250 Mr HOPKINS ON AERIAL VIBRATIONS 



a condition independent of /. Consequently, at every point whose 



distance from the open end is an odd multiple of -, there would be 

 a perfect node. 



20. Put 



f{at-{2l-x)\-y},,{at-{2l + c'-a;)] =x {at- (2l+c,-x)\ (9). 



Then 



v=/{at-x) + x{at-{2l + c,-a;)} (10). 



To find the relation between d and c, we have from equation (8), 

 (proceeding as in Art. 7, and with the same assumptions), 



^(_c)=-x/„(_c'), 



or >|,i(c')= -x|/(c); 



and since >//(») is much larger than >/'i(i8), we shall have c'. considerably 

 larger than c, and affected with a different sign. We may therefore put 



ki being greater than unity. 

 Again from equation (9), 



-^.(-0=x'(-c.), • 



or x'(<=-^)=-Uc'). 



If we suppose x'(«) nearly equal to v//^,(i8), (which probably is not 

 far from the truth), we shall have 



C2= —c' nearly, 



Hence in this case if the phase of the vibration of the extreme section 

 be retarded by a quantity c, that of the actually reflected wave will 

 be retarded by kic; and it will appear by the same reasoning as in the 

 case of the closed tube, that the distance of the points of minimum 



vibration from the open end will be m' -r 1-, {m' being any odd 



number). 



