774 Mr. A. Stephenson on 



for small disturbance of a given character, we seek the force 

 system necessary to maintain the disturbance. If the 

 force when acting alone tends continually to damp it, we 

 conclude that the steady motion is unstable for such variation. 

 Thus in the present problem we seek the conditions under 

 which a small standing train of double wave-length, and 

 therefore of nearly free type, may be maintained by pressure 

 variation of the corresponding period in such phase that its 

 action tends to damp out the train without producing other 

 change. 



As the discussion involves quantities of different orders of 

 smallness, we shall discard the customary stream and velocity- 

 potential functions in favour of more direct coordinates, thus 

 obtaining equations which are readily applicable to all cases 

 in which the magnitude of the amplitude is involved. 



2. Let (#, y + v) be the coordinates of a particle the mean 

 level of which is at distance y, positive upwards, from the 

 undisturbed surface. Then the velocity in the stream-line of 



mean level y is c/\J 1+ (~\ 1 1+ j^-\ where c is a 

 function of y, and its horizontal and vertical components are 



A displacement Sx horizontally is given by a displacement 

 along the stream-line of horizontal component &r, and a 



vertical displacement — ~ . Bas. For irrotational motion 

 therefore 



clrj dr) 



dx d\ °dx ±d 



(d doc d \ dci 



dx~ 1+ d ]1 dy)~^c 



= 



i. e. 



It] 1 j_ dy dy 1 dy 



iy X ' T dy ' dy dy 



drjyd 2 r) _^ / 1 j (dy\ 2 \d 2 y _ 9 /-, _,_ dy\dy d 2 V 



*A 2 *V + h + (p)\ ** _ 2 (i + p)p 



dy) dx z \ \dx! J ay- \ dyjdx 



dxdy 



S(^g)(^(2)>°- • & 



The pressure is given by 



-c 2 \^L + a 



plp = —nv — ^c 2 xu,0/ + a constant. 



4. The forced motion due to a small simple pressure 

 variation of wave-length ir/k is 



7}?=ae 2 *y cos 27cx, 



