INTERFERENCE — RESULTANT UNDULATIONS. 167 



the circle AP]D, &c. At the point C lay off the angle DCA, cr|U;il to the 

 difference of phase of the two components, thus : 



Draw AL perpendicular to the axis at A. Upon this take AK, the devel- 

 opment of the semicirch AED, &c. Join KH, and draw Nd! parallel to it. 

 Kd is the development of the arc AED, which measures the angle DCA, the 

 difference of phase. For HA is to NA as a half undulation is to the difference 

 of phase: — that is, as n scjninrcumfercnce is to the am tchich measvrcs the 

 difference of pTia.sf. But AK is a semicircumference, and ^d being parallel to 

 KH, we have. 



HA : NA::KA : At? : AOx- : arc AED(— A^.) 



Join therefore CD, and complete the parallelogram CDGB, drawing the 

 diagonal CG. Then from what has been before demonstrated, the angle ACE, 

 or the arc AE, is the measure of the interval in phase between the resultant 

 and the component CB zm PP'. The curve of that component crosses the axis 

 in A. Let, then, k.e be the development of AE, and draw eF parallel to KH. 

 F is the point at which the curve of the resultant undulation will cross the axis 

 in ascending. Making FS =: AH, S is the point where the same curve crosses 

 in der^cendnig. And making SR' rzz HP', H' is the point of maximum re^^ultant 

 velocity. Draw K'R perpendicular to the axis, and make it equal to GC, the 

 diagonal of the parallelogram; II is the vertex of the resultant curve. Any 

 other points of this curve may be found by taking the sums of the ordinates of 

 the two components corresponding to the same absciss or point of the axis, 

 with like signs when both components arc above or both below the axis, and 

 with unlike signs when one is above and the other below, for the ordinates of 

 the resultant curve. The curve itself may then be drawn through the points 

 so determined. 



This construction enables us visibly to verify the analytical results which 

 were just now presented. Let the radius, CD, revolve i-ound the point C the 

 parallelogram changing its figure as the revolution advances, and the variations 

 in the value of CG may easily be conceived. 



Thus, whon 6 = DA = 0^, the point D will fall upon CA, and the point 

 G upon BIL CB and BG will then be in a straight line, and CG will equal 

 CB + BG = a -^ a'. W hen B = 90 ^ DCA and GBC are right angles. 

 and CG = VBJ^ + BG^ or, A= ^f a"- + a". When d = 180^ CD falls 

 on the axis to the right of C, and BG falls on BA. Hence, CG = a — a . 

 In this case, if a; = a', A = 0; or if equal waves differ in phase by half an 

 undulation they destroy each other. The two curves intersect the axis in the 

 same points, but the -couvexity of one of them corresponds in position to the 

 concavity of the other. Also, if equal waves differ in phase in any manner, 

 the crest of the resultant will fall half way between the crests of the two com- 

 ponents. 



When Q exceeds 180°, or the waves are more than half an undulation apart 

 the angle of the parallelogram must still ba measured from A through D round 

 to D', and the inclination of the diagonal must be taken in the same way, from 

 A through ED and D' round to E'. These arcs being developed on AL 

 produced, will give the position of the movable component and of the resultant 

 by drawing parallels to cut KH, as before. It will be seen in this case that, in 

 effect, the wave which we have regarded as the precedivg wave becomes the 

 fMoxoing wave, and vive versa; for if we consider the crests that nearest agree 

 in position as forming 'pairs, these pairs will be broken up by a discordance of 

 more than half an undulation, and new pairs will be formed — the lagging crests 

 ceasing to agree with those crests of the other component which are before 

 them, and beginning to agree with those behind them. Hence, as the resultant 

 crest must fall between the two components of a pair, it in this case goes further 



