37 



with (D/L) 2 multiplied by a complicated function r of P. Restricting P to 

 a range -0.6 >■ P > 0.35. ? is monotonically and, on the average, heavily 

 decreasing with decreasing P. Thus any reduction of D/L heavily reduces the 

 wave resistance. 



k.k. LIMITING DEPTH OP IMMERSION 



It is important to know below what depths of immersion f the wave 

 resistance can be neglected. This limit can be established from such cross 

 curves as shown on Figure 11; it obviously depends upon: 



a. The Proude number P or y = 1 /2P 2 



b. The L/D ratio, and 



c. The dimensionless shape of the body, primarily its prismatic coef- 

 ficient 4> , especially outside of the large hump. 



However, some additional simple reasoning may be helpful when curves 

 R = R(f/L) are not available. We can consider the wave resistance as negli- 

 gible either when 



a. It is a small percentage of a given standard resistance, or 



b. It is less than an absolute small value <JR. 



Some obvious differences in results due to the different approach 

 have sometimes been overlooked. 



a. Assume that for f > f the wave resistance becomes less than a 

 given small fraction e of the wave resistance R at a standard depth, for 

 instance at the immersion of one diameter; f is derived from a ratio of the 

 resistances in question. Comparing bodies of equal length, f depends upon 

 the Proude number and upon the dimensionless shape of the body, but only very 

 slightly upon the elongation ratio D/L = b/a, since the latter influences 

 only the constant 47rC 2 pg b 4 /a, which drops out in the comparison. 



b. Assume that the limiting depth f is derived from the condition 

 that the wave resistance is less than an absolute value 6R independent of 

 the standard resistance R . Comparing again bodies of equal length f now 

 becomes highly sensitive to changes in D/L. 



A rough idea of the necessary limiting depth f of immersion can 

 be obtained from the decline of the water disturbance with increasing depth 

 in a plane sinusoidal wave; this estimate normally gives exaggerated values 



V 



Denoting the wave amplitude by h and the amplitude of the distur- 

 bance by h one obtains 



h = \ e ~ X 



