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Details concerning the build up and decay of the envelope
Derivation of on-resonance envelope build up
In this section we derive eqn(100) including the equation for A. We start with (96) and take the case where ω = ω0 = ωdecay . We first repeat (96):
. (96)
We first work on the denominator using (83):
s = (−1/τ) + iωdecay → (−1/τ) + iω (83)
to get:
. (1)
We insert (1) into (96) and multiply by the complex conjugate of (1) in both the numerator and denominator to yield:
. (2)
We next use Euler's Formula 2ndlink and (83) above:
est = e−t/τeiωt = e−t/τ(cosωt + i sinωt) . (3)
We multiply out the denominator of (2) . It is totally real due to our complex conjugate trick. Then substituting (3) into this result we get:
. (4)
More work yields:
. (5)
Remembering that Q = τω0/2 → τω/2, we have:
. (6)
The actual oscillations of the mass/spring resonator are given by the real part of (6). Thus, we next pull out the real part of x(t):
, (7)
which can be rearranged as:
. (8)
Taking the limit of large Q yields:
. (9)
Using Q = τω/2 , yields:
. (10)
Comparing (10) with the first part eqn(100) we see that they are the same, which means we have accomplished our mission here.
For t → ∞ , (10) goes to the value given by:
. (11)
Eqn. (11) agrees with the steady state value given in eqn(50) in an earlier posting repeated here:
, (50)
provided you understand that the 1/i factor in (50) is equivalent to a − 90degrees phase shift, the same phase shift as we have here. That is to say that in deriving (100) we started with a sinωt exciting force and ended up with (11) which has a −cosωt time dependence, a − 90degrees phase shift. The A in (50) is the complex amplitude and the F is a different label for f0, i.e. both are the amplitudes of the driving oscillations.
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