19] 



ELLIPTIC HARMONIC MOTION. 



37 



denominator above vanishes, therefore the numerator must also, and 

 the path is 



X * y* X y 



a* 6 2 a b 



which represents a pair of coincident straight lines 



so that the path of the point is rectilinear and the motion is simple 

 harmonic. Similarly if cc $ = it, the motion is rectilinear. 



The angle (cc /3) is called the phase difference of the two 

 vibrations. If this is a right angle one component reaches its 

 maximum when the other vanishes, we then have 



Fig. 10. 



and the coordinate axes are the principal axes of the ellipse. The 

 amplitudes, a and &, of the component vibrations are in this case 

 the semi -axes of the ellipse. 



It is obvious from 13) that whatever the value of a /3 the 

 maximum values attained by x and y are a and b respectively, so 

 that the ellipse is always inscribed 

 in a rectangle of sides 2 a and 2b 

 (Fig. 10). If we allow the phase 

 difference, a /3, to change its value, 

 the point of tangency will run along 

 the sides of the rectangle, the axes 

 of the ellipse will turn, and it will 

 flatten out, in two positions degener- 

 ating into the straight lines forming the diagonals of the rectangle, 

 as above stated. 



If when the phase difference is a right angle the two amplitudes 

 are equal, the ellipse becomes a circle and the acceleration being 

 toward the center and constant in magnitude the motion must be 

 uniform circular motion. A harmonic motion is often defined as the 

 projection of uniform circular motion on a line in its plane. From 

 the value of the central acceleration in a circle we may by projection 

 obtain the properties of simple harmonic motion. 



The composition of two simple harmonic motions in intersecting 

 perpendicular lines when their frequencies are different gives a class 

 of curves of great interest in acoustics known by the name of 

 Lissajous. 



If the ratio of the frequencies is a rational number the least 

 common multiple of the periods of the component vibrations will be 

 a period for both and the curves are reentrant and algebraic. In 



