振動と波動

Report JS0.5 J Simplicity February 4, 2012

1 J Simplicity HOME http://www.jsimplicity.com/

Preface 2 Report 2

Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3 2 2:............... 9 1.4 3 3:.............. 10 1.5 4 4:....... 12 1.6................................ 14 2 1 17 2.1...................................... 17 2.2...................................... 20 2.3........................... 22 3 2 25 3.1 2................................ 25 II 31 4 32 4.1..................................... 32 4.2..................................... 32 4.3..................................... 34 4.4.................................. 34 4.5.................................. 35 3

CONTENTS 4 5 40 5.1........................................ 40 5.2 1................................... 43 5.3 1................................. 44 5.4 3................................. 48 5.5 3................................. 50 6 56 6.1................................. 56 6.2........................................ 58 6.3......................................... 60 6.4......................................... 64 6.5......................................... 64 6.6................................ 65 7 70 7.1 1.............................. 70 7.2 3.............................. 75 7.3................................. 77

Part I 5

Chapter 1 1.1 Report Chapter 1 m d2 x(t) dt 2 = mω 2 x(t) m[kg] ω[rad/s] x(t) = A sin(ωt + θ 0 ) A[m] θ 0 [rad] x(t)[m] x(t)[m] ψ(t) ψ(t) 4 Section Section 6

CHAPTER 1. 7 1.2 1 1: 4 Section x f (x) 2 f (x) Figure 1.1: f (x + 2) = f (x) 2 f (x) 2 f (x) = a 0 2 + a 1 cos πx + b 1 sin πx + a 2 cos 2πx + b 2 sin 2πx + f (x) = a 0 2 + (a n cos nπx n=1 + b n sin nπx ) (1.1) f (x) a n, b n cos nπx sin nπx cos nπx mπx cos dx = δ nm (n, m = 1, 2, ) (1.2) mπx sin dx = δ nm (n, m = 1, 2, ) (1.3) mπx sin dx = 0 (n, m = 1, 2, ) (1.4) 1 2 1 (n = m) δ nm = 0 (n m)

CHAPTER 1. 8 e ix = cos x + i sin x e ix = cos x i sin x cos x = eix + e ix 2 sin x = eix e ix 2i (1.2) cos nπx mπx cos dx = 1 4 = 1 4 k {exp (i nπx {exp i(n + m)πx ) + exp ( inπx + exp )}{exp (imπx) + exp ( imπx )}dx i(n m)πx + exp i(m n)πx + exp i(n + m)πx }dx exp (i kπx )dx = cos nπx = [ kπ cos kπx dx + i sin kπx dx 1dx + i 0dx = [x] mπx cos dx = = 2 (k = 0) kπx sin ] + i[ kπx cos kπ ] = 0 (k 0) 1 (0 + 2 + 2 + 0) = (n = m) 4 0 (n m) (1.2) (1.3) (1.4) (1.2) (1.3) (1.4) (1.1) a n, b n a n (1.1) cos mπx (m = 0, 1, 2, ) x f (x) cos mπx dx = a 0 2 cos mπx dx+ {a n n=1 1 m 0 m = 0 cos nπx cos mπx dx = [ mπx sin mπ ] = 0 cos 0πx dx = 1dx = [x] = 2 mπx cos dx+b n sin nπx mπx cos dx}

CHAPTER 1. 9 f (x) cos mπx dx = a 0 2 2δ 0m + (a n δ nm + b n 0) n=1 = a 0 δ 0m + a m f (x) cos mπx dx = a m (m = 0, 1, 2, ) m n a n a n = 1 b n b n = 1 f (x) cos nπx dx (n = 0, 1, 2, ) (1.5) f (x) sin nπx dx (n = 1, 2, ) (1.6) (1.5) (1.6) (1.1) f (x) = a 0 2 + (a n cos nπx n=1 + b n sin nπx ) f (x) 1.3 2 2: Section (1.2) (1.3) (1.4) cos nπx sin nπx cos nπx mπx cos dx = δ nm (n, m = 1, 2, ) mπx sin dx = δ nm (n, m = 1, 2, ) mπx sin dx = 0 (n, m = 1, 2, )

CHAPTER 1. 10 cos nπx n = 0 (1.2) (1.3) (1.4) ( 1 2 ) 2 dx = 1 ( 1 cos nπx )( 1 cos mπx )dx = δ nm ( 1 sin nπx )( 1 sin mπx )dx = δ nm ( 1 cos nπx )( 1 sin mπx )dx = 0 3 1 2, 1 cos π x, 1 sin π x, 1 cos 2π x, 1 sin 2π x,, 1 cos nπ x, 1 sin nπ x, 1 2 1 2 0 < x < a < x < b {ϕ n (x)} b a ϕ n (x)ϕ m (x)dx = δ nm {ϕ n (x)} 1.4 3 3: 3 cos x = eix + e ix 2 sin x = eix e ix 2i (1.1) f (x) = a 0 2 + (a n cos nπx n=1 + b n sin nπx )

CHAPTER 1. 11 f (x) = a exp (i nπx 0 2 + {a n n=1 ) + exp ( inπx ) 2 exp (i nπx ) exp ( inπx + b ) n 2i = a 0 2 + { 1 2 (a n ib n ) exp(i nπx ) + 1 2 (a n + ib n ) exp( i nπx )} n=1 c 0 a 0 2 c n 1 2 (a n ib n ) (n = 1, 2, 3, ) c n 1 2 (a n + ib n ) (n = 1, 2, 3, ) } f (x) = c 0 + {c n exp(i nπx ) + c n exp(i ( n)πx )} n=1 f (x) = n= c n exp(i nπx ) (1.7) (1.7) c n, c n (1.5) (1.6) c n 1 2 (a n ib n ) = 1 2 ( 1 = 1 2 = 1 2 f (x) cos nπx dx i 1 f (x)(cos nπx nπx i sin )dx f (x) exp( i nπx )dx (n = 1, 2, ) f (x) sin nπx dx) c n 1 2 (a n + ib n ) = 1 2 ( 1 = 1 2 = 1 2 = 1 2 f (x) cos nπx dx + i 1 f (x)(cos nπx f (x) exp (i nπx )dx nπx + i sin )dx f (x) sin nπx dx) f (x) exp( i ( n)πx )dx (n = 1, 2, )

CHAPTER 1. 12 c n = 1 2 c 0 a 0 2 = 1 2 1 = 1 2 f (x) cos 0πx dx f (x) exp( i 0πx )dx f (x) exp( i nπx )dx (n =, 2, 1, 0, 1, 2, ) (1.8) (1.8) 1.5 4 4: 2 2 f (x) (1.7) (1.8) f (x) = c n = 1 2 n= c n exp(i nπx ) f (ξ) exp( i nπξ )dξ (n =, 2, 1, 0, 1, 2, ) (1.8) (1.7) x ξ (1.8) (1.7) f (x) = n= 1 2 f (ξ) exp( i nπξ )dξ exp(inπx ) k n nπ k = k n+1 k n = π f (x) = 1 2π n= k f (ξ)e iknξ dξ e ik nx

CHAPTER 1. 13 k n k 0 k F(k n ) n= f (x) = 1 2π f (x) = 1 2π dk dk (1.9) (1.9) g(k) = f (x) = 1 2π dk F(k) dξ f (ξ)e ikξ e ikx dξ f (ξ)e ik(x ξ) (1.9) dξ f (ξ)e ikξ (1.10) dk g(k)e ikx (1.11) (1.10) f (ξ) g(k) (1.11) g(k) f (x) (1.10) f (ξ) k g(k) (1.10) (1.11) k g(k) f (x) g(k) f (x) 1 2π g(k) f (x) g(k) = 1 f (ξ)e ikξ dξ 2π f (x) = 1 g(k)e ikx dk 2π (1.10) (1.11) ξ x g(k) = 1 2π f (x)e ikx dx (1.12) f (x) = 1 g(k)e ikx dk (1.13) 2π

CHAPTER 1. 14 3 1 x 3 x = (x, y, z) 1 k 3 k = (k x, k y, k z ) 1 (1.12) g(k x ) = 1 2π g(k y ) = 1 2π g(k z ) = 1 2π g(k x )g(k y )g(k z ) = 1 (2π) 3 dx dy f (x)e ik xx dx f (y)e ik yy dy f (z)e ik zz dz dz f (x) f (y) f (z)e ik xx e ik yy e ik zz g(k x )g(k y )g(k z ) g( k) f (x) f (y) f (z) f ( x) 3 g( k) = 1 (2π) 3 1 (1.13) f (x) = 1 g(k x )e ikxx dk x 2π f (y) = 1 g(k y )e ikyy dk y 2π f (z) = 1 g(k z )e ikzz dk z 2π f (x) f (y) f (z) = 1 (2π) 3 3 f ( x) = 1 (2π) 3 f ( x)e i k x dxdydz (1.14) dk x dk y dk z g(k x )g(k y )g(k z )e ikxx e ikyy e ik zz g( k)e i k x dk x dk y dk z (1.15) 1.6 T[s] ψ(t) f (x)

CHAPTER 1. 15 (1.1) 2 T[s] x t[s] ψ(t) = a 0 2 + (a n cos nπt T/2 + b n sin nπt T/2 ) n=1 = a 0 2 + (a n cos n 2π T t + b n sin n 2π T t) n=1 ψ(t) = a 0 2 + (a n cos nωt + b n sin nωt) n=1 ω = 2π T ω, 2ω, 3ω, (1.5) (1.6) a n = 1 T/2 a n = 2 T b n = 1 T/2 b n = 2 T T/2 T/2 T/2 T/2 T/2 T/2 T/2 T/2 ψ(t) cos nπt T/2 dt ψ(t) cos nωt dt (n = 0, 1, 2, ) ψ(t) sin nπt T/2 dt ψ(t) sin nωt dt (n = 1, 2, ) Section T[s] ψ(t) f (x) (1.7) 2 T[s] x t[s] ψ(t) = = ψ(t) = c n exp(i nπt T/2 ) c n exp(in 2π T t) c n e inωt n= n= n=

CHAPTER 1. 16 ω = 2π T ω, 2ω, 3ω, (1.8) c n = 1 T c n = 1 T T/2 T/2 T/2 T/2 ψ(t) exp ( i nπt T/2 )dt ψ(t)e inωt dt (n =, 2, 1, 0, 1, 2, ) (1.12) (1.13) g(k) = 1 2π f (x)e ikx dx f (x) = 1 g(k)e ikx dk 2π ψ(t) f (x) (1.12) (1.13) k ω[rad/s] x t[s] (1.12) g(ω) = 1 ψ(t)e iωt dt 2π ψ(t) ω[rad/s] g(ω) (1.13) ψ(t) = 1 g(ω)e iωt dω 2π e iωt ω[rad/s] g(ω) ψ(t) J Simplicity HOME http://www.jsimplicity.com/

Chapter 2 1 2.1 1 m d2 x(t) dt 2 = mω 2 x(t) 2mk m d2 x(t) dt 2 d 2 x(t) dt 2 = mω 2 x(t) 2mk dx(t) dt + 2k dx(t) dt + ω 2 x(t) = 0 (2.1) (2.1) d 2 z(t) dt 2 + 2k dz(t) dt + ω 2 z(t) = 0 (z(t) = x(t) + iy(t)) (2.2) (2.1) z(t) (2.2) (2.1) x(t) y(t) x(t) y(t) z(t) = αe λt 17

CHAPTER 2. 1 18 α λ (2.2) λ 2 αe λt + 2kλαe λt + ω 2 αe λt = 0 λ 2 + 2kλ + ω 2 = 0 λ = k ± k 2 ω 2 (2.3) (2.3) 3 k 2 ω 2 < 0 ω 2 ω 2 k 2 (2.3) λ = k ± iω z(t) = α 1 e kt e iω t + α 2 e kt e iω t = (a 1 + ib 1 )e kt (cos ω t + i sin ω t) + (a 2 + ib 2 )e kt (cos ω t i sin ω t) = {( b 1 + b 2 )e kt sin ω t + (a 1 + a 2 )e kt cos ω t} + i{(a 1 a 2 )e kt sin ω t + (b 1 + b 2 )e kt cos ω t} a 1, b 1, a 2, b 2 z(t) (2.1) A 1 b 1 + b 2, A 2 a 1 + a 2 x(t) = A 1 e kt sin ω t + A 2 e kt cos ω t = Ae kt (a sin ω t + b cos ω t) A, a, b x(t) = Ae kt A sin (ω t + θ 0 ) A = a 2 + b 2 B tan θ 0 = b a x(t) = Be kt sin (ω t + θ 0 )

CHAPTER 2. 1 19 k 2 ω 2 > 0 λ 1 = k k 2 ω 2 λ 2 = k + k 2 ω 2 z(t) = A 1 e λ 1t + A 2 e λ 2t A 1, A 2 B 1, B 2 x(t) = B 1 e λ1t + B 2 e λ 2t k 2 ω 2 = 0 z(t) = αe kt z(t) = α(t)e kt (2.2) ( d2 α(t) dt 2 dz(t) = dα(t) e kt kα(t)e kt dt dt d 2 z(t) = d2 α(t) e kt 2k dα(t) e kt + k 2 α(t)e kt dt 2 dt 2 dt 2k dα(t) dt d2 α(t) dt 2 (k 2 ω 2 )α(t) = 0 d2 α(t) dt 2 = 0 α(t) = Ct + D + k 2 α(t))e kt + 2k( dα(t) dt kα(t))e kt + ω 2 α(t)e kt = 0

CHAPTER 2. 1 20 C D (2.2) z(t) = (Ct + D)e kt C D C D (2.1) x(t) = (Ct + D)e kt 2.2 ω 0 [rad/s] ω[rad/s] F cos ωt m d2 x(t) dt 2 = mω 2 0x(t) + F cos ωt d 2 x(t) + ω 2 dt 2 0 x(t) = F cos ωt (2.4) m (2.4) 2 0 x 0 (t) = A sin (ω 0 t + θ 0 ) (2.4) x 1 (t) = b cos ωt (2.4) bω 2 cos ωt + ω 2 0 b cos ωt = F cos ωt m b(ω 2 0 ω2 ) = F m b = F 1 m ω 2 0 ω2 x 1 (t) = F m ω 2 0 1 cos ωt ω2

CHAPTER 2. 1 21 x(t) = A sin (ω 0 t + θ 0 ) + F m ω 2 0 1 cos ωt ω2 1 2 ω = ω 0 Fe iωt m d2 z(t) dt 2 z = mω 2 0z(t) + Feiωt d 2 z(t) dt 2 + ω 2 0 z(t) = F m eiωt (2.5) F cos ωt (2.5) α z(t) = αe iωt ( ω 2 )αe iωt + ω 2 0 αeiωt = F m eiωt α(ω 2 0 ω2 ) = F m α = F 1 m ω 2 0 ω2 α z(t) z(t) = F m = F m 1 ω 2 eiωt 0 ω2 ω 2 0 x 1 (t)[m] x 1 (t) = F m 1 (cos ωt + i sin ωt) ω2 ω 2 0 1 cos ωt ω2

CHAPTER 2. 1 22 2.3 Section Section m d2 x(t) dt 2 d 2 x(t) dt 2 = mω 2 dx(t) 0x(t) 2mk + F cos ωt dt + 2k dx(t) dt + ω 2 0 x(t) = F cos ωt (2.6) m (2.6) 0 Section (2.6) (2.6) x 1 (t) = A cos (ωt δ) Aω 2 cos (ωt δ) 2kAω sin (ωt δ) + ω 2 0 A cos(ωt δ) = F cos ωt m (ω 2 0 ω2 )A cos(ωt δ) 2ωkA sin(ωt δ) = F cos ωt m (ω 2 0 ω2 )A(cos ωt cos δ + sin ωt sin δ) 2ωkA(sin ωt cos δ cos ωt sin δ) = F cos ωt m {(ω 2 0 ω2 )A cos δ + 2ωkA sin δ} cos ωt + {(ω 2 0 ω2 )A sin δ 2ωkA cos δ} sin ωt = F cos ωt (2.7) m (2.7) t = 0[s] (2.7) t[s] t = 0[s] (ω 2 0 ω2 )A cos δ + 2ωkA sin δ = F m (2.8) (2.8) 2ωk+(2.9) (ω 2 0 ω2 ) (ω 2 0 ω2 )A sin δ 2ωkA cos δ = 0 (2.9) 4ω 2 k 2 A sin δ + (ω 2 0 ω2 ) 2 A sin δ = 2ωk F m 2ωk F A sin δ = (ω 2 0 ω2 ) 2 + 4ω 2 k 2 m (2.8) (ω 2 0 ω2 )-(2.9) 2ωk (ω 2 0 ω2 ) 2 A cos δ + 4ω 2 k 2 A cos δ = (ω 2 0 ω2 ) F m A cos δ = ω 2 0 ω2 (ω 2 0 ω2 ) 2 + 4ω 2 k 2 F m

CHAPTER 2. 1 23 A[m] δ[rad] A 2 cos 2 δ + A 2 sin 2 δ = (ω2 0 ω2 ) 2 + 4ω 2 k 2 {(ω 2 0 ω2 ) 2 + 4ω 2 k 2 } ( F 2 m )2 A 2 1 = (ω 2 0 ω2 ) 2 + 4ω 2 k ( F 2 m )2 1 F A = (ω 2 0 ω2 ) 2 + 4ω 2 k 2 m A sin δ A cos δ = tan δ = 2ωk ω 2 0 ω2 2ωk (ω 2 0 ω2 ) 2 + 4ω 2 k 2 F m ω 2 0 ω2 (ω 2 0 ω2 ) 2 + 4ω 2 k 2 F m m d2 z(t) dt 2 d 2 z(t) dt 2 = mω 2 dz(t) 0z(t) 2mk + Fe iωt dt + 2k dz(t) dt α z(t) = αe iωt (2.10) + ω 2 0 z(t) = F m eiωt (2.10) ( ω 2 )αe iωt + 2k iωαe iωt + ω 2 0 αeiωt = F m eiωt α( ω 2 + 2iωk + ω 2 0 ) = F m 1 F α = (ω 2 0 ω2 ) + i 2ωk m β (ω 2 0 ω2 ) + i 2ωk tan δ = 2ωk ω 2 0 ω2 α = 1 F β m α = 1 F β e iδ m

CHAPTER 2. 1 24 Figure 2.1: A 1 F β m α = Ae iδ z(t) = αe iωt = Ae iδ e iωt = Ae i(ωt δ) x 1 (t)[m] x 1 (t) = A cos(ωt δ) J Simplicity HOME http://www.jsimplicity.com/

Chapter 3 2 3.1 2 Chapter 1 3 2 2 (a) 3 Figure 3.1: 2 m[kg] 3 k[n/m] (b) 2 x 1 (t)[m], x 2 (t)[m] (b) 2 m d2 x 1 (t) dt 2 m d2 x 2 (t) dt 2 = kx 1 (t) + k(x 2 (t) x 1 (t)) = kx 2 (t) k(x 2 (t) x 1 (t)) 25

CHAPTER 3. 2 26 m d2 x 1 (t) = 2kx dt 2 1 (t) + kx 2 (t) (3.1) m d2 x 2 (t) = kx dt 2 1 (t) 2kx 2 (t) (3.2) 2 (3.1) +(3.2) (3.1) -(3.2) m d2 dt 2 (x 1(t) + x 2 (t)) = k(x 1 (t) + x 2 (t)) m d2 dt 2 (x 1(t) x 2 (t)) = 3k(x 1 (t) x 2 (t)) q 1 (t) x 1 (t) + x 2 (t) q 2 (t) x 1 (t) x 2 (t) ω 2 1 k m ω 2 2 3k m d 2 q 1 (t) = ω 2 dt 2 1 q 1(t) d 2 q 2 (t) = ω 2 dt 2 2 q 2(t) q 1 (t)[m], q 2 (t)[m] q 1 (t) = A 1 sin(ω 1 t + θ 1 ) q 2 (t) = A 2 sin(ω 2 t + θ 2 ) A 1, A 2, θ 1, θ 2 x 1 (t)[m], x 2 (t)[m] x 1 (t) = 1 2 (q 1(t) + q 2 (t)) x 2 (t) = 1 2 (q 1(t) q 2 (t))

CHAPTER 3. 2 27 2 (3.1) (3.2) 2k m k m k x m 1 (t) 2k x 2 (t) = d2 x 1 (t) dt 2 x 2 (t) m x i (t) = A i e i(ωt+θ i) (i = 1, 2) 2k k x m m 1 (t) k 2k x 2 (t) = x 1 (t) ω2 x 2 (t) m m 2k m k m k x m 1 (t) 2k x 2 (t) = x 1 (t) ω2 x 2 (t) m (3.3) (3.3) B X(t) = ω 2 X(t) (3.4) (3.3) 2k m ω2 k x m 1 (t) k 2k m m ω2 x 2 (t) = 0 (3.5) (3.3) (3.4) (3.5) x 1 (t)[m], x 2 (t)[m] 0 2 2 2 2 0 2k m ω2 k m k 2k m m = 0 ω2

CHAPTER 3. 2 28 ( 2k m ω2 ) 2 ( k m )2 = 0 4k2 m 2 4k m ω2 + ω 4 k2 m 2 = 0 ω 4 4k m ω2 + 3k2 m 2 = 0 (ω 2 k m )(ω2 3k m ) = 0 ω 2 = ω 2 1, ω2 2 ω 2 1 k m ω 2 2 3k m ω 2 = ω 2 1 k m (3.5) k m k m k x m 1 (t) k x 2 (t) = 0 m k m x 1(t) k m x 2(t) = 0 k m x 1(t) + k m x 2(t) = 0 x 1 (t) = x 2 (t) x 1 (t) = x 2 (t) = A e i(ω 1t+θ 1 ) x 1 (t) = x 2 (t) = A cos(ω 1 t + θ 1 ) 1 p 1 = 2 1 2

CHAPTER 3. 2 29 (3.5) k m k m ω 2 = ω 2 2 3k m k x m 1 (t) k x 2 (t) = 0 m k m x 1(t) k m x 2(t) = 0 x 1 (t) = x 2 (t) x 1 (t) = x 2 (t) = A e i(ω 2t+θ 2 ) x 1 (t) = x 2 (t) = A cos(ω 2 t + θ 2 ) 1 p 1 = 2 1 2 (3.3) 4 x 1 (t) = A cos (ω 1 t + θ 1 ) + A cos(ω 2 t + θ 2 ) (3.6) x 2 (t) = A cos (ω 1 t + θ 1 ) A cos(ω 2 t + θ 2 ) (3.7) 2k k x m m 1 (t) k 2k x 2 (t) = d2 x 1 (t) dt 2 x 2 (t) m m 2k k A m m cos (ω 1 t + θ 1 ) + A cos(ω 2 t + θ 2 ) k 2k A cos (ω 1 t + θ 1 ) A cos(ω 2 t + θ 2 ) = ω 2 1 A cos (ω 1 t + θ 1 ) ω 2 2 A cos(ω 2 t + θ 2 ) ω 2 1 m m A cos (ω 1 t + θ 1 ) + ω 2 2 A cos(ω 2 t + θ 2 ) d 2 A cos (ω 1 t + θ 1 ) + A cos(ω 2 t + θ 2 ) dt 2 A cos (ω 1 t + θ 1 ) A cos(ω 2 t + θ 2 ) = ω 2 1 A cos (ω 1 t + θ 1 ) ω 2 2 A cos(ω 2 t + θ 2 ) ω 2 1 A cos (ω 1 t + θ 1 ) + ω 2 2 A cos(ω 2 t + θ 2 )

CHAPTER 3. 2 30 (3.6) (3.7) B 2 P 1 1 2 P = 2 1 1 2 2 B (3.4) B X(t) = ω 2 X(t) P t B P P t X(t) = ω 2 P t X(t) 1 1 2 2k 2 1 1 m k 1 1 1 1 1 1 2 2 m 2 2 x 1 (t) 2 k 2k 1 1 1 1 x 2 (t) = 2 x 1 (t) ω2 1 1 x 2 (t) 2 2 m m 2 2 2 2 2 2 1 k 1 k 1 1 1 1 2 (x 1 (t) + x 2 (t)) (x 1 (t) + x 2 (t)) 2 m 2 m 2 2 1 3k 2 m 1 3k 1 1 1 = ω 2 2 1 (x 1 (t) x 2 (t)) (x 1 (t) x 2 (t)) 2 m 2 2 2 2 k 0 m 3k Q(t) = ω 2 Q(t) 0 m P t P Q(t) = P t X(t) q 1 (t) = q 2 (t) 1 (x 1 (t) + x 2 (t)) = 2 1 (x 1 (t) x 2 (t)) 2 J Simplicity HOME http://www.jsimplicity.com/

Part II 31

Chapter 4 4.1 1 4.2 v[m/s] 1 1 ψ[m] ψ 32

CHAPTER 4. 33 Figure 4.1: 1 ψ ψ ψ J Simplicity x[m] λ[m] A 1 T[s] 1[s] f [Hz][Hz = 1/s] f [Hz] T[s] f = 1 T P 1 T[s] 1 λ[m] 2 Figure 4.2: 2 v = λ T = f λ

CHAPTER 4. 34 4.3 Section ψ ψ Figure 4.3: 1 4.4 2 3

CHAPTER 4. 35 4.5 : v[m/s] : S (Source) : u S [m/s] : O(Observer) : u O [m/s] : f 0 [Hz] : f [Hz] S O S u S [m/s] O 0[s] S 0 S t[s] S 1 S 0 S 1 u S t[m] t[s] S S O S S 1 A 1 vt u S t[m] f 0 t[] λ [m] λ = vt u S t f 0 t λ = v u S f 0 (4.1) S v[m/s]

CHAPTER 4. 36 Figure 4.4: 1 v = f λ f = 1 λ v = f = f 0 v u S v v v u S f 0 (4.2) f [Hz] S S 1 A 2 = vt + u S t[m] f 0 t[] λ [m] λ = vt + u S t f 0 t λ = v + u S f 0 (4.3) S v[m/s] v = f λ f = 1 λ v = f = f 0 v + u S v v v + u S f 0 (4.4) f [Hz]

CHAPTER 4. 37 O S O O O S O (a) Figure 4.5: 2 0[s] S O (b) t[s] vt[m] O u O t[m] t[s] O O O B f [Hz] f t[] O t[s] OB O O OO f t = O B λ = vt + u Ot λ f = v + u O λ (4.5) f = (v + u O ) 1 λ = (v + u O ) f 0 v f = v + u O v f 0 (4.6)

CHAPTER 4. 38 O (a) 0[s] S Figure 4.6: 3 O (b) t[s] vt[m] O u O t[m] t[s] O O O B f [Hz] f t[] O t[s] OB O O OO f t = O B λ = vt u Ot λ f = v u O λ (4.7) f = (v u O ) 1 λ = (v u O ) f 0 v f = v u O v f 0 (4.8) S O S λ [m] (4.1) (4.3) λ = v ± u S f 0 (4.9)

CHAPTER 4. 39 S O f [Hz] (4.5) (4.7) u O [m/s] O f = v ± u O λ = (v ± u O ) 1 λ f 0 = (v ± u O ) v ± u S f = v ± u O v ± u S f 0 (4.10) (4.10) S O (4.10) (4.2) (4.4) (4.6) (4.8) J Simplicity HOME http://www.jsimplicity.com/

Chapter 5 5.1. ψ(t, x = 0) ψ(t, x = 0) = A sin ωt A 0[rad] t [s] ψ(t = t, x) Figure 5.1: 3 x[m] t[s] O x[m] P x v [s] t + x v = t t = t x v 40

CHAPTER 5. 41 Figure 5.2: 4 t[s] P (t x )[s] O v ψ( t[s] P ) = { (t x )[s] O } v ψ(t, x) = A sin ω(t x v ) = A sin(ωt ω v x) ψ(t, x) = A sin(ωt kx) k[rad/m] k = ω v = 2π vt k 2π λ 2π[rad] v = ω k ψ t[s] x[m] 2 1 ψ ψ t = 0[s] ψ(t, x) = A sin( kx) ψ(t, x) = A sin kx

CHAPTER 5. 42 t = 0[s] O ψ(t, x) = A sin ωt ψ ψ (ωt kx)[rad] Figure 5.3: θ[rad] 1 2π[rad] 1 2π[rad] Section v[m/s] v[m/s] ψ(t, x) = A sin ω{t = A sin ω(t + x v ) x ( v) } ψ(t, x) = A sin(ωt + kx)

CHAPTER 5. 43 5.2 1 v[m/s] t [s] ψ(t = t, x = 0) ψ(t = t, x = 0) = f (t ) f (t ) x[m] t[s] O x[m] P x v [s] Figure 5.4: 1 t + x v = t t = t x v t[s] P (t x )[s] O v 1 ψ( t[s] P ) = { (t x )[s] O } v ψ(t, x) = f (t x v ) f (t x v )[s] t[s] ψ ψ

CHAPTER 5. 44 x[m] ψ v[m/s] v[m/s] ψ(t, x) = f (t + x v ) 1 1 π 2 [rad] ψ(t, x) = A sin (ωt kx) ψ(t, x) = A sin (ωt + kx) ψ(t, x) = A cos (ωt kx) ψ(t, x) = A cos (ωt + kx) ψ(t, x) = A e i(ωt kx) ψ(t, x) = A e i(ωt+kx) kx[rad] kx[rad] 5.3 1 Section ψ(t, x) = f (t x v ) ψ(t, x) = f (t + x v ) 1 1 ψ(t, x) ξ t x v

CHAPTER 5. 45 ψ(t, x) t[s] 2 ψ(t, x) t 2 ψ(t, x) t 2 = d f (ξ) ξ dξ t = d f (ξ) dξ = d f (ξ) (d dξ dξ ) ξ t = d2 f (ξ) dξ 2 ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 = d f (ξ) ξ dξ x = 1 d f (ξ) v dξ = d dξ ( 1 d f (ξ) v dξ ) ξ x = 1 d 2 f (ξ) v 2 dξ 2 1 1 2 ψ(t, x) = 2 ψ(t, x) (5.1) v 2 t 2 x 2 η t + x v ψ(t, x) t[s] 2 ψ(t, x) t 2 ψ(t, x) t 2 = d f (η) η dη t = d f (η) dη = d f (η) (d dη dη ) η t = d2 f (η) dη 2 ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 = d f (η) η dη x = 1 d f (η) v dη = d dη (1 d f (η) v dη ) η x = 1 d 2 f (η) v 2 dη 2

CHAPTER 5. 46 1 (5.1) ψ(t, x) = f 1 (t x v ) + f 2(t + x v ) ψ(t, x) 1 (5.1) ψ(t, x) t[s] 2 ψ(t, x) t 2 ψ(t, x) t 2 = d f 1(ξ) ξ dξ = d f 1(ξ) dξ t + d f 2(η) dη + d f 2(η) dη = d dξ (d f 1(ξ) ) ξ dξ = d2 f 1 (ξ) dξ 2 η t t + d dη (d f 2(η) dη + d2 f 2 (η) dη 2 ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 = d f 1(ξ) ξ dξ x + d f 2(η) η dη x = 1 d f 1 (ξ) + 1 d f 2 (η) v dξ v dη = d dξ ( 1 v = 1 v 2 (d2 f 1 (ξ) dξ 2 d f 1 (ξ) ) ξ dξ x + d dη (1 v + d2 f 2 (η) dη 2 ) ) η t d f 2 (η) dη ) η x ψ(t, x) 1 (5.1) 1 (5.1) ψ(t, x) = f (t x v ) ψ(t, x) = f (t + x v ) ψ(t, x) = f 1 (t x v ) + f 2(t + x v ) t = 1 (ξ + η) 2 x = v (η ξ) 2 ψ(t, x) = ψ{ξ(t, x), η(t, x)}

CHAPTER 5. 47 ψ(t, x) t[s] 2 ψ(t, x) t ψ(ξ, η) ξ ψ(ξ, η) = + ξ t η ψ(ξ, η) ψ(ξ, η) = + ξ η 2 ψ(t, x) = η) ( ψ(ξ, + t 2 ξ ξ η t ψ(ξ, η) ) ξ η t + η) ( ψ(ξ, + η ξ = 2 ψ(ξ, η) + 2 ψ(ξ, η) + 2 2 ψ(ξ, η) ξ 2 η 2 ξ η ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 ψ(ξ, η) ξ ψ(ξ, η) η = + ξ x η x = 1 ψ(ξ, η) + 1 ψ(ξ, η) v ξ v η = ξ ( 1 v ψ(ξ, η) ξ + 1 v ψ(ξ, η) ) ξ η x + η ( 1 ψ(ξ, η) + 1 v ξ v = 1 v 2 ( 2 ψ(ξ, η) ξ 2 + 2 ψ(ξ, η) η 2 2 2 ψ(ξ, η) ξ η ) 1 (5.1) ψ(ξ, η) ) η η t ψ(ξ, η) ) η η x 1 ψ(ξ, η) v 2 ( 2 + 2 ψ(ξ, η) + 2 2 ψ(ξ, η) ξ 2 η 2 ξ η ) = 1 ψ(ξ, η) v 2 ( 2 + 2 ψ(ξ, η) 2 2 ψ(ξ, η) ξ 2 η 2 ξ η ) 2 ψ(ξ, η) ξ η = 0 η[s] ψ(ξ, η) ξ = f 1 (ξ) f 1 (ξ) η[s] ξ[s] ξ[s] ψ(ξ, η) = f 1 (ξ)dξ + f 2(η) ψ(ξ, η) = f 1 (ξ) + f 2 (η) ψ(t, x) = f 1 (t x v ) + f 2(t + x v ) f 1, f 2 ξ[s], η[s] 0 1 (5.1) 1 ψ(t, x) = f (t x v ) ψ(t, x) = f (t + x v ) ψ(t, x) = f 1 (t x v ) + f 2(t + x v )

CHAPTER 5. 48 1 1 1 5.4 3 3 u v(= v u)[m/s] t [s] u x = v(t t ) t [s] ψ(t, x) ψ(t, x = 0) = f (t ) f (t ) ψ(t, x) u x[m] t[s] O ψ(t, x) Figure 5.5: 3 u x[m] P u x v [s] t u x + = t v t u x = t v

CHAPTER 5. 49 t[s] P (t O 3 ψ( t[s] P ) = { (t u x )[s] O } v ψ(t, x) = f (t u x v ) u x )[s] v v[m/s] v[m/s] ψ(t, x) = f (t + 3 u x v ) 3 3 ψ(t, x) = A sin ω(t u x v ) = A sin (ωt k u x) k k u ψ(t, x) ψ(t, x) = A sin (ωt k x) 3 3 ψ(t, x) = A sin (ωt k x) ψ(t, x) = A sin (ωt + k x) ψ(t, x) = A cos (ωt k x) ψ(t, x) = A cos (ωt + k x) ψ(t, x) = A e i(ωt k x) ψ(t, x) = A e i(ωt+ k x)

CHAPTER 5. 50 5.5 3 3 Section u x ψ(t, x) = f (t v ) ψ(t, x) = f (t + u x v ) 3 3 ψ(t, x) ξ t u x v ψ(t, x) t[s] 2 ψ(t, x) t 2 ψ(t, x) t 2 = d f (ξ) ξ dξ t = d f (ξ) dξ = d f (ξ) (d dξ dξ ) ξ t = d2 f (ξ) dξ 2 ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 y[m], z[m] 2 ψ(t, x) x 2 + 2 ψ(t, x) y 2 = d f (ξ) ξ dξ x = u x d f (ξ) v dξ = d dξ ( u x v = u2 x d 2 f (ξ) v 2 dξ 2 + 2 ψ(t, x) z 2 d f (ξ) dξ ) ξ x = u2 x + u 2 y + u 2 z v 2 2 f (ξ) ξ 2 u f ξ[s] 2 ψ(t, x) 2 2 ψ(t, x) x 2 + 2 ψ(t, x) y 2 + 2 ψ(t, x) z 2 = 1 v 2 2 ψ(t, x) t 2

CHAPTER 5. 51 3 3 1 2 ψ(t, x) = ( 2 v 2 t 2 x + 2 2 y + 2 )ψ(t, x) 2 z2 ( x, y, z ) 3 2 = ( x, y, z ) ( x, y, z ) = 2 x 2 + 2 y 2 + 2 z 2 1 2 ψ(t, x) = 2 ψ(t, x) (5.2) v 2 t 2 η t + u x v ψ(t, x) t[s] 2 ψ(t, x) t 2 ψ(t, x) t 2 = d f (η) η dη t = d f (η) dη = d f (η) (d dη dη ) η t = d2 f (η) dη 2 ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 = d f (η) η dη x = u x d f (η) v dη = d dη (u x v = u2 x d 2 f (η) v 2 dη 2 d f (η) dη ) η x

CHAPTER 5. 52 y[m], z[m] ψ(t, x) t[s] x[m] 2 ξ[s] η[s] 3 (5.2) ψ(t, x) = f 1 (t u x v ) + f u x 2(t + v ) ψ(t, x) 3 (5.2) ψ(t, x) t[s] 2 ψ(t, x) t 2 ψ(t, x) t 2 = d f 1(ξ) ξ dξ = d f 1(ξ) dξ t + d f 2(η) dη + d f 2(η) dη = d dξ (d f 1(ξ) ) ξ dξ = d2 f 1 (ξ) dξ 2 η t t + d dη (d f 2(η) dη + d2 f 2 (η) dη 2 ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 = d f 1(ξ) ξ dξ x + d f 2(η) η dη x = u x d f 1 (ξ) + u x d f 2 (η) v dξ v dη = d dξ ( u x v y[m], z[m] 2 ψ(t, x) x 2 + 2 ψ(t, x) y 2 d f 1 (ξ) ) ξ dξ x + d dη (u x v = u2 x f 1 (ξ) v 2 (d2 + d2 f 2 (η) ) dξ 2 dη 2 + 2 ψ(t, x) z 2 ) η t d f 2 (η) dη ) η x = u2 x + u 2 y + u 2 z ( d2 f 1 (ξ) + d2 f 2 (η) ) v 2 dξ 2 dη 2 u ψ(t, x) 2 2 ψ(t, x) x 2 + 2 ψ(t, x) y 2 + 2 ψ(t, x) z 2 = 1 v 2 2 ψ(t, x) t 2 ψ(t, x) 3 (5.2) 3 (5.2) u x ψ(t, x) = f (t v ) u x ψ(t, x) = f (t + v ) ψ(t, x) = f 1 (t u x v ) + f 2(t + u x v )

CHAPTER 5. 53 t = 1 (ξ + η) 2 u x = v (η ξ) 2 ψ(t, x) = ψ{ξ(t, x), η(t, x)} ψ(t, x) t[s] 2 ψ(t, x) t ψ(ξ, η) ξ ψ(ξ, η) = + ξ t η ψ(ξ, η) ψ(ξ, η) = + ξ η 2 ψ(t, x) = η) ( ψ(ξ, + t 2 ξ ξ η t ψ(ξ, η) ) ξ η t + η) ( ψ(ξ, + η ξ = 2 ψ(ξ, η) + 2 ψ(ξ, η) + 2 2 ψ(ξ, η) ξ 2 η 2 ξ η ψ(t, x) x[m] 2 ψ(t, x) x 2 ψ(t, x) x 2 ψ(ξ, η) ξ ψ(ξ, η) η = + ξ x η x = u x ψ(ξ, η) + u x ψ(ξ, η) v ξ v η = ξ ( u x v ψ(ξ, η) ξ + u x v = u2 x v 2 ( 2 ψ(ξ, η) ξ 2 + 2 ψ(ξ, η) η 2 2 2 ψ(ξ, η) ξ η ) ψ(ξ, η) ) η η t ψ(ξ, η) ) ξ η x + η ( u x ψ(ξ, η) + u x v ξ v ψ(ξ, η) ) η η x y[m], z[m] 3 (5.2) 1 ψ(ξ, η) v 2 ( 2 + 2 ψ(ξ, η) ξ 2 + 2 2 ψ(ξ, η) η 2 ξ η ) = u2 x + u 2 y + u 2 z v 2 ( 2 ψ(ξ, η) ξ 2 + 2 ψ(ξ, η) η 2 2 2 ψ(ξ, η) ξ η ) 2 ψ(ξ, η) + 2 ψ(ξ, η) + 2 2 ψ(ξ, η) = 2 ψ(ξ, η) + 2 ψ(ξ, η) 2 2 ψ(ξ, η) ξ 2 η 2 ξ η ξ 2 η 2 ξ η 2 ψ(ξ, η) = 0 ξ η η[s] ψ(ξ, η) ξ = f 1 (ξ) f 1 (ξ) η[s] ξ[s] ξ[s]

CHAPTER 5. 54 ψ(ξ, η) = f 1 (ξ)dξ + f 2(η) ψ(ξ, η) = f 1 (ξ) + f 2 (η) ψ(t, x) = f 1 (t u x v ) + f u x 2(t + v ) f 1, f 2 ξ[s], η[s] 0 3 (5.2) 3 u x ψ(t, x) = f (t v ) u x ψ(t, x) = f (t + v ) ψ(t, x) = f 1 (t u x v ) + f 2(t + u x v ) 3 3 1 3 1 (5.1) x[m] r[m] r = x 2 + y 2 + z 2 ψ(t, r) x 2 ψ(t, r) x 2 ψ(t, r) r = r x = x ψ(t, r) r r = 1 ψ(t, r) + x( 1 r r 2 = 1 ψ(t, r) r r 2x r) ) ψ(t, + x r3 r r x2 ψ(t, r) + x2 2 ψ(t, r) r 3 r r 2 r 2 2 ψ(t, r) r 2 y[m], z[m] 2 ψ(t, r) + 2 ψ(t, r) + 2 ψ(t, r) = 3 ψ(t, r) x2 + y 2 + z 2 ψ(t, r) + x2 + y 2 + z 2 2 ψ(t, r) x 2 y 2 z 2 r r r 3 r r 2 r 2 = 2 1 ψ(t, r) + 2 ψ(t, r) r r r 2 = 1 2 (rψ(t, r)) r r2 r x

CHAPTER 5. 55 3 (5.2) 1 2 ψ(t, r) = 1 v 2 t 2 r 2 (rψ(t, r)) r2 1 2 2 (rψ(t, r)) = (rψ(t, r)) v 2 t2 r2 (5.1) ψ(t, x) rψ(t, r) rψ(t, r) = f 1 (t r v ) + f 2(t + r v ) ψ(t, r) = 1 r f 1(t r v ) + 1 r f 2(t + r v ) 1 v[m/s] 2 v[m/s] r[m] J Simplicity HOME http://www.jsimplicity.com/

Chapter 6 6.1 2 2 ψ 1 (t, x) ψ 2 (t, x) ψ(t, x) ψ(t, x) = ψ 1 (t, x) + ψ 2 (t, x) Chapter 3 1 2 ψ(t, x) = 2 ψ(t, x) (6.1) v 2 t 2 3 ψ(t, x) ψ 1 (t, x) ψ 2 (t, x) 56

CHAPTER 6. 57 Figure 6.1: (6.1) c 1, c 2 ψ(t, x) = c 1 ψ 1 (t, x) + c 2 ψ 2 (t, x) (6.1) 1 2 ψ(t, x) = 1 2 (c 1 ψ 1 (t, x) + c 2 ψ 2 (t, x)) v 2 t 2 v 2 t 2 = 1 v 2 (c 1 2 ψ 1 (t, x) t 2 + c 2 2 ψ 2 (t, x) t 2 ) 1 2 ψ 1 (t, x) 1 2 ψ 2 (t, x) = c 1 + c v 2 t 2 2 v 2 t 2 = c 1 2 ψ 1 (t, x) + c 2 2 ψ 2 (t, x) = 2 (c 1 ψ 1 (t, x) + c 2 ψ 2 (t, x)) = 2 ψ(t, x) N

CHAPTER 6. 58 ψ 1 (t, x). ψ N (t, x) (6.1) c 1,, c N ψ(t, x) = N c i ψ i (t, x) = c 1 ψ 1 (t, x) + + c N ψ N (t, x) i=1 1 2 ψ(t, x) = 1 2 (c 1 ψ 1 (t, x) + + c N ψ N (t, x)) v 2 t 2 v 2 t 2 = 1 v (c 2 ψ 1 (t, x) 2 ψ N (t, x) 2 1 + + c t 2 N ) t 2 1 2 ψ 1 (t, x) 1 2 ψ N (t, x) = c 1 + + c v 2 t 2 N v 2 t 2 = c 1 2 ψ 1 (t, x) + + c N 2 ψ N (t, x) = 2 (c 1 ψ 1 (t, x) + + c N ψ N (t, x)) = 2 ψ(t, x) 6.1 () Section 2 6.2 2 T 8 [s]

CHAPTER 6. 59 2 λ[m] 1 x 2 ψ 1 (t, x) = A sin (ωt kx) ψ 2 (t, x) = A sin (ωt + kx + θ 0 ) ψ 1 (t, x) ψ 2 (t, x) 0[s] ψ 1 (t, x) 0[rad] ψ 2 (t, x) θ 0 [rad] ψ(t, x) ψ(t, x) = ψ 1 (t, x) + ψ 2 (t, x) = A sin (ωt kx) + A sin (ωt + kx + θ 0 ) = 2A sin{ (ωt kx) + (ωt + kx + θ 0) 2 = 2A sin(ωt + θ 0 2 ) cos( kx θ 0 2 ) = 2A cos(kx + θ 0 2 ) sin(ωt + θ 0 2 ) ψ(t, x) 2A cos(kx + θ 0 2 ) } cos{ (ωt kx) (ωt + kx + θ 0) } 2 0[m] 2A cos (kx + θ 0 2 ) = 0 kx + θ 0 2 = nπ + π (n = 0, ±1, ±2, ) 2

CHAPTER 6. 60 2A cos(kx + θ 0 2 ) = ±2A cos(kx + θ 0 2 ) = ±1 kx + θ 0 2 = nπ (n = 0, ±1, ±2, ) 6.3 2 2 S 1 S 2 S 1 S 2 P S 1 Figure 6.3: S 2 S 1 P S 2 P = 2λ λ = λ P S 1 P S 2 P = mλ (m = 0, 1, 2, ) r 1 r 2 = mλ (m = 0, ±1, ±2, ) (6.2) S 1 P r 1, S 2 P r 2 Q S 1 S 2

CHAPTER 6. 61 S 1 Q S 2 Q = λ 3 2 λ = 1 2 λ Q S 1 Q S 2 Q = m λ + λ 2 (m = 0, 1, 2, ) r 1 r 2 = m λ + λ 2 (m = 0, ±1, ±2, ) (6.3) S 1 Q r 1, S 2 Q r 2 2 S 1 S 2 2 S 1 S 2 S 1, S 2 (6.2) (6.3) S 1 S 2 θ(t)[rad] (6.2) (6.3) R S 1 θ R1 (t)[rad] R S 2 θ R2 (t)[rad] 1 2π[rad] θ R1 = θ(t) 2π S 1R λ θ R2 = θ(t) 2π S 2R λ R P θ P1 (t)[rad] θ P2 (t)[rad] 2π[rad] θ P1 (t) θ P2 (t) = 2πm (m = 0, 1, 2, ) {θ(t) 2π S 1P λ } {θ(t) 2πS 2P } = 2πm λ 2π λ S 1P + S 2 P = 2πm S 1 P S 2 P = mλ (m = 0, 1, 2, ) r 1 r 2 = mλ (m = 0, ±1, ±2, ) (6.2) R Q θ Q1 (t)[rad] θ Q2 (t)[rad] 2π[rad] π[rad]

CHAPTER 6. 62 θ Q1 (t) θ Q2 (t) = 2πm + π (m = 0, 1, 2, ) {θ(t) 2π S 1Q λ 2π λ S 1Q + S 2 Q = 2πm + π } {θ(t) 2πS 2Q λ } = 2πm + π S 1 Q S 2 Q = m λ + λ (m = 0, 1, 2, ) 2 r 1 r 2 = m λ + λ 2 (m = 0, ±1, ±2, ) (6.3) 2 3 ψ(t, r) 3 Chapter ψ(t, r) = 1 r f 1(t r v ) + 1 r f 2(t + r v ) r[m] 1 R S 1 ψ 1 (t, r 1 ) S 2 ψ 2 (t, r 2 ) ψ 1 (t, r 1 ) = A r 1 sin (ωt kr 1 ) ψ 2 (t, r 2 ) = A r 2 sin (ωt kr 2 ) A R 2 ψ(t, r 1, r 2 ) ψ(t, r 1, r 2 ) = ψ(t, r 1 ) + ψ(t, r 2 ) = A sin (ωt kr 1 ) + A sin (ωt kr 2 ) r 1 r 2 = A (sin ωt cos kr 1 cos ωt sin kr 1 ) + A (sin ωt cos kr 2 cos ωt sin kr 2 ) r 1 r 2 = ( A cos kr 1 + A cos kr 2 ) sin ωt ( A sin kr 1 + A sin kr 2 ) cos ωt r 1 r 2 r 1 r 2 A r 1 cos kr 1 + A r 2 cos kr 2 = A (r 1, r 2 ) sin δ A r 1 sin kr 1 + A r 2 sin kr 2 = A (r 1, r 2 ) cos δ

CHAPTER 6. 63 ψ(t, r 1, r 2 ) = A (r 1, r 2 )(sin δ sin ωt cos δ cos ωt) = A (r 1, r 2 ) cos(ωt + δ) A (r 1, r 2 )[m] A 2 (r 1, r 2 ) sin 2 δ + A 2 (r 1, r 2 ) cos 2 δ = ( A r 1 cos kr 1 + A r 2 cos kr 2 ) 2 + ( A r 1 sin kr 1 + A r 2 sin kr 2 ) 2 A 2 (r 1, r 2 ) = ( A r 1 ) 2 (sin 2 kr 1 + cos 2 kr 1 ) + ( A r 2 ) 2 (sin 2 kr 2 + cos 2 kr 2 ) + 2 A2 r 1 r 2 (cos kr 1 cos kr 2 + sin kr 1 sin kr 2 ) A 2 (r 1, r 2 ) = ( A r 1 ) 2 + ( A r 2 ) 2 + 2 A2 r 1 r 2 cos k(r 1 r 2 ) A (r 1, r 2 ) = {( A r 1 ) 2 + ( A r 2 ) 2 + 2 A2 r 1 r 2 cos k(r 1 r 2 )} 1 2 (6.4) δ[rad] A (r 1, r 2 ) sin δ A (r 1, r 2 ) cos δ = tan δ = A r 1 cos kr 1 + A r 2 cos kr 2 A r 1 sin kr 1 + A r 2 sin kr 2 A r 1 cos kr 1 + A r 2 cos kr 2 A r 1 sin kr 1 + A r 2 sin kr 2 (6.4) (6.2) cos k(r 1 r 2 ) = 1 2π λ (r 1 r 2 ) = 2πm (m = 0, ±1, ±2, ) r 1 r 2 = mλ (m = 0, ±1, ±2, ) (6.3) cos k(r 1 r 2 ) = 1 2π λ (r 1 r 2 ) = 2πm + π (m = 0, ±1, ±2, ) r 1 r 2 = m λ + λ 2 (m = 0, ±1, ±2, )

CHAPTER 6. 64 6.4 6.5 1 2 2 ψ 1 (t, x) = A sin(ω 1 t k 1 x + θ 1 ) ψ 2 (t, x) = A sin(ω 2 t k 2 x + θ 2 ) ψ(t, x) ψ(t, x) = ψ 1 (t, x) + ψ 2 (t, x) = A{sin(ω 1 t k 1 x + θ 1 ) + sin(ω 2 t k 2 x + θ 2 )} = 2A sin (ω 1t k 1 x + θ 1 ) + (ω 2 t k 2 x + θ 2 ) 2 cos (ω 1t k 1 x + θ 1 ) (ω 2 t k 2 x + θ 2 ) 2 = 2A cos{ 1 2 (ω 1 ω 2 )t 1 2 (k 1 k 2 )x + 1 2 (θ 1 θ 2 )} sin{ 1 2 (ω 1 + ω 2 )t 1 2 (k 1 + k 2 )x + 1 2 (θ 1 + θ 2 )} ω 1 2 (ω 1 ω 2 ) k 1 2 (k 1 k 2 ) θ 1 2 (θ 1 θ 2 ) ω 1 2 (ω 1 + ω 2 ) k 1 2 (k 1 + k 2 ) θ 1 2 (θ 1 + θ 2 ) ψ(t, x) = 2A cos( 1 2 ω t 1 2 k x + 1 θ) sin(ωt kx + θ) 2

CHAPTER 6. 65 A(t, x) 2A cos( 1 2 ω t 1 2 k x + 1 2 θ) ξ(t, x) sin(ωt kx + θ) t[s], x[m] ξ(t, x) A(t, x) Figure 6.4: A(t, x) v = ω k v g [m/s] v g = 1 2 ω 1 2 k = ω k v g = dω dk v = ω k 6.6 Chapter Section 2 f (x) f (x) = a 0 2 + (a n cos nπx n=1 + b n sin nπx )

CHAPTER 6. 66 a n = 1 b n = 1 2 c n = 1 2 f (x) cos nπx dx (n = 0, 1, 2, ) f (x) sin nπx dx (n = 1, 2, ) f (x) = n= c n exp(i nπx ) f (x) exp( i nπx )dx (n =, 2, 1, 0, 1, 2, ) Chapter x t[s] 2 T[s] f (x) ψ(t) ψ(t) = a 0 2 + (a n cos nωt + b n sin nωt) a n = 2 T b n = 2 T ψ(t) = c n = 1 T n=1 T/2 T/2 T/2 T/2 c n e inωt n= T/2 T/2 ψ(t) cos nωt dt (n = 0, 1, 2, ) ψ(t) sin nωt dt (n = 1, 2, ) ψ(t)e inωt dt (n =, 2, 1, 0, 1, 2, ) 1 ψ(t, x) t[s] x[m] x[m] ψ(t, x) = Ae i(ωt kx) = e iωt Ae ikx ϕ(x) = Ae ikx

CHAPTER 6. 67 ϕ(x) x x[m] 2 λ[m] f (x) ψ(t, x) ϕ(x) ϕ(x) ϕ(x) = a 0 2 + (a n cos nπx λ/2 + b n sin nπx λ/2 ) n=1 = a 0 2 + (a n cos n 2π λ x + b n sin n 2π λ x) n=1 = a 0 2 + (a n cos nkx + b n sin nkx) k[rad/m] n=1 k, 2k, 3k, a n = 1 λ/2 a n = 2 λ b n = 1 λ/2 b n = 2 λ λ/2 λ/2 λ/2 λ/2 λ/2 λ/2 λ/2 λ/2 ϕ(x) cos nπx λ/2 dx ϕ(x) cos nkx dx (n = 0, 1, 2, ) ϕ(x) sin nπx λ/2 dx ϕ(x) sin nkx dx (n = 1, 2, ) x x[m] 2 λ[m] f (x) ϕ(x) ϕ(x) = = = c n exp(i nπx λ/2 ) c n exp(in 2π λ x) n= n= c n e inkx n= c n = 1 λ c n = 1 λ λ/2 λ/2 λ/2 λ/2 ϕ(x) exp( i nπx λ/2 )dx ϕ(x)e inkx dx (n =, 2, 1, 0, 1, 2, )

CHAPTER 6. 68 g(k) = 1 2π f (x)e ikx dx f (x) = 1 g(k)e ikx dk 2π k ω[rad/s] x t[s] f (x) ψ(t) g(ω) = 1 ψ(t)e iωt dt 2π ψ(t) ω[rad/s] g(ω) ψ(t) = 1 g(ω)e iωt dω 2π e iωt ω[rad/s] g(ω) ψ(t) ψ(t, x) ϕ(x) k k[rad/m] x x[m] f (x) ϕ(x) g(k) = 1 ϕ(x)e ikx dx 2π ϕ(x) k[rad/m] g(k) ϕ(x) = 1 g(k)e ikx dk 2π e ikx k[rad/m] g(k) ϕ(x) 3 3 g( k) = 1 (2π) 3 3 f ( x) = 1 (2π) 3 f ( x)e i k x dxdydz g( k)e i k x dk x dk y dk z

CHAPTER 6. 69 3 ψ(t, x) ϕ( x) ϕ( x) ψ(t, x) = Ae i(ωt k x) = e iωt Ae i k x ϕ(x) = Ae i k x ϕ( x) k k[rad/m] x x[m] f ( x) ϕ( x) 3 g( k) = 1 (2π) 3 3 ϕ( x) = 1 (2π) 3 ϕ( x)e i k x dxdydz g( k)e i k x dk x dk y dk z J Simplicity HOME http://www.jsimplicity.com/

Chapter 7 7.1 1 1 1 2 x ψ(t, x) T[N] ψ(t, x) θ[rad] T sin θ T tan θ ψ(t, x) = T x ψ(t, x) [ ψ(t, x) ] x=0 = 0 (7.1) x ψ(t, x) (7.1) x < 0 ψ i (t, x) = f 1 (t x v ) ψ r (t, x) = g 1 (t + x v ) 70

CHAPTER 7. 71 ψ(t, x) = f 1 (t x v ) + g 1(t + x v ) ξ t x v η t + x v (7.1) [ ψ(t, x) x ] x=0 = 1 v [ f 1(ξ) ] x=0 + 1 ξ v [ g 1(η) η ] x=0 = 0 [ f 1(ξ) ] x=0 = [ g 1(η) ξ η ] x=0 x = 0[m] ξ = η = t[s] f 1 (t) t = g 1(t) t f 1 (t) = g 1 (t) + const f 1 (t) 0 g 1 (t) = 0 const = 0 f 1 (t) = g 1 (t) t[s] η[s] f 1 (η) = g 1 (η) ψ r (t, x) = g 1 (t + x v ) = f 1 (t + x v ) ψ r (t, x) = ψ i (t, x) x > 0

CHAPTER 7. 72 Figure 7.1: ψ(t, x)[m] ψ(t, x = 0) = 0 (7.2) (7.2) ψ(t, x) ψ(t, x) (7.2) x < 0 ψ i (t, x) = f 1 (t x v ) ψ r (t, x) = g 1 (t + x v ) ψ(t, x) = f 1 (t x v ) + g 1(t + x v ) (7.2) ψ(t, x = 0) = f 1 (t) + g 1 (t) = 0 g 1 (t) = f 1 (t)

CHAPTER 7. 73 t[s] η[s] g 1 (η) = f 1 (η) ψ r (t, x) = g 1 (t + x v ) = f 1 (t + x v ) ψ r (t, x) = ψ i (t, x) x > 0 Figure 7.2: 2 1 x < 0 ψ i (t, x) = f 1 (t x v 1 )

CHAPTER 7. 74 ψ r (t, x) = g 1 (t + x v 1 ) x > 0 ψ t (t, x) = f 2 (t x v 2 ) x < 0 1 v 1 [m/s] x > 0 2 v 2 [m/s] ψ(t, x) 1 2 ψ i (t, x = 0) + ψ r (t, x = 0) = ψ t (t, x = 0) ψ(t, x) 1 2 [ ψ i(t, x) x ] x=0 + [ ψ r(t, x) x ] x=0 = [ ψ t(t, x) ] x=0 x f 1 (t) + g 1 (t) = f 2 (t) (7.3) [ f 1(t, x) x ] x=0 + [ g 1(t, x) x ξ 1 [s], η 1 [s], ξ 2 [s] (7.4) ξ 1 t x v 1 η 1 t + x v 1 ξ 2 t x v 2 ] x=0 = [ f 2(t, x) ] x=0 (7.4) x 1 [ f 1(ξ 1 ) ] x=0 + 1 [ g 1(η 1 ) ] x=0 = 1 [ f 2(ξ 2 ) ] x=0 v 1 ξ 1 v 1 η 1 v 2 ξ 2 x = 0 ξ 1 = η 1 = ξ 2 = t 1 ( f 1(t) v 1 t g 1(t) ) = 1 f 2 (t) t v 2 t 1 v 1 { f 1 (t) g 1 (t)} = 1 v 2 f 2 (t) + C

CHAPTER 7. 75 f 1 (t) 0 g 1 (t) = f 2 (t) = 0 C 0 (7.3) v 2 (7.5) 1 v 1 { f 1 (t) g 1 (t)} = 1 v 2 f 2 (t) (7.5) { f 1 (t) + g 1 (t)} v 2 v 1 { f 1 (t) g 1 (t)} = 0 (1 + v 2 v 1 )g 1 (t) = ( 1 + v 2 v 1 ) f 1 (t) g 1 (t) = v 1 v 2 v 1 + v 2 f 1 (t) (7.3) +v 1 (7.5) 2 f 1 (t) = f 2 (t) + v 1 v 2 f 2 (t) v 1 + v 2 v 2 f 2 (t) = 2 f 1 (t) f 2 (t) = 2v 2 v 1 + v 2 f 1 (t) t[s] η 1 [s] ξ 2 [s] g 1 (t + x v 1 ) = v 1 v 2 v 1 + v 2 f 1 (t + x v 1 ) f 2 (t x v 2 ) = 2v 2 v 1 + v 2 f 1 (t x v 2 ) v 1 [m/s] v 2 [m/s] 7.2 3 3 ψ 1 (t, x) = A sin (ωt k 1 x) ψ 1 (t, x) = B sin (ωt k 1 x) ψ 2 (t, x) = C sin (ωt k 2 x)

CHAPTER 7. 76 k xy xz ψ(t, x) Figure 7.3: ψ 1 (t, x) + ψ 1 (t, x) = ψ 2(t, x) A sin (ωt k 1 x) + B sin (ωt k 1 x) = C sin (ωt k 2 x) 3 ωt k 1 x = ωt k 1 x = ωt k 2 x k 1 x = k 1 x = k 2 x (7.6) (7.6) x xy x = (x, y, 0) xz k 1 = (k 1x, 0, k 1z ) k 1 = (k 1x, k 1y, k 1z ) k 2 = (k 2x, k 2y, k 2z )

CHAPTER 7. 77 (7.6) k 1x x = k 1x x + k 1y y = k 2xx + k 2y y x = (x, y, 0) x[m] y[m] k 1x = k 1x = k 2x (7.7) k 1y = k 2y = 0 (7.8) (7.8) xz 1 2 v 1 [m/s], v 2 [m/s] k 1 = ω v 1 k 1 = ω v 1 k 2 = ω v 2 (7.7) k 1 sin θ 1 = k 1 sin θ 1 = k 2 sin θ 2 ω v 1 sin θ 1 = ω v 1 sin θ 1 = ω v 2 sin θ 2 sin θ 1 v 1 = sin θ 1 v 1 = sin θ 2 v 2 θ 1 = θ 1 sin θ 1 sin θ 2 = v 1 v 2 n 1 2 1 v 1 [m/s] 2 v 2 [m/s] n 1 2 1 2 7.3

CHAPTER 7. 78 1 Figure 7.4: θ 1 [rad] θ 1 [rad] AB Figure 7.5: A 2 DC D ABD DCA ABD = DCA = 90

CHAPTER 7. 79 AD AD = DA B D t[s] t[s] A C v[m/s] BD = CA(= vt) BAD = CDA θ 1 = θ 1 1 2 θ 1 [rad] θ 2 [rad] Figure 7.6: AB A 2 CD D B D t[s] A C

CHAPTER 7. 80 BD sin θ 1 = AD sin θ 2 AC AD = BD AC = v 1t v 2 t = v 1 v 2 n 1 2 n 1 2 = sin θ 1 sin θ 2 = v 1 v 2 = λ 1 λ 2 1 2 J Simplicity HOME http://www.jsimplicity.com/