Monochromatic Plane Waves in a Corrugated Tube … Plane Waves in a Corrugated Tube System A Thesis...

Monochromatic Plane Waves in a Corrugated Tube System

A Thesis

Presented to

The Division of Mathematics and Natural Sciences

Reed College

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Arts

Cody R. Myers

May 2009

Approved for the Division(Physics)

Lucas Illing

Table of Contents

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2: General Theory of Wave Propagation in Finitely PeriodicMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 The Single Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Multiple Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Corrugated Tube System . . . . . . . . . . . . . . . . . . . . . . 72.4 The Defect State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Translation to the Physical System . . . . . . . . . . . . . . . . . . . 11

Chapter 3: Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 4: Experimental Procedure and Results . . . . . . . . . . . . 194.1 The Method of Sound Measurement . . . . . . . . . . . . . . . . . . . 194.2 Taking Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Transmission for Best-Measurement Setup . . . . . . . . . . . . . . . 204.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

List of Figures

2.1 The Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Multiple Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Band Gaps with Increasing N . . . . . . . . . . . . . . . . . . . . . . 92.4 Defect Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Defect Transmission 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 61-Cell Defect Transmission . . . . . . . . . . . . . . . . . . . . . . . 122.7 Regular 61-Cell Transmission . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Corrugated Tube System . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 5-Cell Corrugated Tube Transmission . . . . . . . . . . . . . . . . . . 21

Abstract

This thesis focuses on the acoustic transmission properties of a periodically repeatingsystem, simply referred to as a corrugated tube. As with many periodic systems,acoustic wave propagation through the corrugated tube results in frequency “bandgaps,” or large ranges of incident sound frequency over which virtually no sound waveis transmitted through the tube. We first determine the theoretical sound transmis-sion for an N -cell corrugated tube (with cross-sectional area that alternates betweentwo values N times over periodic spatial intervals), also examining a small modi-fication of the same system called an N -cell defect state. Experimentally, soundtransmission data is collected for a specific instance of the corrugated tube systemwith 5 corrugations, where the incident and transmitted sound intensities are com-pared over a 0 to 1000 Hz frequency range for a speaker propagating monochromatic,plane wave sound through the system. We then compare the transmission versusfrequency plot of the observed sound transmission through the system to that of theexpected transmission, which shows some agreement in terms of the location of theband gaps.

Chapter 1

Introduction

When one examines the propagation of waves in a medium with specified boundaryconditions, such as the mechanical wave observed when oscillating a string with oneend tied to an object, two types of wave motion are usually observed over a large rangeof frequencies. For most frequencies, one will observe a traveling wave, in which theoverall shape of the wave is moving. That is, one will observe any point on a givencrest 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 andreflected waves results in a standing wave with no phase velocity; all of the maximaand minima oscillate at fixed x-positions (where the string lies on the x-axis). Whileonly these two forms of wave motion are observed in many simple physical systems,certain systems with periodically repeating boundaries exhibit an entirely differentbehavior, in which there is no transmitted wave over certain frequency ranges, calledband gaps (where “bands” 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 peri-odic crystal lattice will have allowed energy bands with intermittent energy band gapsover which its wave function becomes zero, hence there are certain energies that theelectron will never have. And given an appropriate periodically-repeating medium,one can observe the same phenomenon for any kind of wave propagation. This thesiswill focus on the band gaps observed in monochromatic (of single frequency) acousticpressure waves. The periodic system examined here is a corrugated tube with alter-nating cross-sectional area. While solid state physics deals with wave propagationthrough crystals, which can entail propagation through millions of repetitions of thebasic structural unit (the lattice), a corrugated tube system will have an extremelyfinite number of repetitions of its basic unit (one tube with radius r1 connecting an-other 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 expectationsof monochromatic sound transmission through an arbitrary periodic system. We willthen 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 beimplemented, resulting in a “defect state,” with sound transmission properties thatwill be compared to the original system. After this, the experimental results of sound

2 Chapter 1. Introduction

transmission will be compared to their expected values over a limited frequency range,giving insight into what factors might separate the physical corrugated tube systemfrom the idealized system. All sound used in the experimental study of the corrugatedtube system is monochromatic sound incident from a speaker, and assuming one-dimensionality of the waves, we are dealing with acoustic monochromatic plane wavesthroughout this thesis.

Chapter 2

General Theory of WavePropagation in Finitely PeriodicMedia

In order to treat the propagation of acoustic plane waves in the finite corrugated tubesystem, one can look to a general theory of wave transmission for one-dimensionalwaves in periodic media (systems physically consisting of repeating elements or “cells”),with the goal of describing the outgoing wave in the final cell in terms of the incidentwave in the first cell.

2.1 The Single Cell

A single one-dimensional cell can be described as some object having two boundariesenclosing it, thus wave propagation in a single-cell system will be piecewise, consistingof three separate wave functions: one on each side of the boundaries, and one withinthem. In general, any such system involving monochromatic plane waves will have aspatial wave function described by:

ψ(x) =

Aeikx +Be−ikx, if x < −a;

Feikx +Ge−ikx, if −a < x < a;

Ceikx +De−ikx, if a < x,

(2.1)

where A and C are coefficients for right-propagating waves, and B and D are thecoefficients for left-propagating waves. The variable k is the wavenumber (also calledthe angular wavenumber) for a given frequency of wave, measured in units of recip-rocal meters, m−1. Here k = ω

v, where ω = 2πf for a wave with frequency f traveling

through a medium at speed v, which for a sound wave traveling through air is roughly343 m/s. This spatial wave function originates from a separation of variables appliedto the full wave function Ψ(x, t) = ψ(x)e−iωt, which in turn will be derived from apartial differential equation particular to a given system.

Here the cell’s left and right boundaries are located a distance a from the origin, asdepicted in Fig. 1.1. For relevant systems, two physical conditions at each boundary

4 Chapter 2. General Theory of Wave Propagation in Finitely Periodic Media

Figure 2.1: A unit cell centered at the origin. The three regions are formed by apiecewise 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 expo-nential, allowing for the elimination of F and G. One can therefore express A and Bin terms of C and D, with the relation facilitated by a 2 by 2 transfer matrix M (1):(

AB

)= M

(CD

), (2.2)

where

M =

(M11 M12

M21 M22

). (2.3)

Two conditions allow for the structure of M to be specified in some detail. First,the partial differential equation from which Ψ(x, t) is derived can always be satisfiedby replacing Ψ(x, t) with the time-reversed complex conjugate Ψ∗(x,−t), or ratherany systems studied in this general approach are assumed to satisfy this condition,including the corrugated tube system (1). The transfer matrix obtained from ψ∗(x)tells us that M11 = M∗

22 and M21 = M∗12 (1).

Secondly, one can derive a relation among the wave coefficients that is basedon 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 transfermatrix equation, this informs us that that the transfer matrix must be unimodularwith determinant 1, or

|w|2 − |z|2 = 1, (2.4)

where the elements of the transfer matrix take the form:

M =

(w zz∗ w∗

), (2.5)

so only w and z must be determined (1).

2.2. Multiple Cells 5

2.2 Multiple Cells

The next goal of this analysis is to apply the known single-cell transfer matrix to anarray 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 thedistance between the first and nth cell’s left boundary is ns. Hence the wave functionbetween each cell may be written:

ψn(x) = Aneik(x−ns) +Bne

−ik(x−ns) for (n− 1)s+ a < x < ns− a (2.6)

where n goes from 0 to N (1). This allows for each wave function to be placed in alocal coordinate system, with each origin located at the center of the next cell to theright. Using this transfer matrix approach, one can ignore the wave functions withineach cell, and relate the wave functions between each cell with the use of the knownsingle-cell transfer matrix: (

An

Bn

)= M

(An+1e

−iks

Bn+1eiks

), (2.7)

or (An

Bn

)= P

(An+1

Bn+1

), (2.8)

where

P = M

(e−iks 0

0 eiks

). (2.9)

In effect, the An and Bn coefficients are not the real coefficients of the system forright and left-traveling waves in each given region (except for A0 and B0 which areequal 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 rolesimilar 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 theN-cell system: (

A0

B0

)= PN

(AN

BN

). (2.10)

Therefore evaluating PN will establish a relation between the coefficients of theincident and transmitted wave functions in this system. To begin this evaluation, theCayley-Hamilton theorem states that any matrix will satisfy its own characteristicequation, which for the unimodular 2 by 2 matrix P, tells us

P2 − 2Pξ + I = 0, (2.11)

where ξ = 12Tr(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


I. Recursively, one can therefore reduce any PN to a linear combination of P and I,so in general one can say:

PN = PUN−1(ξ)− IUN−2(ξ) (2.12)

where UN(ξ) is a polynomial of degree N in terms of ξ. Multiplying by P and substi-tuting Eq. (1.11) yields:

PN+1 = (2Pξ − I)UN−1(ξ)−PUN−2(ξ).

Another expression for PN+1 is obtained by replacing N with N + 1 in Eq. (1.12):

PN+1 = PUN(ξ)− IUN−1(ξ).

Now equating these two expressions, we arrive at the following recursion relationconsisting of only the polynomials themselves (1):

0 = UN+2(ξ)− 2ξUN+1(ξ) + UN(ξ). (2.13)

One finds that UN is the Nth Chebychev polynomial of the second kind, so this allowsus to express Eq. (1.12) in closed form for any given N (1). Now PN can be writtenexplicitly, which can be used to find a total transfer matrix MN for the system. Thissimply entails shifting PN back to the first cell’s coordinate system, correcting forthe fact that AN and BN are shifted coefficients:

MN = PN

(eikNs 0

0 e−ikNs

)(2.14)

MN =

([we−iksUN−1(ξ)− UN−2(ξ)]eikNs zUN−1(ξ)e−ik(N−1)s

z∗UN−1(ξ)eik(N−1)s [w∗eiksUN−1(ξ)− UN−2(ξ)]e−ikNs

).(2.15)

With the total transfer matrix at hand, only w and z must be determined for agiven 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-propagatingincident 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(ξ)]2(2.17)

2.3. The Corrugated Tube System 7

Figure 2.2: The corrugated tube system, identical to Fig. 1.1 with a repetition of Ncells.

2.3 The Corrugated Tube System

Now focusing on the subject of interest, we look to sound propagation in the cor-rugated tube system. It consists of two kinds of repeating cylindrical tubes, withrespective cross-sectional areas S1 and S2, and lengths l and 2a. The S2 tube (theunit cell) is centered at the origin, and repeats each distance s a total of N times, withan 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 describedwith the classical wave equation, as would apply to an ordinary cylindrical tube (1).Thus the wave equation for sound within the system is:

∂Ψ(x, t)

∂t2= v2∂Ψ(x, t)

∂x2, (2.18)

where v is the speed of sound in air, and Ψ is a measure of the sound pressure abovethe ambient pressure. Fig. 1.2 provides a full diagram for the system. The solutionfor the single-cell version of this system is equivalent to Eq. (1.1):

ψ(x) =

Aeikx +Be−ikx, if x < −a;

Feikx +Ge−ikx, if −a < x < a;

Ceikx +De−ikx, if a < x.

(2.19)

This ψ is not strictly a function of one dimension throughout the tube, as diffractioninevitably occurs at the boundaries between different cross-sectional areas. However,to good approximation, sound waves with low enough frequency will not generate sig-nificant non-plane wave areas around the boundaries. The cutoff frequency fcutoff, be-low which acoustic waves in this system can be considered essentially one-dimensional,is given by:

fcutoff =v√S, (2.20)

where S is the larger cross-sectional area, and all frequencies dealt with in this researchwill fall under this range (1).


Two boundary conditions allow us to determine the w and z parameters of thissystem. The first condition expresses continuity of pressure,

∆ψ = 0, (2.21)

and the second condition expresses conservation of mass (1),

∆(Sdψ

dx) = 0. (2.22)

Applying these conditions to each of the two boundaries in the single-cell case, oneobtains four equations that algebraic manipulation will show to give the transfermatrix elements:

w = [cos(2ka)− iε+sin(2ka)]e2ika, (2.23)

and

z = iε−sin(2ka), (2.24)

where ε± = 12[S1/S2 ± S2/S1] (1). Using Eq. (1.17), the transmission coefficient is:

TN =1

1 + [ε−sin(2ka)UN−1(ξ)]2. (2.25)

It is this transmission coefficient that leads to the fundamental source of interestin this system; the existence of band gaps across certain frequency ranges over whichno sound is transmitted. As the number of corrugations increases for a given set ofparameters, these gaps become more pronounced. Suppose we take a system withtubes of radii 1 cm and 2 cm, and set both l and a equal to 2 cm. In Fig. 1.3, theresults are plotted for increasing values of N, and one can see the emergence of a largefrequency gap.

2.4 The Defect State

We can modify the multiple cell theory to observe a system with similarly interestingtransmission properties, which will be called the ‘defect state’ version of the multiplecell corrugated tube. We will use the same corrugated tube system, except now thecell centered at the origin will have a length 2b different from the other cells of length2a, with radius and cross sectional area r3 and S3. We take the system to have N cellsto the right of the center defect cell, and K cells to the left. As before, the right andleft-traveling wave coefficients for the farthest-left region of the system are termed Aand B, and our goal is to find the final coefficients C and D on the far right. Thesystem 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:

ψn(x) = Aneik(x−(n−1)s) +Bne

−ik(x−(n−1)s) for b+ (n− 1)s < x < b+ ns− 2a(2.26)

2.4. The Defect State 9

0 1000 2000 3000 4000f

0.2

0.4

0.6

0.8

1.0T

0 1000 2000 3000 4000f

0.2

0.4

0.6

0.8

1.0T

0 1000 2000 3000 4000f

0.2

0.4

0.6

0.8

1.0T

0 1000 2000 3000 4000f

0.2

0.4

0.6

0.8

1.0T

Figure 2.3: These plots show the emergence of a band gap as the number of corruga-tions in the tube system increases from N = 1 to N = 2, to N = 8, to N = 24. They-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 corrugatedtube system, with a cell in the center of differing length and radius.


as n goes from 1 to N . And the equation for the waves between the cells to the leftis given by:

ψn(x) = Aneik(x−(n+1)s) +Bne

−ik(x−(n+1)s) for − b+ ns+ 2a < x < −b+ (n+ 1)s,(2.27)

where n goes from -1 to -K. We have already established the transfer matrix for theregions to the left and the right of the defect. On inspection, one can see that Eq. 2.7holds for all ψn(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 theleft and right regions, and to establish this fact one must solve for the transfer matrixcoefficients 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. Doingthis shows that both transfer matrices are the same as the usual M, with coefficientsgiven in equations 2.23 and 2.24. Intuitively though, one can see that M for the leftand right regions must be the same as before, because after obtaining the single-celltransfer matrix, knowing that Eq. 2.7 still applies to both regions, the same analysisleads one to a total shifted transfer matrix given by Eq. 2.12. And if one was dealingwith only the N cells on the right, then one can see that for transmission throughthe system to be the same as in the non-defect case, the w and z values must bethe same as before. So much like before, we must determine a total transfer matrix,which must take the form PLPMPR, these matrices representing the shifted transfermatrices for the left, middle, and right sides of the system.

We have already deduced that:

PL = PK , (2.28)

and

PR = PN , (2.29)

and now one must determine how to deal with the transfer matrix PM involved inthe center of the system, attributed the defect cell. One might initially suppose thatPM is simply the transfer matrix consisting of the same parameters involving S3 andS1 areas instead of the S2 and S1 areas. Such a modified transfer matrix would begiven by equation 2.5, except now:

M2 =

(w2 z2

z∗2 w∗2

), (2.30)

where

w2 = [cos(2kb)− iε2+sin(2kb)]e2ikb, (2.31)

and

z2 = iε2−sin(2kb), (2.32)

where ε2± = 12[S1/S3 ± S3/S1].

2.5. Translation to the Physical System 11

This is partly correct, but using a modified transfer matrix M2 would only bephysically relevant if one were relating the actual coefficients of the system betweenthe ψ−1 and ψ1 regions. Since PL and PR only relate shifted coefficients, PM mustdo so as well. Hence PM must be equivalent to the single shifted transfer matrix asin the N -cell system. So we have:

P2 = M2

(e−iks2 0

0 eiks2

). (2.33)

where s2 = l + 2b. As before, the total transfer matrix is simply the total shiftedtransfer matrix multiplied by a matrix to account for the shift in coefficients, giventhat the final AN and BN are not the actual coefficients for the final region. For atotal transfer matrix MN(defect), we have:

MN(defect) = PKP2PN

(eik(Ks+Ns+s2) 0

0 e−ik(Ks+Ns+s2)

). (2.34)

Importantly, this total transfer matrix allows one to obtain the same total transfermatrix as in the regular (N+K+1)-cell system, if one models that system by, for exam-ple, setting b = a, s2 = s, and r3 = r2. We can now obtain the transmission functionfor the system. Although the transmission function is slightly too complicated to bemeaningful 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 thefirst row and column of the total transfer matrix.

The transmission function for this system yields some interesting behavior. Onewill still observe band gap regions over essentially the same frequency domains aswould be seen in a similar (N+K+1)-cell system, but for a small total number ofcells, 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 tubestudied. 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 adefect does little to influence the band gap regions, which will no longer display thesesharp peaks. Fig. 2.6 displays a defect state with 61 total cells, while Fig. 2.7 showsthe 61-cell system with the same parameters, but without the defect.

One can imagine that a defect state corrugated tube system would be useful inisolating specific frequencies, if one wanted to select a specific frequency to transmitover a large range, thereby acting as a selective acoustic filter.

2.5 Translation to the Physical System

In 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


0 200 400 600 800 1000f Hz

0.2

0.4

0.6

0.8

1.0T

Figure 2.5: One plots the transmission function versus frequency for a defect systemsimilar to the 5-cell corrugated tube system studied.

0 200 400 600 800 1000f Hz

0.2

0.4

0.6

0.8

1.0T

Figure 2.6: One plots the transmission function versus frequency for the same defectsystem as in Fig. 2.5, but now there are 61 total cells. The band gaps are almostexactly the same as those of Fig. 2.7.

2.5. Translation to the Physical System 13

0 200 400 600 800 1000

0.2

0.4

0.6

0.8

1.0

Figure 2.7: One plots the transmission function versus frequency for the same systemas 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 seemsto put us at a disadvantage, largely because there is no physical barrier accounted foron the left side of the theoretical system. This may affect the incident sound to someextent; sound propagating to the right will reflect off the first boundary, and thenreflect off of the speaker-bounded side, resulting in a right-traveling wave in the firstregion 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 alocation where the spatial wave function, ψ(x), is equal to zero. The mounted speakerdoes not act exactly like a boundary, because although ψ(x) surely should go to zerobehind the speaker (if the speaker were truly only propagating sound to the right),ψ(x) is actually constantly modulated by the speaker at the boundary. So the leftside of the corrugated system is effectively much closer to an open-ended tube as inthe preceding theory, provided that one only defines the wave function on its intendedrange, over the length of the tube.

In measuring the transmission coefficient of the physical corrugated tube systemfor a given frequency, one will obtain an incident and transmitted reading of thesound intensity, measured in decibels (dB), which are dimensionless. If we refer tothe incident intensity reading (obtained from the sound level meter described in thenext section) as PA and the transmitted intensity as PC , then the equations relatingthe intensities to the pressure amplitudes are:

PA = 10log10(A2

X2) (2.36)

PC = 10log10(C2

X2), (2.37)

where X is some reference amplitude. One easily obtains the transmission coefficient,T = C2

A2 , as a function of PC and PA:

T = 10(PC−PA)/10. (2.38)


This will later allow us to plot the transmission of the corrugated tube system as afunction of frequency, referencing only the incident and transmitted dB readings.

Chapter 3

Experimental Setup

A 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 threedimensions? First one should note that the cross-sectional area of each tube is theonly parameter relevant to the extra dimensions; a feature which can essentially beattributed to the symmetry of each tube length on the y-z axis, accounted for in thewave equation. Furthermore, the change in cross-sectional area makes no referenceto the orientation of the long and short tubes, giving one the freedom to constructthe boundaries between each section in whatever way is most convenient. There issome issue regarding the one-dimensionality of the sound source, as it is difficult toconstruct 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 tubewaveguide, 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 largetube made of PVC plastic of inner radius r2 = 3.95 cm, thickness 0.5 cm, and length2.947 m is used to create the larger-radius sections comprising the sides of each unitcell. 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 tubeby 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 almostexactly equal to r2, but slightly smaller (less than 5 mm smaller), allowing them toslide through the long tube with only a small amount of force applied. MD CamperSeal Foam Tape, with thickness .047 cm and width 3.17 cm, is wrapped around eachend of each smaller tube as shown in the diagram. 62 cm of the tape is used for eachwrap, and this effectively increases the sound absorption in the regions between theshorter 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 thedistance 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 untilcoming to a stop at each screw.

The speaker used for the sound source is a Road Gear RGSP54 speaker, which

16 Chapter 3. Experimental Setup

is 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 largerthan r2, allowing for the long tube to fit securely into it. This speaker is driven by aTektronix CFG280 11 MHz function generator, with which one can select the drivingfrequency with a margin of error of ±.1 Hz. A Tektronix TDS 2024 four channeldigital oscilloscope is also sometimes used to monitor the voltage reading across thespeaker, to ensure that only the frequency displayed by the function generator istransmitted by the speaker. A Quest 1200 precision integrating sound pressure levelmeter (SLM) is used to detect the transmitted and incident sound in the system. Itis 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 respectto frequency, so that there is no measurable frequency-dependence of the SLM. TheSLM is set with its microphone end directly in front of the end of the corrugated tubesystem to detect the intensity (dB) level of the system, and each reading it displaysmeasures the maximum dB level detected in the past second at that point in space.

17

Figure 3.1: The individual components of the corrugated tube system are shown, notdrawn to scale for the sake of visual clarity. 1) The long (and short) tube is madefrom PVC plastic with thickness .5 cm. One must note that the long tube has 5 holesdrilled into the top at distances l + 2a, 2(l + 2a), 3(l + 2a), etc. where screws areinserted 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 endsheld 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 isslightly larger. 5) The SLM has several settings, chosen to give a linear dB responseover the 1000 Hz range. 6) The full 5-cell system, where the screws hold the shorttubes in place. One should note that the long pipe is slightly inclined downward fordata runs, so that its end touches the ground.

Chapter 4

Experimental Procedure andResults

4.1 The Method of Sound Measurement

Before 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 trans-mitted and incident sound in regards to the placement of the Sound Pressure LevelMeter (SLM). In the general theory section, it is assumed that the system is entirelylossless, so theoretically, sound measurement in any given region of the tube shouldbe uniform in terms of intensity. However, the first problem with measuring eitherincident or transmitted sound within the tube should be clear; placing the SLM inthe 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 de-tected transmitted intensity in the final section of the pipe, with a tendency towardsincreasing attenuation very close to the end. And yet another problem arises whenone considers the possible resonant frequencies of the pipe, in which standing wavesform. At any given value of x, a standing wave will oscillate up and down over timeas one typically expects of a wave. But unless one happens to be measuring the waveat an x-value upon which one of its maxima lie, one will not be measuring the totalamplitude 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 forthe total amplitude of the wave, observing virtually no intensity.

So in order to best measure the transmitted intensity of the system, the SLM wastypically placed with the end of its microphone on the exact boundary between thefinal section of the tube and outside room, facing perpendicular to the face of thetube. Measuring just outside of the tube allows one to avoid detecting any standingwaves that might develop inside, while also more effectively preventing any influencethat 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, onecannot insert the SLM to get a direct reading of the intensity anywhere in the first

20 Chapter 4. Experimental Procedure and Results

region of the system. And in actuality, with the speaker attached to the left side ofthe system, some reflections will occur on the first interface between the large andsmall cross-sectional areas, resulting in a different A value than that generated by thespeaker alone (either attached to a non-corrugated tube or otherwise). If one thenconsiders reflections from every interface, it is clear that we can only obtain a precisevalue for A with the exact corrugated tube system in place while the speaker is op-erating. Although the incident intensity was measured in a number of different ways,the most effective method (while still feasible using the SLM) was simply measuringthe incident intensity as the sound generated with the SLM close as possible to thespeaker. In this way, one can be reasonably sure that the right-propagating wavesof the highest amplitude in the system are detected, such that there is no risk ofthe transmitted intensity being greater than the incident intensity for any frequency.And this is something of a requirement, because for a lossless system, transmissiongreater than 1 is not theoretically possible, unless outside factors such as resonancein the room are considered.

4.2 Taking Data

First, the corrugated system is ‘constructed’ simply by inserting one screw roughly3 cm into the long tube, at the ending boundary. Then one tilts the long tube atan angle of roughly 30 degrees or higher to slide a short tube down through it, untilhearing the short tube come into contact with the screw. This process is repeateduntil all 5 short tubes are in place. Although there is no method for completelysecuring the short tubes in place, during this process the long tube is never tilted at anegative angle, so one can simply consider the position of the tubes to have an errorequivalent to that of the screw measurement, ±0.2 cm. Placing the system on a longtable, the long tube is then inserted into the hole of the mounted speaker, with itsopposing end left to rest on the top of the table. The SLM is placed along the axisof the tube, with its microphone facing the open end, and the end of the microphoneplaced at the end of the tube. With the function generator driving the speaker andthe 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 betweenexperimental and theoretical transmission.

4.3 Transmission for Best-Measurement Setup

The following transmission results for the 5-cell system were taken with the ‘best-measurement’ setup described above for the positions of the SLM, taking incidentintensity 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 endof 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 400Hz, and 550 to 800 Hz. These regions are for the most part identifiable in the data

4.3. Transmission for Best-Measurement Setup 21

Figure 4.1: This plot displays the transmission coefficient as a function of frequencyfor 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 leftof these two band gaps are also identifiable, however it seems as though both plotswould be much more consistent with theory if the actual transmission was translatedto the left along the frequency axis. A simple explanation for such behavior wouldbe a mechanism causing a shift in the driven frequency for the speaker, that is, adiscrepancy between the wavenumber k for the measured frequency of the functiongenerator and the actual wavenumber, which could possibly be attributed to thespeaker’s mount. But the existence of such a mechanism seems unlikely. A shift inthe wavenumber of a monochromatic wave (in general, think of Eq. 2.1) will occurin a dissimilar medium, any sound traveling through the ABS plastic mount itselfwill have a k different from k in air due to dissimilar speeds of sound propagation ineach medium. However, any sound passing from the speaker through the plastic willregain the same wavenumber once reentering the air, so there should be no k-shift inany 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 thewavenumber 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 roomis highly unlikely for any reason, being an open system (without taking into accountacoustic pressure waves), so the main possible cause for a shift could be attributed toa difference in temperature, due to ventilation in the room or some other factor. Butto account for a frequency shift that would appear to be approximately a constant


50 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, forthe expected wavenumber of the incident sound waves, we have:

kexpected =2πf

vair

, (4.1)

while for the actual wavenumber with a 50-Hz shift, we have:

kactual =2π(f − 50)

vtube

, (4.2)

for velocities of sound in the air and in the tube given by vair, and vtube. The lattervelocity can be found by equating the two wavenumbers, thereby giving the necessaryvtube to correct for the shift in k. For vair = 343 m/s, setting kexpected = kactual yieldsthe 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 possiblefor vtube to be dependent on the frequency of the input wave, this is just a mathe-matical 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 rangesfrom 288.36 to 340.27 m/s over the above data, or an average shift in velocity of25.95 m/s, giving an average velocity vtube = 317.05 m/s. For air at a temperatureof 0 ◦C, the speed of sound in air is 331.5 m/s (2), so assuming that there could bea small temperature change in the tube system of 1 ◦C at most, the resultant changein the velocity of sound in air is not nearly great enough to account for the apparentobserved frequency shift.

It is possible that for some or all frequencies of the function generator, the speakergenerates sound at an additional frequency (or frequencies), but for such a significantfrequency shift to occur, the intensity of those frequencies would have to exceed thatof the function generator frequency, which seems unlikely. Moreover, it is unlikely thata speaker would generate an extra frequency with a constant difference between thedriving frequencies, one usually encounters higher (attenuated) frequencies that arerelated to resonance in the system, taking on values that are some ratio or multiple ofthe driving frequency. Furthermore, using the oscilloscope to observe the sinusoidally-varying driving voltage across the speaker, no additional non-driving frequencies weredetected for the system over the 0 to 1000 Hz range, although this does not completelypreclude the possibility of some physical aspect of the speaker malfunctioning and notreproducing the input voltage at a given time exactly.

It seems that one of the most important factors in accounting for the discrepancyin the predicted and measured transmission is the physical layout of the room inwhich data was taken, which was not modeled analytically or numerically. For thetransmitted wave (and similarly for the incident wave), there will be a left-traveling D

4.4. Conclusion 23

wave 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 bythe speaker that does not enter the tube–this D wave is nonzero. Because the SLMmeter constantly displays the highest intensity (and therefore highest wave amplitude)of sound detected in the previous second, one might expect that these reflections havelittle bearing on its measurement due to their weak intensity relative to the soundemitted from the tube; as long as the SLM finds one maximum from the sound wavedirectly 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 (buta different phase constant, imagine adding two functions Asin(kx) and Bsin(kx +φ)) will create a measured wave with the same frequency, but with a maximumamplitude 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 toresonance in the room, the effects of interference could become pronounced. This isthe most likely explanation for the behavior seen in the middle region, where one seesa much smaller peak in the actual transmission which could be due to destructiveinterference between the C and D waves. This does not account for the apparentfrequency shift though, unless a phase constant φ for the reflected waves somehowvaries continuously as a function of frequency. While there is no clear-cut explanationas to why an apparent frequency shift occurs for the actual transmission, one doesfind that the inability to determine the effect of the left-traveling waves on both theincident and transmitted frequency makes it much more difficult to obtain a truetransmission plot. Particularly, the left traveling waves of each frequency do notnecessarily interfere in the same way for the SLM readings at the speaker and thoseat the end of the system.

4.4 Conclusion

This thesis has thoroughly explored the acoustic transmission properties of the peri-odically repeating corrugated tube system, although given the nature of the generaltheory, many similar periodic systems can make use of the same analysis, applicableto waves outside of the acoustic pressure waves we used. Transmission in this systemwas found to lead to the phenomenon of band gap formation, which is usually re-stricted to fully periodic systems such as crystals. We found that band gap locationsare characteristic of all monochromatic plane wave sound transmitted through thesystem, and these gaps become much more apparent with increasing N . It was alsofound that for a corrugated tube containing one defect cell of abnormal radius andlength, the transmission function will often yield transmission peaks within the bandgap regions, which could result in an effective physical system for selectively filteringindividual frequencies.

The experimental aspect of this thesis explored the transmitted versus incidentintensity of the monochromatic speaker-generated sound, finding that actual trans-mission through the system experiences an apparent frequency shift relative to the


expected transmission, which was not easily explainable. However, it was assumedthat reflected waves at the points of measurement for both the incident and trans-mitted intensity could have significantly altered both readings, given no method tocalculate the reflected sound outside of the tube system.

A number of improvements can be made to the physical corrugated tube systemas pertains to collecting better transmission information. First, it would be ideal forthe shorter tube sections to be completely filled between their outer radii and thelonger tube’s inner radius, so as to ensure that no excess sound can be transmittedthrough those regions. Of course, with the assumption of a lossless medium in thegeneral theory, one cannot recreate the ideal system perfectly with plastic tubing,as some sound will always be transmitted through the plastic itself, so one wouldideally use the most reflective material possible. The foam tape however, probablyprovided a reasonable amount of sound absorption in those regions. If data weretaken in an extremely open area, it seems that reflected waves would become lessof an issue. Ideally, the incident intensity would have a more efficient method ofdetection, possibly with a much smaller microphone inserted into the first region ofthe system as the speaker is attached to the tube, with another microphone measuringthe transmitted intensity, enabling simultaneous and accurate transmission functionsfor each frequency. One might also attempt to determine how one could create atheoretical speaker system to generate completely one-dimensional waves, becausethe attached speaker alone does not guarantee this.

Ultimately, the appearance of band gaps over roughly the expected frequencydomain proves that the theoretical 5-cell corrugated tube system has reasonable cor-respondence to its physical counterpart. For future work, it would be interesting tosee what other kind of band gap phenomenon might occur for different variations onthe corrugated system, aside from the defect cell. Although, the principal utility ofsuch an acoustic band gap system is already found in both the defect and regularsystems, used to selectively filter out or transmit certain frequencies of sound.

References

[1] D. J. Griffiths and C. A. Steinke, “Waves in locally periodic media,” Am. J. Phys.69, 137 (2001).

[2] Trinklein, Frederick E. Modern Physics (New York, 1990), pp 256.

Monochromatic Plane Waves in a Corrugated Tube … Plane Waves in a Corrugated Tube System A Thesis...

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