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Exercise:
Derive formal expressions for oscillations where the envelope is a a linear function; b an exponential function; c a hyperbola. vspacemm For each case derive an expression for the time it takes the amplitude to drop to one half of the initial value. Graph the envelopes for the same initial amplitude and the same half-life time.

Solution:
abcliste abc An oscillation with a linear envelope can be written as y_t A_left-fractt_rightcosomega t where A_ is the initial amplitude and t_ is the time it takes for the amplitude to reach zero. Because of the linear decrease the time for the amplitude to decrease to one half of the initial value is T_/ fract_ abc An oscillation with an exponential envelope can be written as y_t A_ e^-t/taucosomega t where tau is the em half-life of the exponential decay. The half-life time is as usual T_/ tauln abc An oscillation with a hyperbola for the envelope can be written as y_t fracA_+ktcosomega t where k is a positive constant. It is obvious that the amplitude is equal to A_/ for kt so the half-life time is T_/ frack abcliste The diagram shows the graphs of the three envelope functions for the same initial amplitude and the same half-life time. center includegraphicswidthcm#image_path:envelopes# center
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Exercise:
Derive formal expressions for oscillations where the envelope is a a linear function; b an exponential function; c a hyperbola. vspacemm For each case derive an expression for the time it takes the amplitude to drop to one half of the initial value. Graph the envelopes for the same initial amplitude and the same half-life time.

Solution:
abcliste abc An oscillation with a linear envelope can be written as y_t A_left-fractt_rightcosomega t where A_ is the initial amplitude and t_ is the time it takes for the amplitude to reach zero. Because of the linear decrease the time for the amplitude to decrease to one half of the initial value is T_/ fract_ abc An oscillation with an exponential envelope can be written as y_t A_ e^-t/taucosomega t where tau is the em half-life of the exponential decay. The half-life time is as usual T_/ tauln abc An oscillation with a hyperbola for the envelope can be written as y_t fracA_+ktcosomega t where k is a positive constant. It is obvious that the amplitude is equal to A_/ for kt so the half-life time is T_/ frack abcliste The diagram shows the graphs of the three envelope functions for the same initial amplitude and the same half-life time. center includegraphicswidthcm#image_path:envelopes# center
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Attributes & Decorations
Branches
Damped Oscillations
Tags
envelope, half-life
Content image
Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Algebraic
Creator by
Decoration