UMAP 2004 vol. 25 No. 4

TheUMAPJournalPublisherCOMAP, Inc. Vol. 24, No. 4Executive PublisherSolomon A. Garfunkel

ILAP EditorChris ArneyInterim Vice-President

for Academic AffairsThe College of Saint Rose432 Western AvenueAlbany, NY [email protected]

On Jargon EditorYves NievergeltDepartment of MathematicsEastern Washington UniversityCheney, WA [email protected]

Reviews EditorJames M. CargalMathematics Dept.Troy State Univ. Montgomery231 Montgomery St.Montgomery, AL [email protected]

Chief Operating OfcerLaurie W. Aragon

Production ManagerGeorge W. Ward

Director of Educ. TechnologyRoland Cheyney

Production EditorPauline Wright

Copy EditorTimothy McLean

DistributionKevin DarcyJohn Tomicek

Graphic DesignerDaiva Kiliulis

Editor

Paul J. CampbellCampus Box 194Beloit College700 College St.Beloit, WI [email protected]

Associate Editors

Don AdolphsonDavid C. Chris ArneyRon BarnesArthur BenjaminJames M. CargalMurray K. ClaytonCourtney S. ColemanLinda L. DeneenJames P. FinkSolomon A. GarfunkelWilliam B. GearhartWilliam C. GiauqueRichard HabermanCharles E. LienertWalter MeyerYves NievergeltJohn S. RobertsonGarry H. RodrigueNed W. SchillowPhilip D. StrafnJ.T. SutcliffeDonna M. SzottGerald D. TaylorMaynard ThompsonKen TraversRobert E.D. Gene Woolsey

Brigham Young UniversityThe College of St. RoseUniversity of HoustonDowntownHarvey Mudd CollegeTroy State University MontgomeryUniversity of WisconsinMadisonHarvey Mudd CollegeUniversity of Minnesota, DuluthGettysburg CollegeCOMAP, Inc.California State University, FullertonBrigham Young UniversitySouthern Methodist UniversityMetropolitan State CollegeAdelphi UniversityEastern Washington UniversityGeorgia College and State UniversityLawrence Livermore LaboratoryLehigh Carbon Community CollegeBeloit CollegeSt. Marks School, DallasComm. College of Allegheny CountyColorado State UniversityIndiana UniversityUniversity of IllinoisColorado School of Mines

Subscription Rates for 2004 Calendar Year: Volume 25

Individuals subscribe to The UMAP Journal through COMAPs Membership Plus. This subscriptionalso includes print copies of our annual collection UMAP Modules: Tools for Teaching, ourorganizational newsletter Consortium, on-line membership that allows members to download andreproduce COMAP materials, and a 10% discount on all COMAP purchases.

(Domestic) #2420 $90(Outside U.S.) #2421 $105

Institutions can subscribe to the Journal through either Institutional Plus Membership, RegularInstitutional Membership, or a Library Subscription. Institutional Plus Members receive two printcopies of each of the quarterly issues of The UMAP Journal, our annual collection UMAP Modules:Tools for Teaching, our organizational newsletter Consortium, on-line membership that allowsmembers to download and reproduce COMAP materials, and a 10% discount on all COMAP purchases.


Regular Institutional members receive print copies of The UMAP Journal, our annual collectionUMAP Modules: Tools for Teaching, our organizational newsletter Consortium, and a 10% discount onall COMAP purchases.


Web membership does not provide print materials. Web members can download and reproduceCOMAP materials, and receive a 10% discount on all COMAP purchases.


To order, send a check or money order to COMAP, or call toll-free1-800-77-COMAP (1-800-772-6627).

The UMAP Journal is published quarterly by the Consortium for Mathematics and Its Applications(COMAP), Inc., Suite 210, 57 Bedford Street, Lexington, MA, 02420, in cooperation with theAmerican Mathematical Association of Two-Year Colleges (AMATYC), the MathematicalAssociation of America (MAA), the National Council of Teachers of Mathematics (NCTM), theAmerican Statistical Association (ASA), the Society for Industrial and Applied Mathematics (SIAM),and The Institute for Operations Research and the Management Sciences (INFORMS). The Journalacquaints readers with a wide variety of professional applications of the mathematical sciences andprovides a forum for the discussion of new directions in mathematical education (ISSN 0197-3622).

Periodical rate postage paid at Boston, MA and at additional mailing offices.

Send address changes to: [email protected]

COMAP, Inc. 57 Bedford Street, Suite 210, Lexington, MA 02420 Copyright 2003 by COMAP, Inc. All rights reserved.

Membership Plus

Institutional Plus Membership

Web Membership

Institutional Membership

Vol. 24, No. 4 2003

Table of Contents

EditorialWe Have Already Been to the Future

Paul J. Campbell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

ArticlesEfficient Signal Transmission and Wavelet-Based Compression

Susan E. Kelly and James S. Walker . . . . . . . . . . . . . . . . . . . . . . . .395

UMAP ModulesThe Spread of Forest Fires

Emily E. Puckette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

ILAP ModulesGetting the Salt Out

Ethan Berkove, Thomas Hill, Scott Moor . . . . . . . . . . . . . . . . . . . 435

Vehicle EmissionsLei Yu and Don Small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .451

Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .473Annual Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Editorial 393

EditorialWe Have Already Been to the FuturePaul J. CampbellMathematics and Computer ScienceBeloit CollegeBeloit, WI [email protected]

Every 10 years or so, the Mathematical Association of America publishesrecommendations for the undergraduate program in mathematics; long ago,I was part of a panel that prepared such a report. The last report (1991) wasshort; the mathematics community then was in the throes of the calculus wars,computer science was coming on strong, and there was no longer commonagreement on what courses should be part of a mathematics major.

After four years of meetings, panel discussions, focus groups, conferences,surveys of mathematics departments, consultation with partner disciplines,and deliberations, there is a new curriculum guide [CUPM 2003].

The new report distinguishes itself by not focussing primarily on the pro-gram for mathematics majors but addressing the entire college-level mathe-matics curriculum, for all students, even those who take just one course.

The six recommendations of the report are not likely to be controversial,and they are not accompanied by implementation suggestions. Here are para-phrases:

1. Understand and meet the needs of the students, and assess the effectivenessof doing so.

2. Every course should have activities that promote analytical reasoning, problem-solving, communication skills, and mathematical habits of mind.

3. Every course should present various perspectives, employ examples andapplications from contemporary topics, and promote students ability toapply the material.

4. Mathematical sciences departments should collaborate with other depart-ments on courses, majors, research projects, and teaching.

The UMAP Journal 24 (4) (2003) 393394. cCopyright 2003 by COMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for prot or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAP must be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.

394 The UMAP Journal 24.4 (2003)

5. At every level of the curriculum, some courses should have activities thatteach the use of technology in solving problems and understanding mathe-matical ideas.

6. Faculty teaching and curricular efforts should be rewarded.

The report cites

changes: declining numbers of bachelors degrees in mathematics (down 20%

since 1990 while total bachelors degrees rose 20%), corresponding drops in enrollments in calculus and in advanced courses,

and a high proportion (30%) of mathematics majors having a second or joint

major (i.e., they are interested primarily in something other than math-ematics);

a nagging constant: Mathematics has extremely poor public relations. Stu-dents, as well as deans and faculty from other departments,view

mathematicians as insular and excessively inward looking; mathematics departments as disconnected from other disciplines except

through a service component which they believe is accepted only reluc-tantly and executed without inspiration or effectiveness, and uninter-ested in adapting instruction to needs of non-majors; and

(crucially) mathematics itself as irrelevant for them (as in math is toughand too much work anyway, so Ill do without it but keep my self-esteemby deeming it irrelevant).

COMAP has been ahead of the curveor at least, the recommendationsare catching up with what COMAP has been doing all along. For more than25 years, COMAP has played a major role in developing and disseminatingmaterials that promote the use of contemporary applications in teaching math-ematics. ILAPs (see the two in this issue) in particular encourage collaborationwith faculty in other disciplines.

COMAP alone cannot change mathematical instruction or societal attitudes.However, readers of this Journal need to direct the attention of their colleagues(particularly those new to the profession) to the resources that COMAP offersfor implementing the new curricular recommendations.

ReferencesCommittee on the Undergraduate Program in Mathematics (CUPM). 2003. Un-

dergraduate Programs and Courses in the Mathematical Sciences: CUPM Curricu-lum Guide 2004; http://www.maa.org/cupm/welcome.html . Washington,DC: Mathematical Association of America.

Efficient Signal Transmission 395

Efcient Signal Transmissionand Wavelet-Based CompressionSusan E. KellyDepartment of MathematicsUniversity of WisconsinLa CrosseLa Crosse, WI [email protected]

James S. WalkerDepartment of MathematicsUniversity of WisconsinEau ClaireEau Claire, WI [email protected]

Abstract

We describe how digital signals can be transmitted by analog carrier signals. This isthe method of pulse code modulation. Transmission by pulse code modulation is widelyused thanks to its abilities to withstand corruption by noise and to transmit multiple sig-nals simultaneously. Digital signals can also be used to compress analog signals, thusallowing for more rapid transmission, by employing wavelet series. We shall describehow wavelet series are employed and compare them with Fourier series, showing that inmany instances wavelet series provide a better technique.

IntroductionA common method of transmitting digital signals is pulse code modulation (PCM).

A PCM signal is an analog signal that encodes the bits of the digital signal via a carrierwave. The frequency of this carrier wave is modified in such a way that the bits can beread off at the receiving end. This PCM method has a couple of beautiful features.

It can withstand large amounts of noise in the environment that typically corruptthe signal by the time it reaches the receiver.

It allows for simultaneous transmission of several different messages using just onePCM signal.

The UMAP Journal 24 (4) (2003) 395416. cCopyright 2003 by COMAP, Inc. All rights reserved.Permission to make digital or hard copies of part or all of this work for personal or classroom use is grantedwithout fee provided that copies are not made or distributed for profit or commercial advantage and thatcopies bear this notice. Abstracting with credit is permitted, but copyrights for components of this workowned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or toredistribute to lists requires prior permission from COMAP.


An important benefit of digital signals is their ability to compress analog sig-nals. Instead of transmitting the entire analog signal, it is more efficient to trans-mit numbersdigitally encodedthat describe how to reconstruct the signal at thereceiving end. A classical method for producing these numbers is to compute coef-ficients in a Fourier series. Recently, a whole new approach, called wavelet series,has been developed. Coefficients of a wavelet series are computed, digitally encoded,then transmitted in digital form. At the receiving end, these coefficients are used toproduce a wavelet series that provides an excellent approximation to the original sig-nal. Wavelet series are often more efficientfewer coefficients are needed for a goodapproximationthan Fourier series.

In the next section we shall describe the theory behind PCM signals and illustratetheir application to digital signal transmission. We then discuss Fourier series andwavelet series and their application to compression of analog signals. The paper con-cludes with a brief survey of references where further details and extensions of theseideas can be found.

The numerical data and graphs in this paper were produced with the free softwareFAWAV [Walker 2000; 2002].

PCM and Digital Signal TransmissionIn this section we shall explain how an analog carrier signal can be modified to en-

code a sequence of bits. Suppose, for example, that we want to transmit the followingsequence of seven bits:

1 0 0 0 1 1 0. (1)In order to represent any of the bits having value 1, we use a signal of the form:

g(t) = e(tk)2cos 2t. (2)

Figure 1a shows a graph of this signal for k = 6, = 2, and = 6; Figure 1b showsa different graph, using k = 3, = 25, and = 12. It is not hard to see that the

Figure 1. a. Pulse centered at position t = 6. b. Pulse centered at position t = 3.

parameter k controls where the center of the amplitude factor e(tk)2 lies. Also, theparameter controls how quickly this factor damps down toward 0 as t moves away


from k; in Figure 1b, is greater than in Figure 1a, hence the amplitude of the pulsein Figure 1b damps down more quickly to 0.

The parameter in (2) determines the frequency of the factor cos 2t. Hence, thecosine factors for the pulses shown in Figure 1 have frequencies of = 6 and = 12.As you can see from the figure, the oscillations of the pulse in Figure 1b are faster(twice faster, in fact) than the oscillations of the pulse in Figure 1a.

It is a standard task in electrical engineering to design a circuit having maximumresponse centered on a desired frequency [Boylstad 1994; Floyd 2001]. For example,if we want to send the bit 1 using the PCM signal shown in Figure 1b, then we shoulddesign a receiver having an electric circuit element with maximum response at = 12(Figure 2a). The response curve in Figure 2b would provide an excellent detectorfor such a transmission of the bit 1. Likewise, if we wanted to use the PCM signal inFigure 1a, then the maximum response for the receiver should be centered at = 6.

Figure 2. Response curves for electric circuits with maximum response at = 12. a. A broad responsecurve, unsuitable for precise detection. b. A narrow, precisely tuned response curve, suitable for accuratedetection.

By combining both such circuit elements, we could even have a receiver that coulddetect both bits, transmitted simultaneously. We shall return to this point later.

Now that we have discussed how to transmit a single bit having value 1, we canturn to the question of how to transmit a sequence of bits having different values, asin (1).

Transmitting a Bit SequenceThe method of transmitting a bit sequence, such as the one shown in (1), is to

simply add together signals of the type shown in (2). One frequency, 1, is used fortransmitting bits of value 1, and a second frequency 0 = 1 is used for transmittingbits of value 0.

For example, to transmit the sequence of bits 1 0 0 0 1 1 0, we use the followingPCM signal:

f(t) ={e(t2)

2+e(t3)

2+ e(t4)

2+ e(t7)

2}

cos 20t

+{e(t1)

2+ e(t5)

2+ e(t6)

2}

cos 21t.


with = 150, 0 = 232, and 1 = 280 (Figure 3a). In Figures 3 and 4, we illustratethe scheme for decoding at the receiving end the seven bits from the PCM signal.

Figure 3. Filtering of PCM signal to process individual pulses. The filters w1 and w4 in b are used to cutout the individual pulses at positions t = 1 and t = 4. The cutout pulses are shown in c and d.

To isolate an individual pulse at position t = k, the PCM signal f(t) is multipliedby a filter (or window) function wk(t). This function wk(t) is equal to 1 in an interval(k , k + ) centered at k, and decreases smoothly down to 0 for t < k 1 + andt > k + 1 . See Figure 3b, where the filters w1 and w4 are shown. Multiplyingthe PCM signal f by the filter wk produces a single pulse f(t)wk(t) that is nonzeroonly within the interval [k 1 + , k + 1 ] centered at t = k. In Figures 3c and3d, we show the pulses produced by multiplying the PCM signal f by w1 and w4. Themultiplications produce single pulses centered at t = 1 and t = 4.

In Figure 4b, we show the whole set of filters {wk}8k=0 for detection of the pulsesat positions t = 0, 1, 2, . . . , 8. These filters satisfy

8k=0

wk(t) = 1 (3)

for 0 t 8. Equation (3) implies that the PCM signal f(t) is equal to the sum of thefilter functions f(t)wk(t) over the interval [0, 8]. For decoding longer PCM signals,(3) needs to be extended via a larger set of filters, say {wk}Nk=0, satisfying

Nk=0

wk(t) = 1 (4)

for some large positive integer N .


Figure 4. PCM transmission of bit sequence 1 0 0 0 1 1 0. a. PCM signal. b. Filter bank used for detection.c. Detection: [bottom] PCM signal, [top] magnitude of responses to filter bank output. Darker pixelsrepresent more intense response.

While (4) is a handy property, it is not absolutely necessary for the decodingscheme to work. In fact, for decoding to work well it is sufficient that the follow-ing double inequality hold:

A N

k=0

wk(t) B,

where A and B are positive constants. Often a value of A that is significantly largerthan 1 is used, producing an amplification of the PCM signal. Amplification is oftenneeded because PCM signals are significantly reduced in amplitude after transmissionover long distances.

In Figures 3c and 3d, the individual pulses are shown as having restricted domains:[, 2 ] for w1 and [3 + , 5 ] for w4. The domain of each pulse f(t)wk(t)is restricted to facilitate processing by the frequency detector. Since f(t)wk(t) = 0outside of the interval [k1+ , k+1 ], there is no need for the detector to processthis pulse outside of this interval.

In Figure 4c, we show the result of processing each isolated pulse by a frequencydetector. The PCM signal is shown at the bottom of the figure. The top of the figure


shows the magnitude of the response of a frequency detectora frequency detectorthat sweeps through a set of tuning frequencies {2, 4, 6, . . . , 1024}, just like a dial on aradio. This set includes the frequencies 0 = 232 and 1 = 280 used for transmittingbits of value 0 and 1, respectively. (The reason for displaying responses for a largerange of frequencies will be clearer after we discuss simultaneous transmission.) Asthe reader can see, the bit sequence 1 0 0 0 1 1 0 is easily read off from the display ofthe detector response.

Simultaneous Messages and DenoisingThe beauty of PCM signal transmission consists in its wealth of capabilities. In

particular, it provides a simple way of transmitting several messages simultaneouslyusing just one PCM signal. Furthermore, it allows for accurate recovering of signalseven after the addition of large amounts of noise. Sometimes the noise is so severe thatit will not be apparent from the graph of the signal that any transmission has occurredat all; nothing but noise will be visible. Yet, upon decoding, the transmitted bits willstand out clearly from the noise. We now provide examples illustrating these ideas.

Figure 5. Decoding three simultaneous messages.

Example: We show how the three bit sequences 1 0 0 1 1 0 0, 1 1 0 0 1 1 1, and0 1 0 1 0 0 1 can be transmitted simultaneously. The method consists of addingtogether three PCM signals using three different values of 0 and 1: We use0 = 768 32, 1 = 768+32; 0 = 512 32, 1 = 512+32; and 0 = 25632, 1 = 256 + 32, respectively. The resulting signal is shown at the bottom,and its decoding at the top, of Figure 5. The three separate messages have beendecoded from a single PCM signal. With present technology, several thousandseparate messages can be transmitted with just one PCM signal [Catala 1997].


Example: We simulate the recovery of a bit sequence from a PCM signal thathas been severely corrupted by additive noise. In Figure 6a, we show an exam-ple of a PCM signal for transmitting the bit sequence 1 1 0 0 1 0 1. The fre-quencies 0 = 480 and 1 = 544 were used for transmitting the bits. We addedto this signal a simulation of Gaussian (normally distributed) random noise withstandard deviation 1. Gaussian noise is frequently used for modeling real-worldnoise because the Central Limit Theorem implies that sums of noise from manyindependent sources, over the course of transmission, yield a total noise that isapproximately Gaussian. A discrete realization of Gaussian noise with a stan-dard deviation 1 is shown in Figure 6b, and a histogram of its values is shownin Figure 6c. This histogram illustrates the Gaussian distribution underlyingthe random noise. The addition of the noise to the PCM signal is shown in Fig-ure 6d. Notice that it is impossible to tell from this noisy signal that any messagehas been transmitted at all.

Figure 6. Simulation of additive noise corrupting a PCM signal.

The decoding of the noisy PCM signal is shown in Figure 7. In Figure 7a,we can see the responses to the bits in the PCM signal as prominent blacksmudges, which stand out from a light grey background resulting from the noise.These smudges are fairly easy for our eyes to decode as the bits 1 1 0 0 1 0 1encoded by the original PCM signal. For electronic recognition, however, it isbetter if frequency responses having magnitude below some threshold are set tozero. In Figure 7b, we show such a thresholded response. The responses to thebits are easily detectable electronically.


Figure 7. a. Decoding noisy message. b. Thresholded decoding.

Compressing SignalsWe have seen how to transmit a signal if we can represent it as a sequence of

bits. In this section, we consider three different ways to obtain such a sequence, usingpower series, Fourier series, and wavelet series. We shall see why wavelet series oftenprovide the best method.

Power SeriesIn calculus, we see how a function can be represented by a power series. The

Taylor series of a function f(t) at t = a is given by

n=0

f (n)(a)n!

(t a)n.

If limM |f(t)M

n=0fn(a)

n! (t a)n| = 0 for |t a| < R, we say that f is equalto the sum of its Taylor series on the interval described by |t a| < R. For example,the Taylor series for f(t) = sin t + 2 cos 2t centered about the origin is

sin t + 2 cos 2t =

n=0

(cnt

2n+1 + dnt2n),

where

cn =(1)n

(2n + 1)!and dn =

(1)n 22n+1(2n)!


for all real t. Thus, to transmit the function f(t), one need know only the coefficientscn and dn of this power series. These coefficients can be written as 16-bit binary codes.These codes can then be strung together to give bit sequences.

Figure 8. f(t) = sin t + 2 cos 2t.

Of course, in practice only a finite number of coefficients can be used; and whenwe truncate the series by using only the first N terms, we introduce error in our repre-sentation. For each t-value, this error would be

n=N

f (n)(a)n!

(t a)n .

Figure 8 shows the graph of the function f(t) = sin t+2 cos 2t. The Taylor series forthis function at t = 0 converges for all values of t; but when we only use a finite num-ber of terms in the series, we obtain errors. The series works best near t = 0, the pointabout which the series is being expanded. As we add more terms, the reconstructedseries resembles the function for a larger domain; but since we are adding terms ofthe form tn, at some point the reconstructed function will increase or decrease withoutbound. These ideas are illustrated in Figure 9.

Figure 9. Taylor series at t = 0 for sin(t) + 2 cos(2t). Left: 10 terms. Right: 50 terms. The originalfunction is shown in a lighter line.

For a Taylor series about y = a, we can represent a function in terms of powers oft a. Still, even when the power series converges everywhere, a finite sum may notrepresent the function well for values a distance away from t = a (Figure 9).


Another problem with Taylor series in many applications is that computing thederivatives needed for the coefficients is not always practical or possible. What weneed in practical applications is a series with more easily computed coefficients andwhich requires fewer terms to give a good reconstruction of our original signal. Weshall see how Fourier series improve our work in both of these respects, and how tocompress data still further using wavelets.

Fourier SeriesAnother way to represent many functions was introduced by Jean B.J. Fourier in

the early nineteenth century. A Fourier series represents a periodic function in termsof sine and cosine functions. For a 2W -periodic function f(t), its Fourier series is theseries in the following correspondence:

f(t) 12a0 +

n=1

[an cos

(nt

W

)+ bn sin

(nt

W

)],

where the Fourier coefficients are defined to be

an =1W

WW

f(t) cos(nt

W

)dt, for n = 0, 1, 2, ...;

bn =1W

WW

f(t) sin(nt

W

)dt, for n = 1, 2, ....

Note that the 2-periodic signal, sin t + 2 cos 2t, from the previous section can becaptured exactly with only two nonzero Fourier coefficients!

A Fourier series can also be written in the complex-number form

f(t)

n=cne

int/W ,

where the Fourier coefficients are

cn =1

2W

WW

f(t)eint/W dt.

One can view a Fourier series as a sum of terms having single frequencies, the am-plitudes of those terms are the Fourier coefficients for the function. Fourier series areexpansions of functions using the basis functions{

1, sin(

nt

W

), cos

(nt

W

)}n=1

or{ei

ntW

}nZ

,

which are orthogonal bases of periodic functions on [W,W ].


Most signals are not periodic, however, and so we need to broaden the notion ofFourier series. For piecewise smooth functions in L1(R) = {f : |f(t)| dt < },this can be done via the Fourier transform

f() =

f(t)ei2t dt (5)

and the Fourier inversion formula

f(t) =

f()ei2t d.

The discrete form of Fourier series for analog signals is a consequence of the Sam-pling Theorem.

Theorem 1 (Sampling Theorem) If a continuous function f(t) L1(R) is band-limited, that is, f() is nonzero only on some interval [W,W ], then f(t) is completelydetermined by a sequence of points spaced 1/2W units apart. Specifically,

f(t) =

n=f( n2W

)sinc(2Wt n),

where sinc(t) = sin(t)t

.

Discussions of Theorem 1 can be found in Higgins [1985], Kelly [2000], and Walker[1988].

Theorem 1 allows us to use sample values of our function, {f( n2W )}, to createa bit sequence. However, we are also interested in compressing our data by creatingas short a bit sequence as possible while still retaining the ability to generate a goodapproximation to the original function. To explore this issue further, we investigate thediscrete Fourier transform.

In practice, our function f(t) will exist only for a finite duration of time. Therefore,we can scale the function so that f(t) = 0 for t [0, 1]. We can then apply Theorem 1to f() in place of f(t). Hence, f() can be recovered from the set {f(k)}kZ.

Using the definition of the Fourier transform in (5) and the fact that f(t) is zeroaway from [0, 1], we obtain

f(k) = 10

f(t)ei2ktdt

= limN

N1n=0

[f( nN

)ei2k

nN

] 1N

.

This last sum comes from the definition of a Riemann integral. Thus, for N large,

f(k) Fk, (6)


where

Fk =1N

N1n=0

f( nN

)ei2nk/N for all k Z

is the N -point discrete Fourier transform (DFT) of the sequence {fn = f( nN )}N1n=0 .Formula (6) shows how the DFT values are determined by our functions sam-

ple values. Now let us investigate how to recover our sample values, and hence ourfunction, from a finite number of DFT values. We begin with the finite sum

N1k=0

Fk ei2nk/N =

N1k=0

1

N

N1j=0

f

(j

N

)ei2jk/N

ei2nk/N

=1N

N1j=0

[N1k=0

(ei2/N )(jn)k]f

(j

N

),

for n = 0, 1, 2, ..., N 1. Note that (ei2/N )(jn) is a nontrivial root ofxN 1 = (x 1)(xN1 + xN2 + ... + x + 1)

for jn = 0, and thus N1k=0 [(ei2/N )(jn)]k = 0 for j = n. From this, we obtainN1k=0

Fk ei2nk/N = f

( nN

).

Thus to obtain all needed information about our function, we need to store either theN samples {fn = f( nN )}N1n=0 or the N DFT values {Fk}N1k=0 .

Because of the desire to compress our data, we hope to use only a fraction of theDFT valuesthose with largest magnitudesand still get a good reconstructed func-tion. Unfortunately, the magnitudes of the DFT values often damp down very slowly;hence, we often need most, if not all, of these coefficients to obtain a good recon-struction of our original signal. We provide examples in the next subsection, when wecompare this DFT-based method of compression with wavelet-based compression.

WaveletsIn an age when we seek faster and sharper scanners for medical diagnosis, more-

efficient computer storage techniques for larger amounts of computer data, and higher-fidelity audio and visual signals with less information used, a more efficient way toconvert analog signals to digital ones is needed. Many people working in harmonicanalysis helped to provide the groundwork for the wavelet bases, which were intro-duced in the 1980s. Wavelets are beneficial in the areas just mentioned and in manyother applications because of the highly efficient ways a wavelet series can represent afunction. As we illustrate below, they often allow for a better-reproduced analog signalusing less data than with Fourier techniques.


Let us begin by defining a wavelet basis. We will work with wavelet bases for theset of functions L2(R) = {f : |f(t)|2 dt < }. A wavelet basis of L2(R) isa basis of functions generated using dilations by 2m and translations by n2m of asingle function. Here m and n are both integers. That is, if (t) is such a generatingfunction, then {

m,n(t) = 2m/2(2mt n)}

m,nZ

is a wavelet basis of L2(R). By a basis, we mean that for each f(t) L2(R), thefollowing holds:

f(t) =mZ

nZ

mn m,n(t)

where

mn =

f(t)m,n(t) dt

are the wavelet coefficients. We shall refer to as the generating wavelet for thiswavelet basis.

A related way to represent our function f(t) is with scaling functions generatedby a single function . For any integer M , the set {m,n}m


Substituting 2m1t into these identities and multiplying by 2m12 yields

2m12

(2m1t

)=

12

[2

m2 (2mt) + 2

m2 (2mt 1) ]

2m12

(2m1t

)=

12

[2

m2 (2mt) 2m2 (2mt 1) ] .

From these last two identities, we obtain the following relations between Haar scalingcoefficients and wavelet coefficients:

m1k =12m2k +

12m2k+1,

m1k =12m2k

12m2k+1. (8)

These identities show that the (m 1)st level coefficients {m1k }, {m1k } are ob-tained from the mth level scaling coefficients {mk }.

If we assume that N is a power of 2, say N = 2R, then the fact that is supportedon [0, 1) implies that when M = R the support of each function M,n(t) will onlycover one sample point n/N . Hence, in view of (7) we are justified in assuming thatRn = 2

R/2fn. Thus, the scaling coefficients at the Rth level can be set equal to aconstant times fn. In practice, the constant factor 2R/2 is dropped and we assumethat Rn = fn for each n. The reason for dropping the constant factor 2R/2 is thatthen (8), for m = R, defines an orthogonal matrix transformation on each pair ofadjacent sample values f2k and f2k+1.

We have now shown that, given the scaling coefficients {Rn = fn} at the Rthlevel, we can use (8) repeatedly to derive all scaling coefficients {mn } and waveletcoefficients {mn } for all levels m < R. Given the initial data(

R0 , R1 , . . . ,

RN1

)= (f0, f1, . . . , fN1),

we apply (8) to get (R10 , . . . ,

R1N/21,

R10 , . . . ,

R1N/21

).

For example, if (f0, f1, . . . , f7) = (2, 2, 2, 2, 4, 4, 4, 12), then R = 3, and using (8)yields(

20, 21,

22,

23,

20 ,

21 ,

22 ,

23

)=(2

2, 2

2, 4

2, 8

2, 0, 0, 0,4

2).

We may then repeat application of (8) on the scaling coefficients(R10 , . . . ,

R1N/21

)from these last data to get(

R20 , . . . , R2N/41,

R20 , . . . ,

R2N/41,

R10 , . . . ,

R1N/21

).

In the example above, this second application of (8) yields(10,

11,

10 ,

12 ,

20 ,

21 ,

22 ,

23

)=(4, 12, 0,4, 0, 0, 0,4

2).


Continuing in this way, we obtain at the Rth step the data(00,

00 ,

10 ,

11 , . . . ,

R1N/21

).

For the example, we can perform one more application of (8), thereby obtaining(00,

00 ,

10 ,

12 ,

20 ,

21 ,

22 ,

23

)=(8

2,4

2, 0,4, 0, 0, 0,4

2).

Since (8) involves only six arithmetic operations on two adjacent coefficients m2kand m2k+1, it follows that there is a very fast computer algorithm, a fast wavelet trans-form WH , that performs the transformation:

(f0, f1, . . . , fN1)WH

(00,

00 ,

10 ,

11 , . . . ,

R1N/21

). (9)

Note also that (8) is invertible:

m2k =12m1k +

12m1k ,

m2k+1 =12m1k

12m1k .

Hence, the fast wavelet transform in (9) is invertible, by another very fast computeralgorithm, which we denote by W1H :(

00, 00 ,

10 ,

11 , . . . ,

R1N/21

)W1H (f0, f1, . . . , fN1).

Since (t) dt = 0 and is supported on the interval [0, 1), it follows that

when f(t) is constant on the interval [2mn, 2m(n+ 1)), then mn will be zero. Thetransformation

(2, 2, 2, 2, 4, 4, 4, 12) WH(8

2,4

2, 0,4, 0, 0, 0,4

2)

illustrates this phenomenon in an elementary way. A more complex illustration isshown in Figure 10, where in a we show 1,024 samples of the signal

f(t) = 5(sin 8t)(cos 4t)/5, 0 t < 1,

where denotes the greatest integer function. The signal f(t) is a step function,constant over several subintervals of [0, 1). The fast Haar transform of the sample data(fn) will produce a large number of zeros. Consequently, a good approximation of theoriginal signal can be obtained with just a fraction of the nonzero Haar coefficients.Figure 10b shows the graph of W1H applied to the 195 highest magnitude Haar co-efficients (all other coefficients set equal to zero). This reconstructed signal is a goodapproximation of the original.


One way to quantify the accuracy of the approximation is to use relative R.M.S. er-ror. Given a discrete signal {fj}N1j=0 and its reconstruction {gj}N1j=0 , the relativeR.M.S. error is defined by

D(f, g) =N1j=0 |fj gj |2N1

j=0 |fj |2.

For convenience, we refer to D(f, g) as rel. error. As a standard of good approxima-tion, we require that a reconstructed signal approximate the original with a rel. error ofat most 102. The reconstructed Haar signal in Figure 10b approximates the originalsignal with the required rel. error of 102. By comparison, we show a discrete Fourierapproximation in Figure 10c, using the top 195 highest magnitude DFT values. Therel. error is then 2.6102. This increase in rel. error is reflected in the ragged appear-ance of the Fourier approximation. In order to obtain a rel. error of 102, the Fourierapproximation required 448 coefficients. For this step function signal, the Haar seriesclearly outperforms the Fourier method in all respects.

Figure 10. a. Step function data, 1024 points. b. Haar series, 195 highest magnitude coefficients: rel. error102. c. Fourier approximation, 195 highest mag. coefficients: rel. error 2.6 102. d. Coif18 series,195 highest mag. coefficients: rel. error 3.2 102.

The simple nature of the functions in the Haar system makes it easy to study thestructure of Haar wavelets. Their lack of smoothness, however, makes them a poorbasis for many applications. While they performed well in the step function examplejust considered, this same example illustrates that Haar series produce step functionapproximations for all types of signals. If our samples are taken from a smoothlyvarying function, then the step function approximations provided by Haar series will


be unsuitable. In our work, we shall instead use Coiflets, which are wavelet bases ofI. Daubechies [1992] and R. Coifman (see Burrus et al. [1998]).

The Coiflets are an infinite class of wavelets for L2(R). Similar to the Haarwavelets, these wavelets have compact support: for each such system the generatingscaling function (t) and generating wavelet (t) are zero outside of certain intervalsof finite length. Also, just as with the Haar system, we have

(t) dt = 1 and

(t) dt = 0.

These last two integral conditions are true for all wavelets on L2(R). The addedbenefits of Coiflets are their smoothness and the vanishing moment conditions:

tk(t)dt = 0 and

tk(t)dt = 0

for some positive integers k.

Figure 11. a. Coif18 generating scaling signal. b. Coif18 generating wavelet.

For the remainder of our work, we shall use the Coiflets known as the Coif18system. For the Coif18 system, both the generating scaling function (t) and the gen-erating wavelet (t) are supported on the interval [6, 11]. Figure 11 shows graphsof and . These Coif18 generating functions are twice continuously differentiable.That implies that any partial sum for a Coif18 wavelet series will be twice continu-ously differentiable as well. Such smoothness for Coif18 partial sums will providebetter approximations for smoothly varying signals than the step function partial sumsprovided by Haar series. Besides smoothness, the Coif18 generating wavelet (t) hassix vanishing moments,

tk(t) dt = 0, 0 k 5,

and the Coif18 generating scaling function (t) also has six vanishing moments,

tk(t) dt = 0, 1 k 6.


The benefits of vanishing moments can be seen when computing the wavelet coef-ficients. For instance, since

(t) dt = 0, if f(t) is constant on the interval

[2m(n 6), 2m(n + 11)), then mn will be zero. Similarly, all of the vanishingmoments for imply that if f(t) is a polynomial of degree less than 6 on this sameinterval, then mn will also be zero. Put another way, if f can be closely approximatedby a polynomial of degree less than 6 on an interval containing the support of m,n,then mn will be approximately zero. We shall see that these facts are useful for com-pressing signals. The vanishing of the moments of the generating scaling function isimportant for making the approximation Rn 2R/2fn which we discussed above.This approximation is much better justified for Coif18 scaling functions than for Haarscaling functions (see Resnikoff and Wells [1998] for more details), and it justifiesreplacing (Rn ) by the signal samples (fn).

With this background for the Coif18 system, we can now describe how Coifletsallow us to compress signals. As with the Haar wavelet system, the Coif18 systemallows for the computation of {m1n } and {m1n } by applying an orthogonal trans-formation to a small number (18, to be precise) of adjacent values of {mn }. (The smallnumber of adjacent values needed is another advantage of using Coif18 wavelets.) It-erating this process R times yields a fast wavelet transform WC , which converts theinitial data

(R0 , . . . ,

RN1

)= (f0, . . . , fN1) as follows:

(f0, f1, . . . , fN1)WC (00, 00 , 10 , 11 , . . . , R1N/21).

This transformation WC has a fast inverse W1C , as well (see Walker [1997; 1999]for more details). Since the Coif18 wavelet has six vanishing moments, many of thewavelet coefficients mn are often zero or very small. This will give us our desiredcompression. We retain only coefficients that have significant size; all other coeffi-cients are set to zero and are not transmitted. These high-magnitude coefficients willgive us our binary codes for transmission. Binary encoding only a small set of high-magnitude coefficients will often produce a set of data much smaller in size than the Noriginal signal values. At the receiving end, we use only these high-magnitude coeffi-cients, along with zeros inserted for the coefficients we have dropped, and apply W1Cto reconstruct a wavelet series approximation to our signal.

We can illustrate such compression with the alternating frequency sine function

4k=1

{[2kc,(2k+1)c](t) sin(24t) + [(2k+1)c,(2k+2)c](t) sin(48t)},

with c = 0.2, over the interval [0, 2) using 16,384 sample points (Figure 12a). Forconvenience, the function is once again discontinuous; but there also exist continuousfunctions with sharp transitions that would pass through these same sampled data. Toobtain a rel. error of 102, the Coif18 wavelet requires only 301 coefficients (Fig-ure 12b), while the Fourier approximation requires 2,674 (Figure 13b). The errorspointed out in the Fourier case (Figure 13b) are not seen in the wavelet reconstruc-tion (Figure 12b). Using the same number of terms (310) for the Fourier approxima-tion as for the wavelet series produces a much poorer reconstruction: Its rel. error is


Figure 12. a. Original signal. b. Coif18 wavelet series with 310 terms.

Figure 13. a. Fourier approximation with 301 terms for signal in Figure 12a. b. Fourier approximationwith 2,674 terms. The arrows indicate some prominent defects in the Fourier approximation.


8.5102, which is 8.5 times as large as the rel. error for the 301-term Coif18 waveletseries. Considering that the total number of signal values is 16,384, the 301 nonzerocoefficients for the Coif18 reconstruction represents a significant compression of thesignal.

The two compression examples in this section illustrate some of the variety of sig-nal data that can be compressed using wavelets. For step-function data, which arelocally constant, the Haar wavelets performed wellbetter than a Fourier approxima-tion, and also better than Coif18 wavelets (compare Figures 10b and 10d). This is dueto the step nature of the Haar system. For more smoothly varying data, as in samplesfrom speech and music, the Coif18 wavelets give excellent compression, clearly supe-rior to Fourier approximations. This was illustrated with a discretely sampled signalwith sharp transitions.

We have seen how wavelets can often allow us to compress our data quite well.Let us carefully reexamine the properties of wavelets that allowed us to do this. By thedefinition of a wavelet, the basis functions m,n(t) = 2m/2(2mt n) can zoom inon particular areas of a signal. As m increases, the supports of these basis functionsdecrease in size, and varying n allows us to move along to particular areas to examinethe signal. This allows the wavelet coefficients mn =

f(t)m,n(t) dt to pick

up information about local behaviors of the function f(t). The vanishing momentconditions help us to compress the data from smoothly varying signals because manyof the wavelet coefficients mn are either zero or very small. For different types ofsignals, different wavelet systems can be employed. In particular, for step functions,the Haar wavelet system performs very well. While for smoother functions, a smootherwavelet system like Coif18 works much better. The Coif18 system combines a largenumber of vanishing moments for both wavelets and scaling functions (hence goodcompression) with a relatively small support (thus allowing for rapid computation).The Coif18 system is just one of many wavelet systems that can be employed forcompression. Some compression schemes make use of multiple systems, choosingthe best system for a given signal (and transmitting a small amount of overhead tothe receiver to identify which system was used). Besides wavelet series, localizedFourier seriesFourier expansions restricted to smooth time-windowed portions ofthe signalhave been found to be especially effective for compressing audio signals[Malvar 1992]. More details on data compression can be found in Mallat [1998],Vetterli and Kovacevic [1995], and Walker [1999].

ConclusionWe have seen how a PCM signal can carry the bits of a digital signal by carrying

the 1s of a binary code with a different frequency than the 0s. By using more frequen-cies, this PCM signal can carry large numbers of digital signals simultaneously. PCMsignals are also highly resistant to corruption by additive noise.

We next examined three ways to represent a function with such digital codes. Thecodes came from the coefficients for a discrete Fourier transform, a discrete Haarwavelet series, and a discrete Coif18 wavelet series. We saw how wavelets allowed


us to compress signals better so that we needed less data to represent a function.Several of the references would be appropriate reading for undergraduate students.

Students may find Benson [1993] interesting if they wish to read how several radiosignals can be transmitted simultaneously. Walker [1988] and Kelly [2000] give thereader more of the theory and applications of Fourier series. Finally, the reader maywish to learn more about wavelets and their uses by consulting Walker [2000; 2002],Frazier [1999], Walker [1999], and Walker [1997].

ReferencesBoylstad, R.L. 1994. Introductory Circuit Analysis. 9th ed. Upper Saddle River, NJ:

Prentice-Hall,

Benson, C. 1993. How to tune a radio. In Applications of Calculus, vol. 3: Resourcesfor Calculus, edited by Philip D. Straffin, 126134. Washington, DC: Mathemati-cal Association of America.

Burrus, C.S., R.H. Gopinath, and H. Guo. 1998. Introduction to Wavelets and WaveletTransforms, A Primer. Englewood Cliffs, NJ: Prentice-Hall.

Catala, P. (1997). Pulse code modulation. Chap. 23 of The Communications Hand-book. Boca Raton, FL: CRC Press.

Daubechies. I. 1988. Orthonormal bases of compactly supported wavelets. Communi-cations on Pure and Applied Mathematics 51: 909996.

. 1992. Ten Lectures on Wavelets. Philadelphia, PA: SIAM.

Floyd, T.L. 2001. Electronics Fundamentals. 5th ed. Upper Saddle River, NJ: Prentice-Hall.

Frazier, M. 1999. An Introduction to Wavelets through Linear Algebra. New York:Springer-Verlag.

Higgins, J.R. 1985. Five short stories about the cardinal series. Bulletin of the Ameri-can Mathematical Society, New Series 12: 4589.

Kelly, S. 2000. Using the Shannon sampling theorem to design compact discs. TheUMAP Journal 21(2): 157166.

Mallat, S. 1988. Multiresolution approximations and wavelet orthonormal bases ofL2(R). Transactions of the American Mathematical Society 315(1): 6987.

. 1998. A Wavelet Tour of Signal Processing. New York: Academic Press.Malvar, H.S. 1992. Signal Processing with Lapped Transforms. Norwood, MA: Artech

House.

Resnikoff, H.L., and R.O. Wells. 1998. Wavelet Analysis. The Scalable Structure ofInformation. New York: Springer-Verlag.


Vetterli, M., and J. Kovacevic. 1995. Wavelets and Subband Coding. EnglewoodCliffs, NJ: Prentice-Hall.

Walker, J. 1988. Fourier Analysis. New York: Oxford University Press.

. 1996. Fast Fourier Transforms. Boca Raton, FL: CRC Press.

. 1997. Fourier analysis and wavelet analysis. Notices of the AmericanMathematical Society, 44(6): 658670.

. 1999. A Primer on Wavelets and their Scientific Applications. Boca Raton,FL: CRC Press.

. 2000, 2002. FAWAV: A Fourier/Wavelet Analyzer. Freeware for Windows.http://www.uwec.edu/walkerjs/ .

About the AuthorsSusan Kelly received her Ph.D. in mathematics

in 1992 from Washington University in St. Louis,with a thesis in wavelet theory under Richard Rochbergand Mitchell Taibleson. She is a professor at theUniversity of WisconsinLa Crosse. Her hobbiesinclude watercolor and oil painting as well as skiing.She enjoys spending time with her husband and herson Kyle.

James S. Walker received his B.S. from the State Uni-versity of New York at Buffalo (1975), his M.S. from theUniversity of Illinois at UrbanaChampaign (1977), and hisD.A. from the University of IllinoisChicago (1982). He isa professor of mathematics at the University of WisconsinEau Claire. He has published papers on Fourier analysis,complex variables, and wavelet theory. He is the author ofthree books on Fourier analysis, FFTs, and wavelet analysis.

UMAPModules inUndergraduateMathematicsand ItsApplications

Published incooperation with

The Society forIndustrial and Applied Mathematics,

The MathematicalAssociation of America,

The National Council of Teachers ofMathematics,

The AmericanMathematicalAssociation of Two-Year Colleges,

The Institute forOperations Research andthe ManagementSciences, and

The American Statistical Association.

Application Field: Biology, Forestry

COMAP, Inc., Suite 210, 57 Bedford Street, Lexington, MA 02420 (781) 8627878

Module 792The Spread ofForest FiresEmily E. Puckette


INTERMODULAR DESCRIPTION SHEET: UMAP Unit 792

TITLE: The Spread of Forest Fires

AUTHOR: Emily E. PucketteDepartment of Mathematics and Computer ScienceSewanee: The University of the South735 University Ave.Sewanee, TN [email protected]

MATHEMATICAL FIELD: Probability, Calculus

APPLICATION FIELD: Biology, Forestry

TARGET AUDIENCE: Students with a background in innite series, e.g., fromsecond-semester calculus.

ABSTRACT: We create a simple discrete probabilistic model forspread of a forest re. We examine the conditions forwhich the re will either die out or spread indenitely,identifying and bounding a critical value for the prob-ability of transmission of the re to an immediately ad-jacent location.

PREREQUISITES: Summation of geometric series, elementary probabil-ity, intuitive limits; for the Appendix, derivatives andlimits of rational functions.

The UMAP Journal 24 (4) (2003) 417434.cCopyright 2003 by COMAP, Inc. All rights reserved.

Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for prot or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAP must be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.

COMAP, Inc., Suite 210, 57 Bedford Street, Lexington, MA 02420(800) 77-COMAP = (800) 772-6627, or (781) 862-7878; http://www.comap.com

The Spread of Forest Fires 419

The Spread of Forest Fires

Emily E. PucketteDept. of Mathematics and Computer ScienceSewanee: The University of the South735 University Ave.Sewanee, TN [email protected]

Table of Contents1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. FOREST FIRE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3. SPREAD OF THE FOREST FIRE . . . . . . . . . . . . . . . . . . . . . . 3

4. LOWER BOUND FOR pc . . . . . . . . . . . . . . . . . . . . . . . . . 4

5. UPPER BOUND FOR pc . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7. AREAS FOR FURTHER EXPLORATION . . . . . . . . . . . . . . . . . . 9

8. SOLUTIONS TO SELECTED EXERCISES . . . . . . . . . . . . . . . . . . 11

9. APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

ABOUT THE AUTHOR . . . . . . . . . . . . . . . . . . . . . . . . . . 13


MODULES AND MONOGRAPHS IN UNDERGRADUATEMATHEMATICS AND ITS APPLICATIONS (UMAP) PROJECT

The goal of UMAP is to develop, through a community of users and devel-opers, a system of instructional modules in undergraduate mathematics andits applications, to be used to supplement existing courses and from whichcomplete courses may eventually be built.

The Project was guided by a National Advisory Board of mathematicians,scientists, and educators. UMAP was funded by a grant from the NationalScience Foundation and now is supported by the Consortium for Mathemat-ics and Its Applications (COMAP), Inc., a nonprot corporation engaged inresearch and development in mathematics education.

Paul J. CampbellSolomon Garfunkel

EditorExecutive Director, COMAP


1. Introduction What does research in mathematics look like? What do mathematicians do, exactly? Is there any thing new going on in mathematics?

Such questions are exciting, because students asking them are curious aboutmathematics, and frustrating, because the students often do not have enoughbackground in mathematics to tackle a research problem. The challenge to theinstructor is to respond with an easily understood research question whosesolution is within the grasp of a freshman or sophomore. An added bonuswould be to convey the creative mix of ideas that is often crucial in a solution.

The project that follows is one possible response. It begins with a veryapproachable model for the spread of a forest re. The question of interest iswhether the re will die out on its own, and this question leads to a hunt for athreshold value for the probability of further transmission of the re. With useof the formula for the sum of a geometric series plus a minimal amount of prob-ability, a partial answer is attainable: some bounds on the critical probabilityof spread of the re.

The main ideas in this Module have been used

in senior comprehensive projects, as a basis for talks given to undergraduates with a second-semester calculus

background, and

as an independent study for a motivated sophomore.As organized here, the Module is written as an independent project for a studentor a group of students. The time frame for the project is 24 weeks.

2. Forest Fire ModelThe model is constructed on the Z2 lattice, where each lattice point X =

(x, y), x, y Z, is the site of a tree. The tree at the origin is struck by lightningand set ablaze. From any tree, the re can spread in one step to any one of itsfour nearest neighbors, i.e., to any of the trees at (1, 0), (0, 1), (1, 0) or (0,1),independent of one another, each with transmission probability p, 0 p 1.

We keep track of the spread of the re using a discrete time clock. Thus,at time n = 0, the tree at the origin is on re and all other trees in the forestare unburnt. At time n = 1, the re may have spread to one or more nearestneighbors of the tree at the origin. Also at time n = 1, the tree at the originis completely burnt and therefore cannot be set on re again nor transmit refurther.

1

422 UMAP/ILAP Modules 200304

We formalize the re spread by assigning to each lattice site X a state at timen, n = 0, 1, 2, . . . The three possible states for a tree are U (unburnt), F (on re),and B (burnt). The function Sn(X) gives the state of the tree at site X at time n.

Exercise

1. The spread of the re is determined by the following rules. Translate theminto words; for example, Sn(X) = B translates to the tree at site X at timen is burnt.a) If Sn(X) = B, then Sn+1(X) = B.b) If Sn(X) = F , then Sn+1(X) = B.c) If Sn(X) = U , then Sn+1(X) can be either U or F , depending on the

state of the neighboring trees and the transmission of the re.

Case c) requires some exploration. A burning tree spreads the re to neigh-boring trees independently of other burning trees. If events A and B are inde-pendent, then the probability that both events A and B occur, P (AB), is equalto P (A) P (B), the product of the probabilities of each separate event. In otherwords, the proportion of the time that event A occurs is the same when eventB occurs as when B does not occur.

With this precise denition of independence, consider the unburnt tree atsite X with neighboring trees at X1, X2, X3, and X4, as in Figure 1.

XX3 X1

X2

X4

Figure 1. An unburnt tree at X with neighboring trees.

Exercises

2. If Sn(X1) = F , Sn(X2) = F , Sn(X3) = B, and Sn(X4) = U , the tree at Xwill remain unburnt at time n + 1 only if the re at sites X1 and X2 do notspread to X . The probability of not spreading the re from a burning treeto a neighbor is 1 p.

Let A be the event that site X1s re does not spread to site X , andlet B be the event that site X2s re does not spread to site X . Using the

2


independence of the re spread from site to site, the probability that site Xremains in state U at time n + 1 is

P (A B) = P (A) P (B) = (1 p)2.

What is the probability that the site X switches to state F at time n+ 1?That is, what is the probability that the tree at X is on re at time n + 1?

3. Consider this assignment of states to sites: Sn(X1) = F , Sn(X2) = B,Sn(X3) = F , and Sn(X4) = F .a) If Sn(X) = U , what is the probability that Sn+1(X) = U?b) Again, if Sn(X) = U , what is the probability that Sn+1(X) = F ?

4. Consider the case where at time n exactly i neighboring trees are on re,i = 0, 1, . . . , 4.a) If Sn(X) = U , what is the probability that Sn+1(X) = U?b) Again, if Sn(X) = U , what is the probability that Sn+1(X) = F ?

3. Spread of the Forest FireThe spread of the re depends on the value of the transmission probability

p. If p is small, the re will die out eventually. If p is large, then there is somechance that the re will spread innitely, that is, arbitrarily far from the initiallightning strike.

The Big Question: How large does p have to be for there to be a chance forindenite spread of the re?

Essentially, we are hunting for a threshold value or critical value pc. For p [0, pc), the forest re will die out. For p (pc, 1], the re has a chance of innitespread. However, unless p = 1, it is not certain that there will be innite spreadof the re.

The model plays out differently depending on the value of p; the valuepc marks a phase transition in the model. Phase transitions abound in naturalphenomena, too, as in the change of water from a liquid state to a solid state,which (for pure water at sea level) occurs at a critical temperature tc = 0 C.

Exercise

5. Computer simulation can provide some insight into the critical value pc.Write a computer program that starts with one tree burning at the center ofa 100 100 lattice and shows the states of the trees at time n using the rulesfor spread established in the previous section. Explore the re spread withdifferent values of the transmission probability p; to begin, test p = 0.2,p = 0.4, p = 0.6, and p = 0.8. Look at the states of the entire lattice at

3


various times, for example, at n = 20, n = 40, and n = 60, to see how there is spreading over time. For each value of p, run the program severaltimes to observe any trends in how the re spreads. You may also want toincrease the size of the lattice to explore further. What do you notice aboutthe spread of the re? For which values of p does the re appear to die out?For which values does the re seem to continue spreading?

For convenience, let E be the event that the re spreads innitely. Hencefor p < pc, the probability of E is 0, while if p > pc, the probability of E isgreater than 0. The critical probability is therefore dened to be

pc = inf{p | P (E) > 0}where inf denotes the greatest lower bound of the set.

4. Lower Bound for pcTo study innite spread of the re, we start by considering how the re

spreads from the origin to another site at a nite distance away. As the respreads across the lattice, it creates burnt paths through the forest. We considerone such path.

Exercises

6. a) Plot all of the Z2 lattice points (x, y) where4 x, y 4. At time n = 0,the tree at the origin is on re. Suppose that the re spreads to one of theneighboring trees. Pick one neighboring tree, at site X , say. How manyoptions are there for choice of X? On your plot, draw the rst step ofthe res path from the origin to X .

b) For the next step in the spread of the re, pick a tree neighboring X ; callthe site Y . How many options are there for choice of Y ? On your plot,draw the second step of the res spread.

c) For the third step, pick a tree neighboring Y . How many options arethere? On your plot, draw the next step of the res spread.

Because the re cannot revisit a burnt tree, it maps out a self-avoiding path,meaning that the path cannot return to a site it has already visited.

Exercise

7. a) How many self-avoiding paths with a length of 2 steps are there on thislattice? How many with three steps?

b) Are there 4 3 3 3 self-avoiding paths with a length of 4 steps? Sketchseveral paths and think about the question.

4


Depending on where the re is at time n = 3, it might not have three optionsof sites to move to at time n = 4. For example, suppose that the rst step fromthe origin is east, the second step is north, and the third step is west. Then thefourth step must be either north or west.

Counting the number of self-avoiding paths with n steps is a difcult prob-lem, but we can at least establish an upper bound on the number of such paths:

Number of self-avoiding paths of n steps 4 3n1.Then the probability that the re spreads along a self-avoiding path of n stepsis

(Number of n-step self-avoiding paths)(Probability of re spread n times)= (Number of self-avoiding paths of n steps) pn (4 3n1)pn.

Now consider innite spread of the re; re must spread along a self-avoiding path of innite length. So let us consider re spreading along aself-avoiding path with a large length.

Let Q be the probability of re spreading along a self-avoiding path with Nor more steps, where N is large. Then

Q = P (re spreading along a self-avoiding path of length N)

n=N

P (re spreading along a self-avoiding path of length n)

n=N

(4 3n1)pn

= 4p

n=N

(3p)n1.

Exercise

8. If p < 13 , then 0 3p < 1, and the sum in Q is the tail of a convergentgeometric series. Use the closed form of the series to simplify the upperbound of Q. Hint: The geometric series

n=0 ax

n converges to a/(1 x)for 1 < x < 1 and a R. For more assistance, see the Appendix.

For innite spread of the forest re, we are interested in the limiting proba-bility of re spreading along a self-avoiding path of length N , that is, the limitof Q as N .

Exercises

9. Assume that p < 13 . Use the simplied upper bound of Q to nd an upperbound for limNQ.

5


10. Note that

P (E) = limN

Q,

so we can conclude that if p < 13 , then the probability of innite re spreadis 0. What then can be said about the value of the critical probability pc?

5. Upper Bound for pcTo obtain an upper bound forpc, we consider the spread of a re at timen = 0

from a set I of M initial adjacent sites in a row: (0, 0), (1, 0), (2, 0), . . . , (M, 0).The re spreads along self-avoiding paths; so let J be the set of sites X suchthat there is a self-avoiding path from a site in I to X . That is, J consists of allsites that are burned by the res spread; note that I J . See Figure 2 for anillustration.

0 M

- site in J

Figure 2. Illustration of a re starting from a row of trees from 0 to M .

For each site Y in J , construct a closed unit square centered at Y . So ifY = (x, y), then the unit square has vertices at (x 12 , y 12 ). Let A be theregion of the plane covered by these squares, and let A have boundary B. SeeFigure 3 for an illustration.

If there is indeed a boundary around all the sites that are burnt, then there did not spread innitely from the initial sites. So our attention turns to theprobability of the existence of such a boundary B.

In constructing the unit squares around sites inJ , we have effectively createda dual lattice W2, where

W2 ={(u, v) : u = x + 12 , v = y +

12 for all (x, y) Z2

}.

6


0 M

boundary B

Figure 3. The boundary B of the region A corresponding to the re in Figure 2.

The boundary B connects lattice points in this dual lattice.The spread of the re in the Z2 lattice induces a conguration in the W2

lattice in the following way. In the Z2 lattice, let X and Y be neighboring sites.We call the segment of the lattice between the sites an edge. If the re has spreadfrom one site to a neighboring site, we say the edge is open; otherwise, the edgeis closed. Thus, an edge in the Z2 lattice is open with probability p, and as themodel runs its course, each edge in Z2 is classied as either open or closed.

Every edge in the Z2 lattice crosses an edge in the W2 lattice. If the edgeis open in Z2, the corresponding edge is declared to be open in the W2 lattice.In our example (Figure 3), the edge in Z2 from the origin to the point (0, 1) isopen because the re has spread from the origin to this site. Therefore, the edgein W2 from ( 12 , 12 ) to ( 12 , 12 ) is also open. Thus a conguration in W2 of openand closed edges is determined by the conguration of open and closed edgesin Z2.

The boundary B consists of closed edges in W2. Furthermore, B is a self-avoiding path until it closes the loop around the burnt area in its last step.

Exercises

11. What is the smallest possible number N of edges in B? Consider the mini-mum size of the set J , and determine the minimum number of edges in W2

that would be required to bound J .

12. Consider a boundary B of some length k, where k N ; it must be a pathformed of k closed edges. What is the probability that all k edges are closed?

7


How many boundaries of length k exist? Since a boundary is a self-avoidingpath, the question is difcult to answer precisely. However, we can establishan upper bound for the number of boundaries of length k. Since the initial setI is on the horizontal axis of Z2, a boundary of length k must contain a verticaledge between W2 lattice points of the form (j + 12 , 12 ) and (j + 12 , 12 ) for someinteger j where 0 j < k. Thus, there are at most k options for an edge of thisform; and moving in a self-avoiding fashion from that edge onward, there areat most 3k1 options for the following k 1 steps. Therefore, the number ofboundaries of length k is less than or equal to k 3k1.

Let R be the probability of the existence of a closed boundary B aroundset J . Then we have

R = P (closed boundary B exists)

=

k=N

P (closed boundary B of length k exists)

k=N

(k 3k1)(1 p)k

= (1 p)

k=N

k[3(1 p)]k1,

where N = 2M + 4 is the minimum number of edges in the boundary.

Exercise

13. If p > 23 , then 0 3(1p) < 1, and the sum is the tail of a convergent series.(The ratio test shows that the series

k=1 k[3(1 p)]k1 is convergent, and

Exercise A2 calculates what it converges to; but here we are concerned justwith the tail of the series, the sum from the N th term on.) What happens toR if N is increased? Hint: See Exercise A2 in the Appendix.

Assume that p > 23 . If N = 2M + 4 is large enough, then R < 1. But if theprobability of the existence of a closed boundary B around set J is less than 1,then the probability of no closed boundary around set J is greater than 0. Thatimplies that at least one of the trees in the initial set I was the starting pointof an innite open self-avoiding path. Since it is equally likely for any one ofthese trees to be the starting point of such a path, we have

8


0 < P (closed boundary B does not exist)= P (set I generates innite re spread)

P (innite open self-avoiding path starting at (0,0) exists)+ P

(innite open self-avoiding path starting at (1,0) exists

)+ . . .

+ P(innite open self-avoiding path starting at (M,0) exists

)= (M + 1)P

(innite open self-avoiding path starting at (0,0) exists

)= (M + 1)P (E).

Exercise

14. So we can conclude that if p > 23 , then the probability of innite re spreadP (E) is greater than 0. What then can be said about the value of the criticalprobability pc?

6. ConclusionsWe now have bounds on the critical probability pc:

13 pc 23 .

You should review your computer simulation results to see how those resultsagree. The next mathematical step would be to improve the techniques so asto narrow the bounds of pc, and ultimately nd the exact value of the criticalprobability. The proof that

pc

actually equals 12

was rst published by Kesten [1980], and other proofs have subsequently ap-peared (e.g., Grimmett [1989]).

7. Areas for Further ExplorationThere are other interesting questions besides innite spread. For example,

for transmission probabilities below the critical probability, we know that there will die out.

How far will it spread on average before it dies out? What can be determined about the shape of the burnt region? Is it circular,

or does it exhibit some other pattern?

9


The answers depend on the size of the transmission probability p.We could explore variations on the model. The lattice for the model is

square, with each tree having four nearest neighbors. However, we could alsomodel using a lattice that is triangular or hexagonal, with each tree having threeor six nearest neighbors. How can the methods for nding upper and lowerbounds of pc be adjusted for a different lattice? Do the bounds change?

In the model, we held the transmission probability p xed. However, it isreasonable to consider that the probability of spread may vary. In particular,a constant wind would make it more likely for a re to spread in one direc-tion. Consider a constant wind from the south that makes the probability ofre spreading to the north more likely than toward any other direction. Usecomputer simulations to explore this with the probability of northward spreadabove the threshold value pc = 12 and the probability of eastern, western andsouthern spread below the threshold value. What do you predict will happen?

Another adjustment is to consider variable winds. A wind coming fromthe south with variable speed can be worked into the computer model as arandom choice from several transmission probability values; the stronger thewind, the higher the probability of spread to the north. One could make itmore complicated by having random winds from any direction at any time, orstronger winds as the re grows.

There are many other models with similar structures and questions. Forexample, the model for the spread of forest re can also be used to study thespread through a population of an immunity-granting illness such as measlesor chicken pox (e.g., Durrett [1988a]). A closely related model is the percolationmodel. The model is set on the Z2 lattice with edges between sites open withprobability p, and a central question is whether the origin is in an innite opencluster. That is, is there is an open path for uid (such as water or oil) to movefrom the origin to innity?

This project is an entry point into the study of interacting particle systemsand cellular automata. Those models all follow simple rules to determine thestates of sites, and yet they produce complex behaviors for study. Consider themost famous cellular automaton of all, John Conways Game of Life. In thismodel, sites have only two states, alive and dead, and at each time step, siteschange states depending on the states of their nearest neighbors. Even withthese simplest of transition rules, interesting evolutions occur (see Gardner[1970; 1983] and computer implementations such as Schaffhauser [1996]).

10


8. Solutions to Selected Exercises

1. a) If the tree at site X at time n is burnt, then the tree at that site is burnt atthe following time n + 1.

b) If the tree at site X at time n is on re, then the tree at that site is burntat the following time n + 1.

c) If the tree at site X at time n is unburnt, then the tree at that site is eitherunburnt or on re at the following time n + 1.

2. The probability that the tree at site X is on re at time n + 1 is 1 (1 p)2.3. a) If Sn(X) = U , the probability that Sn+1(X) = U is (1 p)3.

b) If Sn(X) = U , the probability that Sn+1(X) = U is 1 (1 p)3.4. a) If Sn(X) = U , the probability that Sn+1(X) = U is (1 p)i.

b) If Sn(X) = U , the probability that Sn+1(X) = U is 1 (1 p)i.6. a) There are 4 options for the choice of X .

b) There are 3 options for the choice of Y .c) There are 3 options for the third step.

7. There are 4 3 = 12 self-avoiding paths of length 2 and 4 3 3 = 36 self-avoiding paths of length 3.a) There are fewer than 4 3 3 3 self-avoiding paths of length 4.

8. Let x = 3p. Then Q 4p

n=N

(3p)n1 = 4p(3p)N1

1 3p .

9. limN

Q 4p limN

(3p)N1

1 3p = 0.

10. Since pc = inf{p : P (E) > 0}, pc cannot be less than 13 . Thus pc 13 .11. The minimum set is I itself, and for this set, N = 2(M + 1) + 2 = 2M + 4.

12. The probability of k edges being closed is (1 p)k by independence.13. If N is increased, the probability R decreases.

14. Since pc = inf{p : P (E) > 0}, pc cannot be greater than 23 . Thus pc 23 .

11


9. AppendixWe give more details on the geometric series used in several exercises. Let

x be a real number. For some integer M , dene the sum SM to be

SM = 1 + x + x2 + x3 + + xM =M

k=0

xk.

ExercisesA1. Consider the derivative of SM :

dSMdx

= 0 + 1 + 2x + 3x2 + + MxM1 =M

n=1

n xn1.

Take the derivative of the closed form of SM to nd another expression ofthe derivative:

dSMdx

=d

dx

(1 xM+1

1 x)

= .

Let S = limM

dSMdx

. Does this limit exist for 1 < x < 1?

A2. For 1 < x < 1, let N = M + 1 and dene the tail of the series

TN = S d SMdx

=

n=N

n xn1.

What is the limit of TN as N ?

Solutions

A1.dSMdx

=(M + 1)xM (1 x) + (1 xM+1)

(1 x)2 =(M + 1)xM

1 x +SM

1 x .For 1 < x < 1, the limit does exist, and S = 1/(1 x)2.

A2. limN

TN = 0.

12


ReferencesDurrett, R. 1988a. Crabgrass, measles, and gypsy moths: An introduction to

interacting particle systems. Mathematical Intelligencer 10: 3747.

. 1988b. Lecture Notes on Particle Systems and Percolation. CA: Wadsworthand Brooks/Cole.

Gardner, Martin. 1970. The fantastic combinations of John Conways new soli-taire game Life. Scientic American 223 (4): 120ff. 1983. Reprinted withsubstantial additional commentary in Wheels, Life and Other MathematicalAmusements, 214257. NY: W.H. Freeman.

Grimmett, G. 1989. Percolation. NY: Springer-Verlag.

Kesten, H. 1980. The critical probability of bond percolation on the squarelattice equals 12 . Communications in Mathematical Physics 74: 4159.

Liggett, T.M. 1985. Interacting Particle Systems. NY: Springer-Verlag.

Schaffhauser, Christoph. 1996. Game of Life. Version 1.7.0. Computer programfor PowerPC and Motorola 680x0 Macintosh computers. Available at http://www.datacomm.ch/darkeagl/ .

About the Author

Emily Puckette is an associate professor ofmathematics at the University of the South in Se-wanee, TN. She earned her B.A. in mathematicsat Smith College and her Ph.D. at Duke Univer-sity, focusing on critical probabilities and ran-dom walks. One of her favorite extracurricularactivities is to go walking in the woods, ponder-ing probabilities.

13


14

I N T E R D I S C I P L I N A R Y L I V E L Y A P P L I C A T I O N S P R O J E C T

AUTHORS:Ethan Berkove(Mathematics)[email protected]

Thomas Hill(Mathematics)

Scott Moor(Chemical Engineering)

Lafayette CollegeEaston, PA 18042

CONTENTS1. Setting the Scene2. Your Job3. Part 1: Numerical and

Graphical Analyses4. Part 2: Let's Get Explicit5. Part 3: What Do You

Recommend?ReferencesSample SolutionNotes for the InstructorAbout the Authors

The UMAP Journal 24 (4) 435-449. Copyright 2003 by COMAP, Inc. All rightsreserved. Permission to make digital or hard copies of part or all of this work for per-sonal or classroom use is granted without fee provided that copies are not made ordistributed for profit or commercial advantage and that copies bear this notice.Abstracting with credit is permitted, but copyrights for components of this workowned by others than COMAP must be honored. To copy otherwise, to republish, topost on servers, or to redistribute to lists requires prior permission from COMAP.

Getting the Salt Out 435

Getting the Salt OutMATHEMATICS CLASSIFICATIONS:

Calculus, Differential Equations

DISCIPLINARY CLASSIFICATIONS: Chemistry, Engineering

PREREQUISITE SKILLS:1. Modeling with an ordinary differential equation with initial

condition.2. Sketching a solution to a differential equation on its direction

field.3. Solving an ordinary differential equation by separation of vari-

ables.

PHYSICAL CONCEPTS EXAMINED: Osmotic pressure, permeability, desalination.

COMPUTING REQUIREMENT: A program to plot direction fields. (A Mathematica program is

available from the authors.)


Contents1. Setting the Scene2. Your Job3. Part 1: Numerical and Graphical Analyses4. Part 2: Lets Get Explicit5. Part 3: What Do You Recommend?ReferencesSample SolutionNotes for the InstructorAbout the Authors

1. Setting the SceneWhen two water (or other solvent) volumes are separated by a semi-permeable

membrane, water will ow from the side of low solute concentration to the sideof high solute concentration. This is known as osmosis (see Figure 1). The owof solvent across the membrane may be stopped, or even reversed, by applyingexternal pressure on the side of higher solute concentration. This process iscalled reverse osmosis (see Figure 2).

Figure 1. Osmosis. Figure 2. Reverse osmosis.

We will use vant Hoffs equation to model osmosis. Jacobus Henricusvant Hoff (18521911) determined that the osmotic pressure is given by theequation

= cRT,

where

c is the molar solute concentration,

R is the universal gas constant, and

T is the absolute temperature.

Notice that c = n/V , where n is the number of moles of solute and V is thevolume of solution. Although vant Hoffs equation looks like a restatementof the ideal gas law, PV = nRT , it is special because it is being applied to a


liquid rather than to a gas. This equation was a signicant development; in1901 vant Hoff received the Nobel Prize in Chemistry in recognition of theextraordinary services he has rendered by the discovery of the laws of chemicaldynamics and osmotic pressure in solutions [Nobel Foundation 2000].

Consider the case in Figure 2, where an external pressure, P , is applied tothe side of the membrane with higher solute concentration. The resulting uxthrough the membrane is proportional to the difference between the osmoticpressure and the applied pressure. That is, if x represents the volume of waterthat has been extracted from the solution, then

dx

dt= A(P ), (1)

where is the water permeability constant andA is the membrane area perpen-dicular to the ow.

It is important to note that is not a constant here. If the initial volume ofsolution is V , and if x units of water have been extracted from the solution attime t, then

(t) =n

V x RT.

Substituting this expression into (1), we get the desalination differential equa-tion,

dx

dt= A

(P n

V x RT)

= A(P cV RT

V x)

. (2)

2. Your JobClearwater, Inc., is planning to build a new, portable water purier to re-

move salt from seawater. They have designed a machine, shown below inFigure 3; but before they go to the expense of building a prototype, they wantyou to perform a theoretical analysis of their design.

Your task is to analyze how the values of the design parameters for thedesalination machine will affect the performance of the machine. Keeping inmind that the machine must be portable:

How should the membrane coefcient , the membrane size A, the volume V ofthe desalination chamber, and the applied pressure P be chosen to produce amachine that is both efcient and economical?

Clearwater is counting on you to design a good product, so dont let themdown!

Desalinated WaterRevese Osmosis Membrane

Applied Load

Salt Water

Reverse Osmosis


Figure 3. Clearwaters desalination machine.

3. Part 1: Numerical and GraphicalAnalyses

Requirement 1Use direction elds and Eulers method (of following along the directions

in a direction eld) to sketch solutions of (2). Of course, to do that you willneed to assign values to the various constants and parameters in the equation.For the physical constants in the problem, use the values

c = 0.103moles

L,

R = 0.082Lbar

moleK ,T = 293 K.

For the design parameters, begin with

= 0.1m3

m2daybar ,

A = 1.5 m2,P = 15 bars,V = 4 L.

Plot a direction eld for the desalination differential equation where t (mea-sured in days) goes from 0 to 5 and x (measured in L) goes from 0 to 4. Also,


plot the solution curve for x that passes through the point (0, 0).According to your direction eld and solution, how much fresh water can

the machine extract from 4 L of sea water?Approximately how long does it take to extract 2 L of fresh water?As time goes by, what happens to the rate at which the machine produces

fresh water? Does this make sense to you? Can you think of a physical reasonwhy this would happen?

Requirement 2Suppose that we change the capacity of the machine. For example, suppose

that the initial volume of brine is 6 L instead of 4. How much fresh watercan we extract? How long does it take to extract 2 L of fresh water? Whathappens if we begin with 8 L of brine? How sensitive is the time required toextract 2 L of fresh water to the initial volume of sea water in the machine?What relationship seems to exist between the initial volume of sea water andthe total amount of fresh water that can be extracted from it? The graphs inFigure 4 show examples of proportionality relationships that you might lookfor in your investigations.

Figure 4. Three proportionality relationships: v u, v u2, and v 1/u.

Requirement 3Now x the value of V at 6 L and investigate how the performance of the

machine depends on the other design parameters. What effect does increasingor decreasing A have on the amount of fresh water that we can extract froma given volume of brine? On the time required to extract 2 L of fresh water?Answer the same questions for and P . Of the three variables, which one,when varied, causes the greatest change in the amount of fresh water extracted?On the time required to extract 2 L of fresh water?

Requirement 4What is the equilibrium solution of (2)that is, the solution when dx/dt =

0? Discuss how that solution depends on the design parameters and compareyour conclusions to the observations you made in the earlier parts above.


4. Part 2: Lets Get Explicit

Requirement 5Use separation of variables to obtain the solution of (2) that satises the

initial condition that x = 0 when t = 0. Hint: Solve for t as a function of x.

Requirement 6Use the solution of the differential equation to determine how the perfor-

mance of the Clearwater desalination machine depends on the values of theparameters , A, and P . Do your results agree with the conclusions that youdrew earlier using direction elds?

5. Part 3: What Do You Recommend?

Requirement 7Prepare a report for the engineers working on the desalination project at

Clearwater, Inc. An important issue is how quickly the proposed machine willbe able to produce 2 L of fresh water. Your report should include an analysisof how this time depends on each of the design parameters , A, and P , andon V .

Clearwaters marketing people have reported that, if their machine is tobe competitive in the marketplace, it must be able to produce at least 2 L offresh water per day. If the machine is to be portable, it can handle at most8 L of seawater at a time. Furthermore, because the walls of the pressurevessel are quite thin, the upper limit on the applied pressure, P , is 30 bars.Filters for this machine are extremely expensive; consequently, the area of thelter is to be no greater than 1.2 m2. Finally, the best lter available has apermeability coefcient of = 0.08. Is it theoretically possible to build a reverseosmosis desalination machine that meets all of these specications? What isthe approximate size of the smallest lter that can be used in this machine tomeet the specications?

To nish your report, give your recommendation for values of design pa-rameters that yield the best machine. A best machine should be builtwith some margin for error, so pick your parameter values with some room tomaneuver (is 510% possible?) i

Documents

UMAP 2004 vol. 25 No. 4