What Differentiates Simple Harmonic Motion from ψ = Amod(t) cos (wavt)?

[whitehorsey]

1. What is the difference between ψ = A_mod(t) cos (w_avt)and the simple harmonic oscillator?

3. A. The amplitude is time dependent
B. The amplitude,A_mod , is twice the amplitude of the simple harmonic oscillator, A.
C. The oscillatory behavior is a function of ? instead of the period, T.

I'm not sure what the difference is. I thought that the equations are similar but that was incorrect.

 

[BvU]

What is ψ for a simple harmonic oscillator ?

 

[whitehorsey]

Simple harmonic oscillator:
ψ = 2A[cos(w_avt) cos(1/2 w_beatt)]

A_mod (t) = 2Acos(1/2 w_beatt)

In the book, it says the maximum amplitude A_mod of the wave changes with time. Would that mean it is A?

 

[BvU]

My simple harmonic oscillator $\ddot x + \omega^2 x = 0$ doesn't have a beat frequency. Only $A\cos(\omega t + \phi)$ with $A$ and $\phi$ constant.

 

[HalfLostAndSearching]

I can clarify the difference between the two equations and the simple harmonic oscillator. The simple harmonic oscillator refers to a system where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. This results in an oscillatory motion with a constant amplitude and a period, T, which is the time it takes for one complete cycle.

On the other hand, the equation ψ = Amod(t) cos (wavt) represents a more complex system where the amplitude, A, is not constant but varies with time. This means that the oscillatory behavior is not solely dependent on the period, T, but also on the function Amod(t). This can result in a more complicated oscillatory motion with varying amplitudes and periods.

In summary, the main difference between the simple harmonic oscillator and the given equation is the time dependence of the amplitude and the resulting oscillatory behavior. While the simple harmonic oscillator has a constant amplitude and period, the given equation has a time-varying amplitude and a more complex oscillatory behavior.