There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis

Friday, June 22, 2012

Details concerning the build up and decay of the envelope

All postings by author back: 3.10 Build up and decay of the envelope up: Contents next: Various Q's

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) + decay →   (−1/τ) +      (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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