140 EKMAN. ON DEA D-WATER. [n orw. pol. EXP. 



Under these circumstances, and if Kelvin's notation be slightly altered, 

 the height h of the water-surface above its mean level, is expressed by the 

 integral 



cos(ga) 



i. A 



<p(o) 



da , (1) 



where 



—a A n —a 



e -\- e 1 e — c 



and where 



A = the cross-section of the ridge producing the waves, 

 D = the depth of the channel; 



x = the horizontal distance from the ridge to the point in the water- 

 surface under consideration, counted positively in the direction 

 opposite to the motion of the ridge relative to the water; and 

 s = the ratio between the velocity of the ridge through the water 

 (or of the water relative to the immovable ridge) and the maxi- 

 mum velocity V</D of waves in the channel. 



1 

 To calculate the integral (1), Lord Kelvin breaks up the function -r-r 



into partial fractions which put its singularities in evidence, and h is then 

 obtained as a series of exponential functions. When s >• 1 all the sin- 

 gularities are imaginary values of a, and in this case the expression of 

 the level-disturbance (1) is infinitesimal at some distance from the ridge, and 

 represents a constant quantity of energy. There is then no wave-making 



resistance. When s< 1, —t-x is infinite for one positive value of a, and to 



cp(a) 



this value corresponds a term in (1) representing a series of harmonic waves, 

 and which may be calculated by help of an indeterminate definite integral 

 evaluated by Cauchy, namely 



cos px , 

 q* — x* 



= + g- sin pq for p > 



n (3) 



= — g- sin pq for p < 



When the sole object is to determine the resistance, the calculation may 

 be made much simpler, since we can leave out of account any term in the 

 expression for h of the form 



