Bow and Stern Waves and Small-Wave-Ship Singularities 



physical quantities related to the bow only, and Eq. (17'), the stern only, to deal 

 separately with Eq. (16') from Eq. (17') is quite dangerous for the range of 

 higher Froude numbers. The following example may demonstrate this. 



Let us consider a source distribution 



"^(^i-^i) , , (18) 



= a cos (ttXj) ^ ' 



in < Xj < 1 and < Zj < H, which produces approximately a ship of sine wa- 

 terlines. Then from Eq. (14) we have the regular wave height at x > l , and y = 0, 



p"/2 1 _ exp(-koH sec2<9) 



C = 4ak„ I :; ~ — sec 9 [sec d sin (k„x sec 9) 



° J../2 (^0 sec 9y-n^ 



+ sec 9 sin (kgX- 1 sec 9] 69 , 



where the first term in the brackets corresponds to Eq. (16') and the second 

 term corresponds to Eq. (17'). However each term becomes singular when kg 

 approaches to n (or the wavelength is 2 and Froude number F = 0.564), although 

 the total regular wave height is always finite. If we examine the regular wave 

 height on < X < 1 and y = 0, from Eq. (13), 



■'"^^ 1 - exp(-koH sec^^) 

 4a I sec 9 [k^ sec 6* sin (kgX sec 9) - v sin ttx] d9 , 



-77/2 



'0' 



(k„ sec 9)' 



which is also finite for any k^ . This expresses physical bow waves defined in 

 Eq. (16). 



It is easily seen here that dealing with only bow waves, Eq. (16'), without 

 the stern waves, Eq. (17'), in the neighborhood of Froude number l/yJrF is 

 meaningless. 



LOCAL DISTURBANCE AND IMAGE SYSTEM FOR THE 

 ZERO WAVE SHIP 



When we observe the wave height on a ship hull, it is the superposition of 

 the regular wave height and the local disturbance. Although the local disturb- 

 ance does not contribute to the wave resistance in the linear theory, a large 

 local disturbance may induce sizable higher order effects. Therefore, it is of 

 interest to check the behavior of the local disturbance of the zero wave ship. In 

 addition, by this we can understand the nature of zero wave singularities better. 



For convenience, first, we will consider the simplest case of the zero- 

 bow-wave submerged body (2). A uniform source distribution of the strength 

 m/v= a^ on a line o<xi<i,y = o, and Zj = f produces the x component of the 

 perturbation velocity (3): 



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