Computer Algebra Recipes

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260 CHAPTER 6. LINEAR PDE MODELS. PART 2 > for m from 0 to 1 do > sol[m]:=seq(p[m,s]=fsolve(f(m),p,s..s+1),s=0..4); > assign(sol[m]): > end do; sol 0 := p0; 0 =0:; p0; 1 =1:219669891; p0; 2 =2:233130594; p0; 3 =3:238315484;p0; 4 =4:241062864 sol 1 := p1; 0 =0:5860669999; p1; 1 =1:697050942; p1; 2 =2:717193891; p1; 3 =3:726137088; p1; 4 =4:731227206 Recalling that ! = ck = c p ® 2 + q 2 , the eigenfrequencies are given by !n;m;s = ¼c r ³ ´ pm;s 2 + a ³ n ´ 2 h radians per second; or, on using ! =2¼º, ºn;m;s = c r ³pm;s ´ 2 ³ n ´ 2 + hertz: 2 a h A functional operator F is formed to calculate the ºn;m;s. > F:=(n,m,s)->(c/2)*sqrt((p[m,s]/a)^2+(n/h)^2): The given values of the cylinder radius a, cylinder height h, and speed of sound c are entered, > a:=1.83: h:=3.04: c:=344: and the three lowest allowed frequencies calculated using F. > nu[0,1,0]:=F(0,1,0); º0; 1; 0 := 55:08389289 > nu[1,0,0]:=F(1,0,0); º1; 0; 0 := 56:57894736 > nu[1,1,0]:=F(1,1,0); º1; 1; 0 := 78:96462843 Consulting The Acoustical Foundations of Music, by John Backus [Bac69], Vectoria determines that the closest musical notes are A1 (55.000 Hz) for the ¯rst two frequencies and D2 # (77.782 Hz) for the third. Not having much of a feeling for these numbers, Vectoria goes to the Internet and ¯nds that the lowest-frequency note on a piano is 27.500 Hz and the highest frequency is 4186.0 Hz. So the allowed frequencies in this \shower stall" example are at the low-frequency end of the piano keyboard. PROBLEMS: Problem 6-7: Acoustical waveguide A sound wave of frequency ! is generated at one end (z =0)ofaverylong straight cylindrical pipe of radius a having rigid walls.
6.2. SEMI-INFINITE AND INFINITE DOMAINS 261 (a) Determine and discuss in detail the allowed modes of propagation and the cuto® frequency for wave propagation. The cuto® frequency is the minimum frequency for propagation of a speci¯c mode. (b) If the speed of sound in air is 1100 feet per second and if the frequency of the sound wave is 500 Hz, show that only a plane wave will be propagated if a
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Richard H. Enns George C. McGuire C
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PREFACE A computer algebra system (
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viii CONTENTS 2.3.1 Finite Di®eren
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x CONTENTS 8.3 The Bifurcation Diag
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2 INTRODUCTION B. Computer Algebra
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4 INTRODUCTION (a) At what angle Á
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6 INTRODUCTION The negative answer
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8 INTRODUCTION D. Maple Help We tea
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Part I THE APPETIZERS A man ceases
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14 CHAPTER 1. PHASE-PLANE PORTRAITS
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48 CHAPTER 2. PHASE-PLANE ANALYSIS
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Chapter 3 Linear ODE Models Among a
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3.1. FIRST-ORDER MODELS 111 Therate
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3.1. FIRST-ORDER MODELS 113 Pressur
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3.1. FIRST-ORDER MODELS 115 3.1.2 G
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3.1. FIRST-ORDER MODELS 117 Evil Kn
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3.2. SECOND-ORDER MODELS 119 > tf:=
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3.2. SECOND-ORDER MODELS 121 3.2.2
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3.2. SECOND-ORDER MODELS 123 Loadin
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3.2. SECOND-ORDER MODELS 125 0.4 0.
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3.2. SECOND-ORDER MODELS 127 On ent
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3.2. SECOND-ORDER MODELS 129 Using
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3.3. SPECIAL FUNCTION MODELS 131 3.
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3.3. SPECIAL FUNCTION MODELS 133 1
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3.3. SPECIAL FUNCTION MODELS 135 Th
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3.3. SPECIAL FUNCTION MODELS 137 3.
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3.3. SPECIAL FUNCTION MODELS 139 th
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3.3. SPECIAL FUNCTION MODELS 141 PR
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3.3. SPECIAL FUNCTION MODELS 143 ¡
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3.3. SPECIAL FUNCTION MODELS 145 μ
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3.3. SPECIAL FUNCTION MODELS 147 >
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150 CHAPTER 4. NONLINEAR ODE MODELS
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312 CHAPTER 7. THE HUNT FOR SOLITON
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320 CHAPTER 8. NONLINEAR DIAGNOSTIC
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356 BIBLIOGRAPHY [Dav62] H. T. Davi
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358 BIBLIOGRAPHY [Nyq28] H. Nyquist
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Index acoustical waveguide, 261 ade
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INDEX 363 eigenvalue, 130 Eiram Eir
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INDEX 365 Help, Full Text Search, 8
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INDEX 367 laplace, 121,267,268 lhs,
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INDEX 369 ¯sh harvesting, 100 ¯xe
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INDEX 371 quartic map, 345 Queen Di
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Computer Algebra Recipes

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