Let's define:
$f(t) = A_1 \cos(\omega_1t) + A_2 \cos(\omega_2t) $
I am interested in finding an expression for the peak of this function. It is not true in general that this peak will have the value:
$max{f(t)} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2}$
To find the value of max(f), I did the following manipulations:
$\omega_2 = \omega_1 + \Delta \omega_1$
so I can express the second cosine as that of a sum of a single radian frequency:
$f(t) = A_1 \cos(\omega_1t) + A_2 \cos (\omega_1 t + \Delta \omega t)$
and after a little algebra:
$f(t) = [A_1 + A_2 \cos(\Delta \omega t)]\cos(\omega_1t) – A_2 \sin(\Delta \omega t) \sin(\omega_1t )$
I can then transform the sum of two isochronic $\sin$ and $\cos$ into a single $\cos$ with a certain amplitude and phase:
$f(t) = \sqrt{A_1^2 A_2^2 + A_1 A_2 \cos(\Delta \omega t)} \; \cos \left[\omega_1t – \tan^{-1} \left( \frac{A_1 + A_2 \cos (\Delta \omega t)}{A_2 \sin(\Delta \omega t)} \right) \right]$
But that's about how far I can drive it: since a trig function of $\Delta \omega$ is present both in the amplitude and phase of the cosine, I am not sure how to proceed. For sure, it is not said that the maximum of the function will be that of the amplitude part.
I find the straight approach of taking the derivative of f(t) and find its zero would be probably too cumbersome, however I would ask more experienced people what they think the best way to proceed would be.
Best Answer
If the ratio $r=\dfrac{\omega_1}{\omega_2}$ is not rational, function $f$ can reach values arbitrarily close to $|A_1|+|A_2|$.
A graphical example in the case $f(t)=2 \cos(1.5t)-3 \cos(\sqrt{2}t)$ with values arbitrarily close to $|2|+|-3|=5$.
A sketch of proof: As remarked by Yves Daoust, the case $A_1>0$ and $A_2>0$ is evident. Let us consider, for the sake of definiteness, the case $A_1>0$ and $A_2<0$.
It suffices to find a value of $t$ simultaneously:
very close to $\dfrac{2k\pi}{\omega_1}$, giving a value of the first cosine is very close to $1$, and
very close to $\dfrac{(2\ell+1)\pi}{\omega_2}$, so as the second cosine is very close to $-1$.
(with with $k,\ell \in \mathbb{Z}$). These approximate identities:
$$t \approx \dfrac{2k\pi}{\omega_1} \approx \dfrac{(2\ell+1)\pi}{\omega_2} \ \ \ (1)$$
can be reinterpreted by saying that it suffices to be able to find integers $k$ and $\ell$ such that:
$$\dfrac{\omega_1}{\omega_2}\approx\dfrac{2k}{2\ell+1}$$
otherwise said, to be able to find a "very good" rational approximation of $r=\dfrac{\omega_1}{\omega_2}$ of the form $\dfrac{2k}{2\ell+1}$. This is a consequence of the density of the rationals in the reals (even of the rationals of the form $\dfrac{2k}{2\ell+1}$).