UNDULATION. I53 



wave Ijij rarefaction ; tlie former, tlic wave hy condcnsatirm. Tin; two arr; com- 

 plementary to each other, and Loth arc to be understood as included when iu 

 optics wc speak of "an undulation." 



A visible illustration of the relation between vibration and undulation, which 

 we have been endeavoring, in what precedes, to explain, may serve a us(;fLil 

 purpose in facilitating the formation of correct conceptions. Let AB represent 



Ficc 31. 



the rano-e of movement of a vibrating body, C being the centre of force and 

 the position of equilibrium. Upon AB describe the circle ADBE. Divide 

 the semi-circumference A D B into any convenient number of equal parts. 

 Bay ei'^'lit, and from the points of division drop perpendiculars upon AB. Take 

 AA' equal to the entire length of a Avave, and bisect it at B'. Divide AB' 

 and B'A' each into eight equal parts. From the points so found draw per- 

 pendiculars, upward on B'A', and downward on AB'. Cut off these perpen- 

 diculars to equality Avith those before drawn in the circle, in reversed order for 

 A'B', and in the same order for AB', and through their extremities, as thus 

 determined, draw the curve AE'B'D'A'. The ordinates to this curve will 

 represent the velocities of the molecular movements which will bo in activity 

 alon^'- the line AB'A', at the end of one complete double vibration of the excit- 

 ing body in AB. 



The point iu which this illustration fails t') convey a correct iuipression is in 

 the immense exa^-geration of the extent of the range AB, of the vibrating 

 body, as compared with AA', the length of the undulation. In sonorous 

 undulations this latter usually exceeds AB some hundreds of times. Of course, 

 no fi'^^ure which could be introduced here could preserve these proportions 

 without making AB imperceptible. 



Let us now return for a momeuf to the assumption, which, as w(; have said, 

 experiment, in the case of sound, coniirms, that the velocity of wave propaga- 

 tion will be uniform. This assumption, antecedently to any experiment on the 

 subject, mi^'ht be justified by the consideration that each successi%'c molecule, 

 or stratum of molecules, commences its motion under precisely the same cir- 

 cumstances as those of the molecules before it; taken along with t!ie ascertained 

 fact that the forces of recoil of elastic bodies in general are proportional to the 

 extent of disturbance. There is no circumstance Avhich could niaice the time 

 occupii;d by one stratum of panicles in transmitting the movement to the next, 

 longer or shorter than that similarly occupied by any other ; unless it should 

 be tiie decreasing amplitude of the molecular excursions^ as, in the expanding 

 wave, the motion is divided atnong a greater number of particles; and this cir- 

 cumstance, under the observed law of elastic force just stated, caji have no effect 

 upon the time of completing the oscillation. 



But the proposition admits of a more direct and not a difficult jjroof. If AB 



CDEF be a tube containing an elastic fluid — ^say air — open at CD, but with a 



, ^, movable piston at AF closing the tube completely; 



^A el Jf o _- ^^^^ j^-^ -j^ .^ ^^.^^ minute portion of time, t, this piston 



I be advanced from AF to A'F', proj)agating a tn^mor 



Avhich just reaches BE when AF reaches A'F', then, 



JB'F' B b' Ji for the instant, the fluid between A'F' and BE will 

 Fig- 32. }je more dense than before, in the same ratio as its 



volume is less, or (the diameter of the tube being constant^ iu the ratio of AB 



