Ratio of amplitudes in a damped oscillator

[tiago23]

Homework Statement

Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.(Note: The maxima do not occur at the points of contact of the displacement curve with the curve Aeˆ(-yt) where y is supposed to be gamma.

2. Homework Equations

The Attempt at a Solution

I was going through Fowles' Analytical Mechanics and found this exercise in the oscillations chapter. My problem isn't so much with doing the exercise as it is with the note of the authors in the end about the maxima not being in the points of contact between the two curves: Why not? Where else would the maxima of displacement be? I don't get that. Where is my mistake?

 

[berkeman]

Homework Statement

Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.(Note: The maxima do not occur at the points of contact of the displacement curve with the curve Aeˆ(-yt) where y is supposed to be gamma.

2. Homework Equations

The Attempt at a Solution

I was going through Fowles' Analytical Mechanics and found this exercise in the oscillations chapter. My problem isn't so much with doing the exercise as it is with the note of the authors in the end about the maxima not being in the points of contact between the two curves: Why not? Where else would the maxima of displacement be? I don't get that. Where is my mistake?

This is one of the reasons we have a "Relevant Equations" section in the Template. It would help us a lot if you posted the Relevant Equations for damped harmonic motion, showed some graphs (use Google Images with attribution if necessary), and described how your perception is different from the Relevant Equations and those graphs. Can you do that for us please? Thanks.

 

[epenguin]

Have you got this now? Well you might well, remembering what these curves look like, think for a moment that the points of contact were at the maxima. But then as soon as somebody tells you they aren't you think a few seconds and say doh, oh yes, right! So I almost don't like to spell anything further out. What can you say about slopes at the points of contact between the two curves?