Resonances, waves and fields: 3.15 Resonant scattering of waves

previous: 3.14 Waveguide excited resonator with circulator

This posting includes flash animations showing the physics discussed. Most computers have a flash player already installed, but if yours does not, download the free Adobe flash player here.

Flash animation of Scattering circuit with resonator and circulator.
Flash animation of Scattering circuit with resonator and no circulator.

3.15 Resonant scattering of waves - one dimensional cases.

Summary: Two animations are presented and discussed showing examples of wave driven resonators. The animations are arranged to emphasize that this is a scattering process, similar to scattering of infrared, visible and ultraviolet light off resonant molecules.

Keywords: scattering, resonant, circulator, flash animation

Topics covered in this posting

We explain and discuss two animations, which are variations of the animation in the last posting.
The animations are constructed to emphasize the concepts of resonant scattering in a one dimensional setting.
The animations are qualitatively discussed.
The mathematical analysis of the first animation was done in the last posting.
The analysis of the second animation is done qualitatively in this posting.
Mathematical analyses of the second animation are done in future postings 3.17 and 3.19.

Contents of this posting

Case 1 - with a circulator - single channel output
- Fig. 35 - animation of resonant scattering with a circulator
Case 2 - without a circulator - two output channels
- Fig. 36 - animation of resonant scattering without a circulator
Description of case 2 animation
Overview and comparison of the two cases
- Fig. 37 - illustrations of case 1 - one output channel
- Fig. 38 - illustrations of case 2 - two output channels - simple analysis

The process of scattering waves off objects (and atoms and molecules) is heavily used in today's imaging technology. Such scattering processes are used in radar, sonar, ultrasonic security systems, as well as IR, visible and UV detection of molecules and atoms. In this posting we shall delve into one dimensional scattering of waves off a resonator.

The first animation of this posting is structurally the same as the animation of the previous posting, but this one is reconfigured to stress the scattering aspect of waves interacting with a resonator. Because this first animation is structurally the same as that in the previous posting, the math is the same and we can cite results from that posting.

The first animation relates to one dimensional situations such as the scattering of waves in a waveguide due to a resonator. The "resonator" may be a physical structure similar to that shown in the animations of this posting, or it may be ions, atoms, or molecules that resonate in response to infrared, visible, or ultraviolet light waves in a microwave or optical waveguide or optical fiber.

In addition to being applicable to resonant scatterers in a waveguide, this teaching also relates to the resonant scattering of free waves in cases where resonant and non-resonant scattered waves have the same angular distribution. Such would be the case of waves scattered off an object which is large compared with a wavelength and has a uniform resonant layer coating it. See Section 4 for illustrations of this.

1. Case 1 - resonant scattering with a circulator - one output channel

Below we have an animation similar to the one of the last posting

As with the previous animations, mouse over the animation to see the action, mouse off it to suspend the action and click on it to restart it.
The coupling is initially set at unity coupling but can be changed by clicking on the buttons in the top right hand corner. After clicking on a button, mouse off the button (but still over the animation) to see the action.
The acoustical waves are shown as red and blue color shading inside the piping.
Pipes serve as acoustical waveguides.
There is a circulator which separates the waves coming from the sound source from the waves coming from the direction of the resonator. This circulator functions similar to circulators used in microwave circuits.
The resonator is a half wave length long closed pipe section with a small hole joining it to the nearest waveguide/pipe. The length of the resonator is exaggerated compared with the wavelength in the adjacent waveguide to make the resonator more visible.
The graphs just below and to the right of the waveguides plot the acoustical pressure versus position along the waveguides. These change as the waves progress. The graphs for the acoustical pressure inside the resonator and the adjacent waveguide are rotated 90 degrees because these items are tipped up from the orientation of the adjacent waveguide.
The acoustical waves, as sensed by three microphones, are plotted versus time in the lower "oscilloscope" graphs. The signals on these graphs appear reversed in the horizontal direction from the ones above the oscilloscope. Otherwise they look very similar, however they do have different independent variables: time for the oscilloscope and position for the upper graphs. Most of the signals plotted are traveling waves which are functions of the variable u = x ± ct and as such can be plotted versus x, t or u.

Fig. 35. Animation of 1D acoustical scattering by a resonator using a circulator. Mouse over this animation to activate it or mouse off it to suspend it. Clicking on it will restart it. This animation is really just a rearrangement of the one in the previous posting, but reconfigured to stress scattering of waves by a resonator. More details are discussed in the text above and below the animation.

As stated before, the above animation is just a rearrangement of the animation of the previous posting, rearranged to appear more as a scattering setup. The waves come in from the left, interact in the center with the resonator then travel on to the right. With the circulator present, no waves are scattered back towards the sound source no matter what the coupling strength is. The incident wave is partially absorbed by the resonator's losses and partially allowed to continue on to the right absorber. The fraction of the wave amplitude absorbed compared with that allowed to continue depends on the coupling to the resonator and the Q₀ of the resonator. If the coupling is adjusted so that the coupling Q equals Q₀ (based on only the losses internal to the resonator) so that there is unity coupling, all the wave amplitude is absorbed by the resonator (after the resonator fills with energy and steady state is achieved). If Q₀ is very high, the coupling hole needed to achieve unity coupling is very tiny but yet still allows the resonator to absorb all the wave amplitude.

As we pointed out in the previous posting, the wave radiated from the resonator can be thought to cancel out the wave which would propagate to the rightmost waveguide if the resonator were not there, or if the driving frequency did not match the resonator's resonant frequency to within a bandwidth.

Another interesting aspect is that the resonator gives up some of its energy to the waveguide on the right after the driving source is turned off no matter what the coupling strength is.

2. Case 2 - resonant scattering without a circulator - two output channels

Next we repeat the above animation but without the circulator. Without the circulator, the waves are free to reflect back towards the source as well as travel on through to the right waveguide. As we show in the animation below, the resonator is attached to the side of the waveguide via a small coupling hole.

In this case, unless the coupling is zero some waves will always be reflected at the coupling hole and returned to the sound source even at unity coupling. Mouse over the animation to activate it and observe the action.

Fig. 36. Animation of one dimensional acoustical scattering by a resonator. Concerning the wave graph just below the waveguide: the red wave is the total wave versus position in the waveguide. It changes with time as the wave in the waveguide propagates. The blue and green waves are the components of the total wave to the right of the resonator. The green component, the transmitted wave, is essentially the whole original wave. It is the wave that will pass by the resonator if the resonator does not respond. The blue component, the radiated wave, is the component radiated from inside the resonator when the resonator does respond to the incident wave.

This last wave component (the radiated wave) builds up with time as the resonator becomes excited. If the excitation frequency equals the resonant frequency of the resonator, these two components will be phased to partially cancel each other, thus reducing the size of the wave on the right side of the resonator as compared with that on the left side. This is the resonator's way of stealing power from the incident wave.

In the arrangement shown above, with two output channels, unity coupling is not as striking as in the previous arrangement, i.e. the incident wave is not totally absorbed by the resonator. On the other hand, unity coupling does result in the maximum resonator response, everything else being the same. For the animation, β approximately equals 0.3, 1, and 4 for the three couplings. The wave amplitudes are all approximate. In postings 3.17 and 3.19 we analyze the waves more thoroughly. Below we briefly discuss the amplitudes of the various waves.

The "resonator oscillations" plotted in the middle of the lower "oscilloscope" graph have been scaled by a factor of 2 compared with the other graphs, so that the oscillations will fit.

3. Description of Case 2 animation

Below is a qualitative description of the processes in the second case, i.e. that in Fig. 36 above. For a mathematical analysis of this see postings 3.17 and 3.19.

While the physics in Fig. 36 is related to that in Fig. 35, it is not identical. Next we shall qualitatively morph the results for Fig. 35 into results appropriate for Fig. 36.

In Fig. 35, the incident wave reflects upon hitting the wall with the coupling hole. This doubles the amplitude of the acoustic pressure applied to the waveguide end of the coupling hole and doubles the amplitude of excitation inside the resonator, compared with the case in Fig. 36 where there is no reflection and no doubling.
In Fig. 36, the waves radiated from inside the resonator back into the waveguide will propagate both to the left and to the right. Taken together these two waveguide directions present half the acoustical impedance that the radiated waves see in Fig. 35 where radiated waves only propagate in one direction. In the case of a small coupling hole, this will mean that the amplitude of the two radiated waves in Fig. 36 will both be half the amplitude of the radiated waves in Fig. 35, all else being the same.
Taken together, both effects (items 1 and 2 above) mean that the radiated waves in Fig. 36 will be about one quarter that seen in Fig. 35. Item 1 means that the amplitude inside the resonator in Fig. 36 should be about one half that seen in Fig. 35.

We can examine the power in the radiated waves just after the incident pulse has left the coupling hole area. This will allow us to ascertain the coupling Q of the resonator in Fig. 36 compared with that in Fig. 35.

Item #2 above means that a coupling hole of a certain size in Fig. 36 with the same field level inside the resonator as in Fig. 35 will radiate ½ the power total (and ¼ the power in each direction). Since the coupling Q is inversely proportional to the radiated power, this means the coupling Q in Fig. 36 will be twice that in Fig. 35 with the same size coupling hole. This also means the coupling coefficient β will be half that in Fig. 35 for the same coupling hole.

4. Overview and comparison of the two above cases

Wave scattering setups for case 1 above - one output channel

↑ Fig. 37a. One dimensional acoustical circuit of case 1 above. Most of the incident wave amplitude is reflected at the entrance of the resonator. The waves radiated back out the coupling hole travel in the negative direction and become inseparably mixed with the reflected waves.	↑ Fig. 37b. Schematic diagram of the wave interactions of case 1. The resonator and coupling is connected to one source (or channel) of waves and one output channel for the negative going waves.

↑ Fig. 37c. Three dimensional scattering setup that is closely related to case 1. The waves simply reflected by the reflective object are in the same directions as are those radiated by the resonant layer. The scatterer is large compare with the wavelength of the incident waves. A common example of this arrangement is a reflective object that is painted with resonant coating containing molecules that selectively resonate at particular wavelengths of light. Another example would be a metal ship having a uniform resonant coating that resonates particular radar frequencies.	↑ Fig. 37d. A more general case of that shown in Fig. 37c. Here the scattering object is not a single plane shape, but a more general shape having curved and flat areas. As was the case in Fig. 37c the object is large compared with the wavelength of the incident waves and a uniform resonant layer coats the entire object. The simply reflected waves have the same scattering pattern as do the waves radiated from the resonant layer and so the two become inseparably mixed with each other.

Wave scattering setups for case 2 above - two output channels

↑ Fig. 38a. One dimensional acoustical circuit seen in case 2 above. Part of the waves radiated from the resonator travel in the negative direction and are easily separated from the incident waves which travel in the positive direction down the waveguide. This represents one of the output channels of the resonator and coupler. The other output channel is the continuation of the waveguide on past the resonator/coupler to the right side of the figure.	↑ Fig. 38b. A schematic diagram of the wave interactions with the coupler and resonator in case 2. The resonator/coupler is connected to one input channel for the incident waves and two output channels, one for the negative going waves and one for the "transmitted" waves.

↑ Fig. 38c. Three dimensional setup that is closely related to that of case 2. Here the scatterer is similar sized or smaller than the wavelength of the incident waves. The resonator(s) radiate in a angular pattern different than would occur if the resonance feature of the scatterer was missing. In effect there are two different angular radiation patterns or output channels.	↑ Fig. 38d. A different illustration of a similar resonant scatterer as shown in Fig. 38c showing an asymmetric scattering object. The scattering object has two components of its scattering: An opaque, partially translucent and/or reflective external surface which is non-resonant a resonant structure These two components are such that they have different radiation patterns and these differing radiation patterns are the two output channels.