`Monochromatic Plane Waves in a Corrugated Tube SystemA Thesis Presented to The Division of Mathematics and Natural Sciences Reed CollegeIn Partial Fulfillment of the Requirements for the Degree Bachelor of ArtsCody R. Myers May 2009Approved for the Division (Physics)Lucas IllingTable of ContentsChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2: General Theory of Wave Propagation Media . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Single Cell . . . . . . . . . . . . . . . . . . 2.2 Multiple Cells . . . . . . . . . . . . . . . . . . . 2.3 The Corrugated Tube System . . . . . . . . . . 2.4 The Defect State . . . . . . . . . . . . . . . . . 2.5 Translation to the Physical System . . . . . . . in . . . . . . . . . . . . 1Finitely Periodic . . . . . . . . . . 3 . . . . . . . . . . 3 . . . . . . . . . . 5 . . . . . . . . . . 7 . . . . . . . . . . 8 . . . . . . . . . . 11 15 19 19 20 20 23 25Chapter 3: Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . Chapter 4: Experimental Procedure and Results 4.1 The Method of Sound Measurement . . . . . . . 4.2 Taking Data . . . . . . . . . . . . . . . . . . . . 4.3 Transmission for Best-Measurement Setup . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .List of Figures2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 4.1 The Unit Cell . . . . . . . . . Multiple Cells . . . . . . . . . Band Gaps with Increasing N Defect Cell . . . . . . . . . . . Defect Transmission 1 . . . . 61-Cell Defect Transmission . Regular 61-Cell Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7 9 9 12 12 13 17 21Corrugated Tube System . . . . . . . . . . . . . . . . . . . . . . . . . 5-Cell Corrugated Tube Transmission . . . . . . . . . . . . . . . . . .AbstractThis thesis focuses on the acoustic transmission properties of a periodically repeating system, simply referred to as a corrugated tube. As with many periodic systems, acoustic wave propagation through the corrugated tube results in frequency &quot;band gaps,&quot; or large ranges of incident sound frequency over which virtually no sound wave is transmitted through the tube. We first determine the theoretical sound transmission for an N -cell corrugated tube (with cross-sectional area that alternates between two values N times over periodic spatial intervals), also examining a small modification of the same system called an N -cell defect state. Experimentally, sound transmission data is collected for a specific instance of the corrugated tube system with 5 corrugations, where the incident and transmitted sound intensities are compared over a 0 to 1000 Hz frequency range for a speaker propagating monochromatic, plane wave sound through the system. We then compare the transmission versus frequency plot of the observed sound transmission through the system to that of the expected transmission, which shows some agreement in terms of the location of the band gaps.Chapter 1 IntroductionWhen one examines the propagation of waves in a medium with specified boundary conditions, such as the mechanical wave observed when oscillating a string with one end tied to an object, two types of wave motion are usually observed over a large range of frequencies. For most frequencies, one will observe a traveling wave, in which the overall shape of the wave is moving. That is, one will observe any point on a given crest or trough of the wave moving with a constant velocity called the phase velocity. For very specific frequencies, one will find that superpositioning of the incident and reflected waves results in a standing wave with no phase velocity; all of the maxima and minima oscillate at fixed x-positions (where the string lies on the x-axis). While only these two forms of wave motion are observed in many simple physical systems, certain systems with periodically repeating boundaries exhibit an entirely different behavior, in which there is no transmitted wave over certain frequency ranges, called band gaps (where &quot;bands&quot; are the ranges over which there is some transmitted wave). This effect is a crucial subject of focus in solid state physics; an electron in a periodic crystal lattice will have allowed energy bands with intermittent energy band gaps over which its wave function becomes zero, hence there are certain energies that the electron will never have. And given an appropriate periodically-repeating medium, one can observe the same phenomenon for any kind of wave propagation. This thesis will focus on the band gaps observed in monochromatic (of single frequency) acoustic pressure waves. The periodic system examined here is a corrugated tube with alternating cross-sectional area. While solid state physics deals with wave propagation through crystals, which can entail propagation through millions of repetitions of the basic structural unit (the lattice), a corrugated tube system will have an extremely finite number of repetitions of its basic unit (one tube with radius r1 connecting another with radius r2 ), and so it is referred to as a finitely, or locally, periodic medium, to quote the paper upon which most of this research is based (1). This thesis will begin with the general theory detailing the theoretical expectations of monochromatic sound transmission through an arbitrary periodic system. We will then describe the physical system of interest as an application of this general theory: the corrugated tube system. In addition, a slight alteration to the system will be implemented, resulting in a &quot;defect state,&quot; with sound transmission properties that will be compared to the original system. After this, the experimental results of sound2Chapter 1. Introductiontransmission will be compared to their expected values over a limited frequency range, giving insight into what factors might separate the physical corrugated tube system from the idealized system. All sound used in the experimental study of the corrugated tube system is monochromatic sound incident from a speaker, and assuming onedimensionality of the waves, we are dealing with acoustic monochromatic plane waves throughout this thesis.Chapter 2 General Theory of Wave Propagation in Finitely Periodic MediaIn order to treat the propagation of acoustic plane waves in the finite corrugated tube system, one can look to a general theory of wave transmission for one-dimensional waves in periodic media (systems physically consisting of repeating elements or &quot;cells&quot;), with the goal of describing the outgoing wave in the final cell in terms of the incident wave in the first cell.2.1The Single CellA single one-dimensional cell can be described as some object having two boundaries enclosing it, thus wave propagation in a single-cell system will be piecewise, consisting of three separate wave functions: one on each side of the boundaries, and one within them. In general, any such system involving monochromatic plane waves will have a spatial wave function described by:  Aeikx + Be-ikx , if x &lt; -a;  (x) = F eikx + Ge-ikx , if -a &lt; x &lt; a; (2.1)  ikx  Ce + De-ikx , if a &lt; x, where A and C are coefficients for right-propagating waves, and B and D are the coefficients for left-propagating waves. The variable k is the wavenumber (also called the angular wavenumber) for a given frequency of wave, measured in units of reciprocal meters, m-1 . Here k =  , where  = 2f for a wave with frequency f traveling v through a medium at speed v, which for a sound wave traveling through air is roughly 343 m/s. This spatial wave function originates from a separation of variables applied to the full wave function (x, t) = (x)e-it , which in turn will be derived from a partial differential equation particular to a given system. Here the cell's left and right boundaries are located a distance a from the origin, as depicted in Fig. 1.1. For relevant systems, two physical conditions at each boundary4Chapter 2. General Theory of Wave Propagation in Finitely Periodic MediaFigure 2.1: A unit cell centered at the origin. The three regions are formed by a piecewise function S(x), with the regions surrounding the unit cell identical. will provide four equations that are linear in terms of the coefficients of each exponential, allowing for the elimination of F and G. One can therefore express A and B in terms of C and D, with the relation facilitated by a 2 by 2 transfer matrix M (1): A B where M= M11 M12 . M21 M22 (2.3) =M C , D (2.2)Two conditions allow for the structure of M to be specified in some detail. First, the partial differential equation from which (x, t) is derived can always be satisfied by replacing (x, t) with the time-reversed complex conjugate  (x, -t), or rather any systems studied in this general approach are assumed to satisfy this condition, including the corrugated tube system (1). The transfer matrix obtained from   (x)   tells us that M11 = M22 and M21 = M12 (1). Secondly, one can derive a relation among the wave coefficients that is based on the continuity equation for a pressure wave (1). The second condition becomes |A|2 + |D|2 = |B|2 + |C|2 , or simply AA + DD = BB  + CC  . Using the transfer matrix equation, this informs us that that the transfer matrix must be unimodular with determinant 1, or |w|2 - |z|2 = 1, where the elements of the transfer matrix take the form: M= w z , z  w (2.5) (2.4)so only w and z must be determined (1).2.2. Multiple Cells52.2Multiple CellsThe next goal of this analysis is to apply the known single-cell transfer matrix to an array of N uniformly-repeating cells, giving the transfer matrix for an entire system. The multiple cell system begins with a single cell defined on the x-interval (-a, a). The distance between the left boundary of the first and second cell is s, and the distance between the first and nth cell's left boundary is ns. Hence the wave function between each cell may be written: n (x) = An eik(x-ns) + Bn e-ik(x-ns) for (n - 1)s + a &lt; x &lt; ns - a (2.6)where n goes from 0 to N (1). This allows for each wave function to be placed in a local coordinate system, with each origin located at the center of the next cell to the right. Using this transfer matrix approach, one can ignore the wave functions within each cell, and relate the wave functions between each cell with the use of the known single-cell transfer matrix: An Bn or An Bn where P=M e-iks 0 . 0 eiks (2.9) =P An+1 , Bn+1 (2.8) =M An+1 e-iks , Bn+1 eiks (2.7)In effect, the An and Bn coefficients are not the real coefficients of the system for right and left-traveling waves in each given region (except for A0 and B0 which are equal to A and B), but are related to the real coefficients by a simple factor of e±ikns . The matrix P will be referred to as the shifted transfer matrix (1), playing a role similar to M in the single cell case. By recursive application of the above equation, one obtains the relation between the coefficients on the far left and far right of the N-cell system: A0 B0 = PN AN BN . (2.10)Therefore evaluating PN will establish a relation between the coefficients of the incident and transmitted wave functions in this system. To begin this evaluation, the Cayley-Hamilton theorem states that any matrix will satisfy its own characteristic equation, which for the unimodular 2 by 2 matrix P, tells us P2 - 2P + I = 0, (2.11)1 where  = 2 Tr(P) (1). To find P3 , one would multiply this equation by P, and then substitute the known expression for P2 to obtain P3 as a linear combination of P and6Chapter 2. General Theory of Wave Propagation in Finitely Periodic MediaI. Recursively, one can therefore reduce any PN to a linear combination of P and I, so in general one can say: PN = PUN -1 () - IUN -2 () (2.12)where UN () is a polynomial of degree N in terms of . Multiplying by P and substituting Eq. (1.11) yields: PN +1 = (2P - I)UN -1 () - PUN -2 (). Another expression for PN +1 is obtained by replacing N with N + 1 in Eq. (1.12): PN +1 = PUN () - IUN -1 (). Now equating these two expressions, we arrive at the following recursion relation consisting of only the polynomials themselves (1): 0 = UN +2 () - 2UN +1 () + UN (). (2.13)One finds that UN is the N th Chebychev polynomial of the second kind, so this allows us to express Eq. (1.12) in closed form for any given N (1). Now PN can be written explicitly, which can be used to find a total transfer matrix MN for the system. This simply entails shifting PN back to the first cell's coordinate system, correcting for the fact that AN and BN are shifted coefficients: M N = PN eikN s 0 0 e-ikN s (2.14)MN =[we-iks UN -1 () - UN -2 ()]eikN s zUN -1 ()e-ik(N -1)s . 2.15) (  ik(N -1)s  iks z UN -1 ()e [w e UN -1 () - UN -2 ()]e-ikN sWith the total transfer matrix at hand, only w and z must be determined for a given system in order to relate the incident and transmitted wave function coefficients. If one assumes a right-propagating wave comes from the left with no left-propagating incident or transmitted wave, the transmission coefficient for the system is T = |C/A|2 = 1 , |M11 |2 (2.16)and then using Eq. (1.4), the exponential terms cancel, giving: TN = 1 1 + [|z|UN -1 ()]2 (2.17)2.3. The Corrugated Tube System7Figure 2.2: The corrugated tube system, identical to Fig. 1.1 with a repetition of N cells.2.3The Corrugated Tube SystemNow focusing on the subject of interest, we look to sound propagation in the corrugated tube system. It consists of two kinds of repeating cylindrical tubes, with respective cross-sectional areas S1 and S2 , and lengths l and 2a. The S2 tube (the unit cell) is centered at the origin, and repeats each distance s a total of N times, with an S1 pipe connecting the left and right sides of each S2 pipe. As one might expect, the propagation of an acoustic pressure wave through the system can be described with the classical wave equation, as would apply to an ordinary cylindrical tube (1). Thus the wave equation for sound within the system is: (x, t) (x, t) = v2 , 2 t x2 (2.18)where v is the speed of sound in air, and  is a measure of the sound pressure above the ambient pressure. Fig. 1.2 provides a full diagram for the system. The solution for the single-cell version of this system is equivalent to Eq. (1.1):  Aeikx + Be-ikx , if x &lt; -a;  (x) = F eikx + Ge-ikx , if -a &lt; x &lt; a; (2.19)  ikx  -ikx Ce + De , if a &lt; x. This  is not strictly a function of one dimension throughout the tube, as diffraction inevitably occurs at the boundaries between different cross-sectional areas. However, to good approximation, sound waves with low enough frequency will not generate significant non-plane wave areas around the boundaries. The cutoff frequency fcutoff , below which acoustic waves in this system can be considered essentially one-dimensional, is given by: v fcutoff =  , S (2.20)where S is the larger cross-sectional area, and all frequencies dealt with in this research will fall under this range (1).8Chapter 2. General Theory of Wave Propagation in Finitely Periodic MediaTwo boundary conditions allow us to determine the w and z parameters of this system. The first condition expresses continuity of pressure,  = 0, and the second condition expresses conservation of mass (1), (S d ) = 0. dx (2.22) (2.21)Applying these conditions to each of the two boundaries in the single-cell case, one obtains four equations that algebraic manipulation will show to give the transfer matrix elements: w = [cos(2ka) - i + sin(2ka)]e2ika , and z = i - sin(2ka), where±(2.23)(2.24)= 1 [S1 /S2 ± S2 /S1 ] (1). Using Eq. (1.17), the transmission coefficient is: 2 TN = 1 . 1 + [ - sin(2ka)UN -1 ()]2 (2.25)It is this transmission coefficient that leads to the fundamental source of interest in this system; the existence of band gaps across certain frequency ranges over which no sound is transmitted. As the number of corrugations increases for a given set of parameters, these gaps become more pronounced. Suppose we take a system with tubes of radii 1 cm and 2 cm, and set both l and a equal to 2 cm. In Fig. 1.3, the results are plotted for increasing values of N, and one can see the emergence of a large frequency gap.2.4The Defect StateWe can modify the multiple cell theory to observe a system with similarly interesting transmission properties, which will be called the `defect state' version of the multiple cell corrugated tube. We will use the same corrugated tube system, except now the cell centered at the origin will have a length 2b different from the other cells of length 2a, with radius and cross sectional area r3 and S3 . We take the system to have N cells to the right of the center defect cell, and K cells to the left. As before, the right and left-traveling wave coefficients for the farthest-left region of the system are termed A and B, and our goal is to find the final coefficients C and D on the far right. The system is diagrammed below, with essentially the same notation as before. The equation for waves in the regions between the cells to the right is given by: n (x) = An eik(x-(n-1)s) + Bn e-ik(x-(n-1)s) for b + (n - 1)s &lt; x &lt; b + ns - 2a(2.26)2.4. The Defect State9T 1.0 0.8 0.6 0.4 0.2 f 1.0 0.8 0.6 0.4 0.2 f 1.0 0.8 0.6 0.4 0.2T0 T 1.0 0.8 0.6 0.4 0.210002000300040000 T1000200030004000f0100020003000400001000200030004000fFigure 2.3: These plots show the emergence of a band gap as the number of corrugations in the tube system increases from N = 1 to N = 2, to N = 8, to N = 24. The y-axis displays transmission and the x-axis is frequency in Hertz.Figure 2.4: This diagram represents the defect state of the previous N-cell corrugated tube system, with a cell in the center of differing length and radius.10Chapter 2. General Theory of Wave Propagation in Finitely Periodic Mediaas n goes from 1 to N . And the equation for the waves between the cells to the left is given by: n (x) = An eik(x-(n+1)s) + Bn e-ik(x-(n+1)s) for - b + ns + 2a &lt; x &lt; -b + (n + 1)s, (2.27) where n goes from -1 to -K. We have already established the transfer matrix for the regions to the left and the right of the defect. On inspection, one can see that Eq. 2.7 holds for all n (x) defined here, so the transfer matrix equation Eq. 2.9 holds as well. This of course depends on the single-cell transfer matrix being the same for both the left and right regions, and to establish this fact one must solve for the transfer matrix coefficients for the cells immediately to the right and left of the center defect cell, which due to the x-coordinate shift now have shifted boundary conditions. Doing this shows that both transfer matrices are the same as the usual M, with coefficients given in equations 2.23 and 2.24. Intuitively though, one can see that M for the left and right regions must be the same as before, because after obtaining the single-cell transfer matrix, knowing that Eq. 2.7 still applies to both regions, the same analysis leads one to a total shifted transfer matrix given by Eq. 2.12. And if one was dealing with only the N cells on the right, then one can see that for transmission through the system to be the same as in the non-defect case, the w and z values must be the same as before. So much like before, we must determine a total transfer matrix, which must take the form PL PM PR , these matrices representing the shifted transfer matrices for the left, middle, and right sides of the system. We have already deduced that: PL = PK , and PR = PN , (2.29) (2.28)and now one must determine how to deal with the transfer matrix PM involved in the center of the system, attributed the defect cell. One might initially suppose that PM is simply the transfer matrix consisting of the same parameters involving S3 and S1 areas instead of the S2 and S1 areas. Such a modified transfer matrix would be given by equation 2.5, except now: M2 = where w2 = [cos(2kb) - i and z2 = i where2± 2- sin(2kb), 2+ sin(2kb)]e 2ikbw2 z2   , z2 w2(2.30),(2.31)(2.32)= 1 [S1 /S3 ± S3 /S1 ]. 22.5. Translation to the Physical System11This is partly correct, but using a modified transfer matrix M2 would only be physically relevant if one were relating the actual coefficients of the system between the -1 and 1 regions. Since PL and PR only relate shifted coefficients, PM must do so as well. Hence PM must be equivalent to the single shifted transfer matrix as in the N -cell system. So we have: P2 = M2 0 e-iks2 iks2 . 0 e (2.33)where s2 = l + 2b. As before, the total transfer matrix is simply the total shifted transfer matrix multiplied by a matrix to account for the shift in coefficients, given that the final AN and BN are not the actual coefficients for the final region. For a total transfer matrix MN (def ect) , we have: MN (defect) = PK P2 PN eik(Ks+N s+s2 ) 0 -ik(Ks+N s+s2 ) . 0 e (2.34)Importantly, this total transfer matrix allows one to obtain the same total transfer matrix as in the regular (N+K+1)-cell system, if one models that system by, for example, setting b = a, s2 = s, and r3 = r2 . We can now obtain the transmission function for the system. Although the transmission function is slightly too complicated to be meaningful when written down explicitly, one can simply use the relation: T = 1 |MN (defect)11 |2 (2.35)to signify the transmission function, where MN (defect)11 is the element located at the first row and column of the total transfer matrix. The transmission function for this system yields some interesting behavior. One will still observe band gap regions over essentially the same frequency domains as would be seen in a similar (N+K+1)-cell system, but for a small total number of cells, one will observe isolated peaks within these gaps. To illustrate this behavior, we take a system with parameters that will be similar to the physical corrugated tube studied. We have N = K = 2, l = .3668, a = .0746, b = 0.2, r1 = .0395, r2 = .0195, and r3 = .03, and the transmission function is plotted in Fig. 2.5. Of course, as the number of cells goes to infinity, one finds that the addition of a defect does little to influence the band gap regions, which will no longer display these sharp peaks. Fig. 2.6 displays a defect state with 61 total cells, while Fig. 2.7 shows the 61-cell system with the same parameters, but without the defect. One can imagine that a defect state corrugated tube system would be useful in isolating specific frequencies, if one wanted to select a specific frequency to transmit over a large range, thereby acting as a selective acoustic filter.2.5Translation to the Physical SystemIn constructing a corrugated tube system, as will be discussed in the following chapter, it will be necessary to mount a speaker on the left end as the sound source, effectively12Chapter 2. General Theory of Wave Propagation in Finitely Periodic MediaT 1.00.80.60.40.202004006008001000f HzFigure 2.5: One plots the transmission function versus frequency for a defect system similar to the 5-cell corrugated tube system studied.T 1.00.80.60.40.202004006008001000f HzFigure 2.6: One plots the transmission function versus frequency for the same defect system as in Fig. 2.5, but now there are 61 total cells. The band gaps are almost exactly the same as those of Fig. 2.7.2.5. Translation to the Physical System1.0130.80.60.40.202004006008001000Figure 2.7: One plots the transmission function versus frequency for the same system as in Fig. 2.5, but with 61 cells and no defect. closing off one end of the tube. The necessity of having a physical sound source seems to put us at a disadvantage, largely because there is no physical barrier accounted for on the left side of the theoretical system. This may affect the incident sound to some extent; sound propagating to the right will reflect off the first boundary, and then reflect off of the speaker-bounded side, resulting in a right-traveling wave in the first region that is different from the right-traveling wave directly emitted by the speaker. However, a true closed-end boundary for a pressure wave would be defined as a location where the spatial wave function, (x), is equal to zero. The mounted speaker does not act exactly like a boundary, because although (x) surely should go to zero behind the speaker (if the speaker were truly only propagating sound to the right), (x) is actually constantly modulated by the speaker at the boundary. So the left side of the corrugated system is effectively much closer to an open-ended tube as in the preceding theory, provided that one only defines the wave function on its intended range, over the length of the tube. In measuring the transmission coefficient of the physical corrugated tube system for a given frequency, one will obtain an incident and transmitted reading of the sound intensity, measured in decibels (dB), which are dimensionless. If we refer to the incident intensity reading (obtained from the sound level meter described in the next section) as PA and the transmitted intensity as PC , then the equations relating the intensities to the pressure amplitudes are: A2 ) X2 C2 PC = 10log10 ( 2 ), X PA = 10log10 ( (2.36) (2.37)where X is some reference amplitude. One easily obtains the transmission coefficient, 2 T = C 2 , as a function of PC and PA : A T = 10(PC -PA )/10 . (2.38)14Chapter 2. General Theory of Wave Propagation in Finitely Periodic MediaThis will later allow us to plot the transmission of the corrugated tube system as a function of frequency, referencing only the incident and transmitted dB readings.Chapter 3 Experimental SetupA goal of this thesis is to model the preceding corrugated tube system for small N , to verify that the transmission as a function of frequency varies according to theory. But as this model is essentially one-dimensional, how does one construct it in three dimensions? First one should note that the cross-sectional area of each tube is the only parameter relevant to the extra dimensions; a feature which can essentially be attributed to the symmetry of each tube length on the y-z axis, accounted for in the wave equation. Furthermore, the change in cross-sectional area makes no reference to the orientation of the long and short tubes, giving one the freedom to construct the boundaries between each section in whatever way is most convenient. There is some issue regarding the one-dimensionality of the sound source, as it is difficult to construct a source to emit only x-dependent pressure waves with a normal speaker. But assuming that the speaker emits sound somewhat spherically, and given a tube waveguide, the speaker acts like a piston that oscillates minutely in one dimension, just enough to create slight acoustic pressure waves. The physical layout of the corrugated tube system is shown in Fig. 2.1. A large tube made of PVC plastic of inner radius r2 = 3.95 cm, thickness 0.5 cm, and length 2.947 m is used to create the larger-radius sections comprising the sides of each unit cell. Smaller tubes also made of PVC plastic, each with inner radius r1 = 1.95 cm, length 2a = 14.92 cm, and thickness 0.5 cm, are sectioned off from the larger tube by ABS plastic discs that are 0.5 cm thick, glued to the ends of each smaller tube. These discs were machined using a lathe in order to make their outer radii almost exactly equal to r2 , but slightly smaller (less than 5 mm smaller), allowing them to slide through the long tube with only a small amount of force applied. MD Camper Seal Foam Tape, with thickness .047 cm and width 3.17 cm, is wrapped around each end of each smaller tube as shown in the diagram. 62 cm of the tape is used for each wrap, and this effectively increases the sound absorption in the regions between the shorter tube's outer radius and the inner radius of the long tube. Screws of diameter .4 cm are inserted into each of 5 locations along the large tube in order to make the distance l between each smaller tube uniform. Then inserting one screw at a time, the long tube is tilted, and the smaller tubes are inserted and let to slide down until coming to a stop at each screw. The speaker used for the sound source is a Road Gear RGSP54 speaker, which16Chapter 3. Experimental Setupis mounted to a 15.3 cm by 1.2 cm by 15.4 cm piece of ABS plastic via four screws. The circular hole in the mounting block is machined to have radius slightly larger than r2 , allowing for the long tube to fit securely into it. This speaker is driven by a Tektronix CFG280 11 MHz function generator, with which one can select the driving frequency with a margin of error of ±.1 Hz. A Tektronix TDS 2024 four channel digital oscilloscope is also sometimes used to monitor the voltage reading across the speaker, to ensure that only the frequency displayed by the function generator is transmitted by the speaker. A Quest 1200 precision integrating sound pressure level meter (SLM) is used to detect the transmitted and incident sound in the system. It is set to the 50-120 dB range, with Z weighting and its response time set to slow. These parameters ensure that the intensity readings are essentially linear with respect to frequency, so that there is no measurable frequency-dependence of the SLM. The SLM is set with its microphone end directly in front of the end of the corrugated tube system to detect the intensity (dB) level of the system, and each reading it displays measures the maximum dB level detected in the past second at that point in space.17Figure 3.1: The individual components of the corrugated tube system are shown, not drawn to scale for the sake of visual clarity. 1) The long (and short) tube is made from PVC plastic with thickness .5 cm. One must note that the long tube has 5 holes drilled into the top at distances l + 2a, 2(l + 2a), 3(l + 2a), etc. where screws are inserted during data runs. 2) The short tube has ABS plastic discs glued to each end. 3) Foam tape is wrapped tightly around both sides of each short tube, with the ends held down by scotch tape, while the inside stickiness of the tape holds it in place. 4) The hole in the mounted speaker has radius 3.95 cm, while the speaker radius is slightly larger. 5) The SLM has several settings, chosen to give a linear dB response over the 1000 Hz range. 6) The full 5-cell system, where the screws hold the short tubes in place. One should note that the long pipe is slightly inclined downward for data runs, so that its end touches the ground.Chapter 4 Experimental Procedure and Results4.1 The Method of Sound MeasurementBefore observing the actual transmission through the 5-cell system for varying N , it is important to identify the correct method of measurement for both the transmitted and incident sound in regards to the placement of the Sound Pressure Level Meter (SLM). In the general theory section, it is assumed that the system is entirely lossless, so theoretically, sound measurement in any given region of the tube should be uniform in terms of intensity. However, the first problem with measuring either incident or transmitted sound within the tube should be clear; placing the SLM in the tube to do this will result in some minor reflection of waves due to the SLM itself. Secondly, as was detected experimentally, there was noticeable variation in the detected transmitted intensity in the final section of the pipe, with a tendency towards increasing attenuation very close to the end. And yet another problem arises when one considers the possible resonant frequencies of the pipe, in which standing waves form. At any given value of x, a standing wave will oscillate up and down over time as one typically expects of a wave. But unless one happens to be measuring the wave at an x-value upon which one of its maxima lie, one will not be measuring the total amplitude of the wave. And if one is measuring a standing wave at one of its nodes, where there is only destructive interference, one will obtain a very poor estimate for the total amplitude of the wave, observing virtually no intensity. So in order to best measure the transmitted intensity of the system, the SLM was typically placed with the end of its microphone on the exact boundary between the final section of the tube and outside room, facing perpendicular to the face of the tube. Measuring just outside of the tube allows one to avoid detecting any standing waves that might develop inside, while also more effectively preventing any influence that the SLM's presence might exert on the transmitted wave. Measuring the incident intensity of the speaker proves to be much more difficult. First, due to the length of the tube and the speaker's tight attachment to it, one cannot insert the SLM to get a direct reading of the intensity anywhere in the first20Chapter 4. Experimental Procedure and Resultsregion of the system. And in actuality, with the speaker attached to the left side of the system, some reflections will occur on the first interface between the large and small cross-sectional areas, resulting in a different A value than that generated by the speaker alone (either attached to a non-corrugated tube or otherwise). If one then considers reflections from every interface, it is clear that we can only obtain a precise value for A with the exact corrugated tube system in place while the speaker is operating. Although the incident intensity was measured in a number of different ways, the most effective method (while still feasible using the SLM) was simply measuring the incident intensity as the sound generated with the SLM close as possible to the speaker. In this way, one can be reasonably sure that the right-propagating waves of the highest amplitude in the system are detected, such that there is no risk of the transmitted intensity being greater than the incident intensity for any frequency. And this is something of a requirement, because for a lossless system, transmission greater than 1 is not theoretically possible, unless outside factors such as resonance in the room are considered.4.2Taking DataFirst, the corrugated system is `constructed' simply by inserting one screw roughly 3 cm into the long tube, at the ending boundary. Then one tilts the long tube at an angle of roughly 30 degrees or higher to slide a short tube down through it, until hearing the short tube come into contact with the screw. This process is repeated until all 5 short tubes are in place. Although there is no method for completely securing the short tubes in place, during this process the long tube is never tilted at a negative angle, so one can simply consider the position of the tubes to have an error equivalent to that of the screw measurement, ±0.2 cm. Placing the system on a long table, the long tube is then inserted into the hole of the mounted speaker, with its opposing end left to rest on the top of the table. The SLM is placed along the axis of the tube, with its microphone facing the open end, and the end of the microphone placed at the end of the tube. With the function generator driving the speaker and the SLM set to the `Run' mode, dB readings of the SLM are taken at 50 Hz intervals, from 0 to 1000 Hz. The cutoff frequency for this system is 4900 Hz according to Eq. 2.20, so this frequency range ensures that a correct comparison can be made between experimental and theoretical transmission.4.3Transmission for Best-Measurement SetupThe following transmission results for the 5-cell system were taken with the `bestmeasurement' setup described above for the positions of the SLM, taking incident intensity with the microphone as close as possible to the center of the mounted speaker (within .5 cm of touching it), and taking transmitted intensity just outside the end of the tube. Eq. 2.28 is used to obtain the transmission coefficient at each frequency. In the expected transmission plot, two bandgap regions exist between 200 to 400 Hz, and 550 to 800 Hz. These regions are for the most part identifiable in the data4.3. Transmission for Best-Measurement Setup21Figure 4.1: This plot displays the transmission coefficient as a function of frequency for the 5-cell corrugated tube system, using the best-measurement setup.taken, however there is very little evidence of transmission peaks between these gaps, which we should see based on the theoretical plot. The peaks to the right and left of these two band gaps are also identifiable, however it seems as though both plots would be much more consistent with theory if the actual transmission was translated to the left along the frequency axis. A simple explanation for such behavior would be a mechanism causing a shift in the driven frequency for the speaker, that is, a discrepancy between the wavenumber k for the measured frequency of the function generator and the actual wavenumber, which could possibly be attributed to the speaker's mount. But the existence of such a mechanism seems unlikely. A shift in the wavenumber of a monochromatic wave (in general, think of Eq. 2.1) will occur in a dissimilar medium, any sound traveling through the ABS plastic mount itself will have a k different from k in air due to dissimilar speeds of sound propagation in each medium. However, any sound passing from the speaker through the plastic will regain the same wavenumber once reentering the air, so there should be no k-shift in any monochromatically-driven pressure wave if it is measured in the air. In regards to dissimilar media, the only other tenable approach for a shift in the wavenumber would be a decrease in the speed of sound through air within the tube, due to a difference in the temperature, density, or pressure of air in the tube. However, a difference in the pressure or density of air between the tube and the outside room is highly unlikely for any reason, being an open system (without taking into account acoustic pressure waves), so the main possible cause for a shift could be attributed to a difference in temperature, due to ventilation in the room or some other factor. But to account for a frequency shift that would appear to be approximately a constant22Chapter 4. Experimental Procedure and Results50 Hz based on the displacement of the expected and measured transmission peaks, the speed of sound in the tube would have to vary unreasonably. To show this, for the expected wavenumber of the incident sound waves, we have: kexpected = 2f , vair (4.1)while for the actual wavenumber with a 50-Hz shift, we have: kactual = 2(f - 50) , vtube (4.2)for velocities of sound in the air and in the tube given by vair , and vtube . The latter velocity can be found by equating the two wavenumbers, thereby giving the necessary vtube to correct for the shift in k. For vair = 343 m/s, setting kexpected = kactual yields the velocity of sound in the tube, vtube = 343 - 2732.24 , f (4.3)where f must be greater than or equal to 50 Hz. Clearly, it is not actually possible for vtube to be dependent on the frequency of the input wave, this is just a mathematical consequence of a frequency shift in the data that we suppose to be constant. Nonetheless, one can obtain an estimate for the average shift in the velocity: at f = 50 Hz, vtube = 288.36 m/s, and at f = 1000 Hz, vtube = 340.27 m/s so vtube ranges from 288.36 to 340.27 m/s over the above data, or an average shift in velocity of 25.95 m/s, giving an average velocity vtube = 317.05 m/s. For air at a temperature of 0  C, the speed of sound in air is 331.5 m/s (2), so assuming that there could be a small temperature change in the tube system of 1  C at most, the resultant change in the velocity of sound in air is not nearly great enough to account for the apparent observed frequency shift. It is possible that for some or all frequencies of the function generator, the speaker generates sound at an additional frequency (or frequencies), but for such a significant frequency shift to occur, the intensity of those frequencies would have to exceed that of the function generator frequency, which seems unlikely. Moreover, it is unlikely that a speaker would generate an extra frequency with a constant difference between the driving frequencies, one usually encounters higher (attenuated) frequencies that are related to resonance in the system, taking on values that are some ratio or multiple of the driving frequency. Furthermore, using the oscilloscope to observe the sinusoidallyvarying driving voltage across the speaker, no additional non-driving frequencies were detected for the system over the 0 to 1000 Hz range, although this does not completely preclude the possibility of some physical aspect of the speaker malfunctioning and not reproducing the input voltage at a given time exactly. It seems that one of the most important factors in accounting for the discrepancy in the predicted and measured transmission is the physical layout of the room in which data was taken, which was not modeled analytically or numerically. For the transmitted wave (and similarly for the incident wave), there will be a left-traveling D4.4. Conclusion23wave which we assume to be 0 in theory. But due to reflections on walls in the room­ both from the sound exiting the tube and from the non-plane-wave sound emitted by the speaker that does not enter the tube­this D wave is nonzero. Because the SLM meter constantly displays the highest intensity (and therefore highest wave amplitude) of sound detected in the previous second, one might expect that these reflections have little bearing on its measurement due to their weak intensity relative to the sound emitted from the tube; as long as the SLM finds one maximum from the sound wave directly emitted from the tube, then one is only detecting this right-traveling wave. While this is partly true for very weak reflected waves and waves of differing frequency, one must note that the superpositioning of two waves with the same frequency (but a different phase constant, imagine adding two functions Asin(kx) and Bsin(kx + )) will create a measured wave with the same frequency, but with a maximum amplitude that could be anywhere from 0 to the sum of the two wave coefficients. So if the reflected waves have high amplitude for certain frequencies, possibly due to resonance in the room, the effects of interference could become pronounced. This is the most likely explanation for the behavior seen in the middle region, where one sees a much smaller peak in the actual transmission which could be due to destructive interference between the C and D waves. This does not account for the apparent frequency shift though, unless a phase constant  for the reflected waves somehow varies continuously as a function of frequency. While there is no clear-cut explanation as to why an apparent frequency shift occurs for the actual transmission, one does find that the inability to determine the effect of the left-traveling waves on both the incident and transmitted frequency makes it much more difficult to obtain a true transmission plot. Particularly, the left traveling waves of each frequency do not necessarily interfere in the same way for the SLM readings at the speaker and those at the end of the system.4.4ConclusionThis thesis has thoroughly explored the acoustic transmission properties of the periodically repeating corrugated tube system, although given the nature of the general theory, many similar periodic systems can make use of the same analysis, applicable to waves outside of the acoustic pressure waves we used. Transmission in this system was found to lead to the phenomenon of band gap formation, which is usually restricted to fully periodic systems such as crystals. We found that band gap locations are characteristic of all monochromatic plane wave sound transmitted through the system, and these gaps become much more apparent with increasing N . It was also found that for a corrugated tube containing one defect cell of abnormal radius and length, the transmission function will often yield transmission peaks within the band gap regions, which could result in an effective physical system for selectively filtering individual frequencies. The experimental aspect of this thesis explored the transmitted versus incident intensity of the monochromatic speaker-generated sound, finding that actual transmission through the system experiences an apparent frequency shift relative to the24Chapter 4. Experimental Procedure and Resultsexpected transmission, which was not easily explainable. However, it was assumed that reflected waves at the points of measurement for both the incident and transmitted intensity could have significantly altered both readings, given no method to calculate the reflected sound outside of the tube system. A number of improvements can be made to the physical corrugated tube system as pertains to collecting better transmission information. First, it would be ideal for the shorter tube sections to be completely filled between their outer radii and the longer tube's inner radius, so as to ensure that no excess sound can be transmitted through those regions. Of course, with the assumption of a lossless medium in the general theory, one cannot recreate the ideal system perfectly with plastic tubing, as some sound will always be transmitted through the plastic itself, so one would ideally use the most reflective material possible. The foam tape however, probably provided a reasonable amount of sound absorption in those regions. If data were taken in an extremely open area, it seems that reflected waves would become less of an issue. Ideally, the incident intensity would have a more efficient method of detection, possibly with a much smaller microphone inserted into the first region of the system as the speaker is attached to the tube, with another microphone measuring the transmitted intensity, enabling simultaneous and accurate transmission functions for each frequency. One might also attempt to determine how one could create a theoretical speaker system to generate completely one-dimensional waves, because the attached speaker alone does not guarantee this. Ultimately, the appearance of band gaps over roughly the expected frequency domain proves that the theoretical 5-cell corrugated tube system has reasonable correspondence to its physical counterpart. For future work, it would be interesting to see what other kind of band gap phenomenon might occur for different variations on the corrugated system, aside from the defect cell. Although, the principal utility of such an acoustic band gap system is already found in both the defect and regular systems, used to selectively filter out or transmit certain frequencies of sound.References[1] D. J. Griffiths and C. A. Steinke, &quot;Waves in locally periodic media,&quot; Am. J. Phys. 69, 137 (2001). [2] Trinklein, Frederick E. Modern Physics (New York, 1990), pp 256.`

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