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3.12 Coupling of waves to a resonator
There are many examples in nature and in technology of waves exciting resonators. A sampling of these are:
 Light exciting (resonances) in flourescent material, as in a flourescent light bulb.
 Tidal waves exciting resonance in certain bays, such as the Bay of Fundy.
 A piano singing back at a person singing to it.
 Seismic waves from an earthquake exciting resonance in a building.
 Light of a particular wavelength exciting molecules in a sample to make an atomic transition.
 Infrared radiation exciting gas molecules into rotation.
Waves in a waveguide exciting a resonator
Rather than the coupling of a resonator to a freely propagating wave as in the cases listed above, an easier case to analyze is the coupling to a wave constrained to travel in a linear fashion, i.e. in a waveguide, a structure defining a path for the waves. Some technological applications of this situation are shown in the following figure:
In the cases shown in Fig. 24, waves travel to the resonator, at which point part of the wave's amplitude is absorbed by the resonator and some of it is reflected. Additional waves are often radiated from inside the resonator back into the waveguide where they propagate back towards the source, as shown in Fig. 25a.
To be consistent with our earlier postings (and because it makes nice, easytounderstand animations), we will consider the case of a massspring resonator with a string as the transmission line as shown in Figs. 25b and 26. Note that the concepts and math will apply to a large number of other setups, such as those shown in Fig. 24. We will send transverse waves down the string to the resonator. The far end of the string is loosely coupled to the resonator as shown in Figs. 25b and 26 (below).
We will actually do the analysis, out of a host of possibilities, for the setup of Fig. 26 because this particular coupling mechanism, a lever, is conceptually cleaner. It separates the coupling from the resonator's resonant frequency and from the properties of the string transmission line. We adjust the coupling by moving the lever's fulcrum. In a real setup, the lever would be aligned perpendicular to the string to avoid influencing the resonator's resonant frequency, however we will show it as in Fig. 26 for ease of illustration and ignore this influence.
Let's put some math behind this idea. With all simple wave guides and transmission lines, we can define a characteristic impedance which is the ratio of a carefully selected potential parameter to a selected kinetic parameter. If we choose the right parameters, the characteristic impedance or Z_{0} is constant for pure traveling waves of a particular frequency propagating on the waveguide, even though the two parameters are sinusoids oscillating as a function of both time and distance. The characteristic impedance is best known in the case of electronic waveguides and transmission lines, but this concept also applies to various mechanical transmission lines as well (see this reference page 12, section 4.3.1 and Elmore and Heald). In the case of mechanical waves on a string, the correct parameters to use are the vertical force f(x,t) (a sheer force) that is transmitted from one segement of the string to the next, AND the vertical velocity v(x,t) of a segment of the string. The characteristic impedance is thus defined as:
for a wave propagating in the positive x direction. The ratio of this force to velocity equals −Z_{0} for a negatively propagating wave:
The vertical velocity v(x,t) is just the time derivative of the vertical displacement from equilibrium of the string at any position x and time t:Dividing the wave into its components
With the right equipment, we can experimentally separately observe and measure two waves traveling on the string, one propagating in the positive direction and one traveling in the negative direction. These two components are also mathematically independent. The one traveling in the positive direction is known: it is the wave we launch towards the resonator via the waveguide which we will call the "incident wave".The wave traveling away from the resonator is not known to us at this point in our derivation. It is convenient to mentally divide this second wave into two parts:
 One part being the reflected wave, equal to exactly minus the incident wave at the right end of the string. Since the incident wave is known, this first reflected part is also known. It is the wave which would be passively reflected from the string's right end if we rigidly clamped the lever and mass so they could not move.
We put the above concepts into equations (realizing that the vertical velocity v is simply the time derivate of vertical displacement y. The following are true at the right end of the string only:y_{ref} = − y_{inc} and v_{ref} = − v_{inc} at x = x_{end} . (133)  The second part of the wave traveling away from the resonator (which contains the unknown part of the negatively propagating wave) is a result of the lever and mass moving. We will call this the radiated wave, since in a real sense it is launched on the string by the motion of the lever and mass, i.e. it represents radiation from the resonator.
 The total wave field on the string is the sum of the incident wave, the reflected wave, and the radiated wave. Mathematically we write:
y_{wave}(x,t) = y_{inc}(x,t) + y_{ref}(x,t) + y_{rad}(x,t) , (134)
where y_{wave} is the total wave field on the string, what we actually see.
At one location, at x = x_{end} , we can use (133) to write:
y_{wave} = y_{inc} + (− y_{inc}) + y_{rad} = y_{rad} at x = x_{end} . (135)
Taking the time derivate of (135) makes the displacements, i.e. the y's, become vertical velocities v's. Thus:
v_{wave} = v_{rad} at x = x_{end} . (136)
The animation below shows a wave driven resonator, breaking out the various components just discussed. We continue our mathematical derivation after the animation. For more on the animation, read its caption.
Fig. 26. Animation of a onedimensional wave driven resonator. The wave propagates on a string and excites the resonator on the right. The resonator is a variation of a massspring resonator and is coupled to the string by a lightweight lever. The animation provides for three different coupling strengths, selected by the buttons at the lower left side of the animation. Mouse over the animation to activate it and off to suspend it. The actual physical wave on the string is shown in black. Also shown are the leftgoing component of the wave (i.e. incident wave, in red) and the rightgoing component (negative going, in dark blue). For more on breaking a wave into two components see this earlier posting or this other earlier posting. The graphs in the lower central region of the animation reflect the oscillations versus time occuring at the left end of the coupling lever (i.e. right end of the string) due to various wave components. As explained above, the "reflected" wave is defined to be the incident wave reflected at the lever as if the lever and resonator are held stationary. The "radiated" wave is defined to be the component of the negativegoing wave due to the motion of the resonator. These two components add to yield the total negativegoing wave shown in dark blue. The values used in this animation are:

Newton's second law
In order to apply Newton's second law to the mass of our resonator (shown in Fig. 26) we first need to examine all the forces applied to this mass.
The various forces applied to the mass are: The spring force, −kx_{mass} , (137a)
 The damping force, −Rv_{mass} , (137b)
 The force exerted by the coupling lever, f_{coupling} . (137c)
where the reflected and radiated waves are propagating in the negative direction and so warrant the use of (130b) instead of (130a). We also divided the wave forces at the end of the string by r the mechanical advantage of the lever to get the forces applied to the mass. Note that r is negative as shown in Fig. 26 because it inverts the direction of vertical motion.
Newton's second law F = ma is:
Substituting (138) into this yields:
Equation (136) indicates that v_{rad} = v_{wave} = v_{mass}/r . Substituting this into (140) and rearranging to the form of (30) of a previous posting gives:
We repeat (30) here for easy comparison:
We see that except for slightly different styles and variable names, these equations (141) and (30) are the same, with the proviso that the damping coefficient R is replaced by R + Z_{0}/r^{2} AND the driving force f_{driver} is replaced by the incident wave vertical velocity v_{inc} times 2Z_{0}/r .
From (120) we see that in a simple resonator, the Q was related to the damping factor R by Q = √km/R . Making the substitution for R, this becomes:
where Q_{0} is the "intrinsic Q" of the resonator just by itself (involving the loss mechanisms inside the resonator itself) and Q_{L} is the "loaded" Q loaded down by the coupling to the transmission line (and other external effects we have yet to consider).
We invert (142) to get:
where Q_{coupling} is that portion of the quality factor attributed to coupling to the transmission line. Q_{coupling} is given by:
Consistent with (122) we see from (143) that the Q's add as inverses. Generally if one wants to measure the inherent losses in a resonator, they should either minimize the coupling so that 1/Q_{coupling} in (143) is a small effect, OR figure out how to measure Q_{coupling} to be able to subtract off the effect of the coupling.
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