Intersection of arbitrary oscillation

Question

I am looking for a solution to calculate the intersections of arbitrary (harmonic) oscillations. For example Biorythm with constant amplitude and different periods. Especially my problem with oscillations havin different periods and amplitudes.

Answer 1

With arbitrary periods and phases, we have $$\sin(at+b)=\sin(ct+d).$$ Taking the $\arcsin$, $$at+b=ct+d+2k\pi\text{, or }at+b=\pi-(ct+d)+2k\pi,$$ $$t=\frac{d-b+2k\pi}{a-c}\text{, or }t=\frac{\pi-d-b+2k\pi}{a+c}.$$ In the specific case of physical vs. emotional rythms, $a=2\pi/23,b=0,c=2\pi/28,d=0$, and $$t=\frac{644k}{5}\text{, or }t=\frac{322(2k+1)}{51}.$$