schlichting 's hypothesis has some theoretical foundation in 

 that, within the^subcritical range, the most important phenomena can be deduced 

 from it." 



We summarize briefly some important points: 



a. An additional Froude number F, = -^ is useful; F, = 1 for v = Vgh, 



^ Vgh h c 



the critical wave speed. 



b. Shallow water effects become appreciable only when Fu ^ 0.7; gen- 

 erally below this limit the water can be considered as infinitely deep (in so 

 far as wave resistance is concerned). 



c. Obviously the common Froude number F = -7-— • is connected with P, by 

 the depth-length ratio 



r L F. 



