Two periodic wave motions with the same frequency and phase are propagated at right angles from a source. What is the resulting motion?

It is an ellipse. :)
The speed of sound is for fun and not physically accurate. The marked points are the intersections of lines spaced at equal increments of time. Here the two functions are
f1(x) = .75 sin(2 pi t + pi/6), propagating upward
f2(x) = .5 cos(2 pi t + pi/6), propagating to the side
In general, if the two signals have the same frequency and phase, and differ only in amplitude, the resulting motion is elliptic. Variation of the other parameters introduces more complications...