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My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$ x = Ae^{- \gamma t} \cos (\omega \, t)$$ for $\gamma = \dfrac b {2m}$.

It says this implies the amplitude follows the relationship: $$x_{max} = Ae^{-\gamma t}$$

I am not understanding where this second equation comes from. Why does the cosine term disappear if $\omega$ is a constant?

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$\cos (\omega t)$ behaves like any other cosine term. $\cos (c)$ where $c$ is any constant is itself a constant. Since cosine oscillates between $[ -1 , 1]$ then when cosine is at it's maximum value it can be replaced with $1$.

Therefore: $$x_{max} = max(Ae^{- \gamma t} \cos (\omega t)) = Ae^{- \gamma t} \cdot max(\cos (\omega \, t)) = Ae^{- \gamma t} \cdot 1 = Ae^{- \gamma t}$$

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