Amplitude for sum of sinusoids

Question

I am trying to plot the following function: $$ \max_t \left| \frac{1}{1-r^2} \big( \sin(r \omega_n t) - r \sin(\omega_n t)\big) \right| $$ By inspection, I have determined that the amplitude of the sum of the sin terms is independent of $\omega_n$, but I haven't been able to find an expression for the amplitude.

Answer 1

If $f(t) = \sin(r a t) - r \sin(a t) $, then the extreme values will be at $t$ such that $f'(t) = 0$.

Find these values, and look at $f(\text{these values)}$.

Answer 2

If $r^2 \lt 1$ the first factor is positive and you have $\max_t \left| \frac{1}{1-r^2} \big( \sin(r \omega_n t) - r \sin(\omega_n t)\big) \right|=\frac{1}{1-r^2}\max_t \left| \big( \sin(r \omega_n t) - r \sin(\omega_n t)\big) \right|$ and the absolute value will be maximal when the sines are properly $\pm 1$ depending on the sign of $r$. The absolute value will then be $1+|r|$ and the overall value will be $\frac 1{1-|r|}$. You need to convince yourself that you can get the sines to work out this way. If $r^2 \gt 1$ the factor you pull out is $\frac 1{r^2-1}$ and you get a final answer of $\frac 1{|r|-1}$ for the same reason.