Question

Mathematically describe the particular waves of a Lissajous figure, it’s superposition and the resulting figure.

Answer 1

In Physics,Harmonic vibration is a type of periodic motion where the restoring force is directly proportional to displacement.

Lissajous Curves:These are beautiful patterns formed when two harmonic vibrations along perpendicular lines are superimposed.Mathematically speaking since the motion of point P consists of two component vibrations whose equations of motions are already known to us,we can easily get the parametric equations of P ‘s motion. A and B are magnitude of two harmonic vibrations.   

Mathematical Explanation Of Lissajous Curves:

*Reason for the appearance of Lissajous Curves , Step1-Direct Elimination of t,   

Since Lissajous Curves are defined by the following parametric equations:

In principle, one can use trigonometric formulas to eliminate t from these equations and get a relationship between x and y Example : Straight Line:

  If in addition to A = B = 1, we have a = b = 1, and φ = 0, then the parametric equations will become

This set of equations tells us:

Moreover, since the range of sin(x ) is from -1 to 1, we have:

   This shows the figure (I) Example 2: Circle Taking

a = b = 1. But instead of leting φ = 0, we change it to :

Using the trigonometric identity , we can get:

Since , we can eliminate t and get the following equation:

given in figure (iii)