SOLUTION:
Three approaches.
1. Using equally spaced, perpendicular lines...
Here, the equations are
Since ω1 = ω2 α = β, if the motion differs from a circle, it will be due only to a difference in amplitude of the two signals.
The cosine function travels to the right; the sine function travels upward. Successive lines are equally spaced increments of time. The black dots join two perpendicular lines which correspond to the same moment in time. The intersection is thus the predicted motion of a particle equidistant from both originating motions. The red dot is the first such intersection. The black line is an equal time curve. Every point on this curve will experience the same motion as the red dot, delayed in time.
The speed of sound is for fun and is very slow. For small values of c, the signal cascades over itself in space, which is unrealistic. As c increases, the response of the medium becomes increasingly linear, and the black dots form a nearly straight line. The motion of an individual particle is unaffected.
Trace has been turned on for the red dot. How does it move?
The motion is elliptic.
Result 1: The ratio of amplitudes of the two signals is the ratio of the major and minor radii of the resulting elliptic motion.
2. The resulting motion can be determined by projection. Place one circle in the second quadrant and a second circle in the fourth. Rotate a vector around each circle following ω1t and ω2t. Mark equal time intervals of the two motions on the circumference of the circle. Project these points across the axes and connect the intersections of equal increments of time. The resulting figure is the motion of the particle.
3. Find the locus of this projection.
[i.e. find the locus of the parametric (vector) curve (A cos(w1 t + α), B sin(w2 t + β))]
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