Details concerning the build up and decay of the envelope

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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 = ω_decay . We first repeat (96):

    .   (96)

We first work on the denominator using (83):

   s = (−1/τ) + iω_decay →   (−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 2^ndlink and (83) above:

e^st = e^−t/τe^iωt = 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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