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  1. R. W. Wood, Phil. Mag. 16, p. 184; 1908.
    [Crossref]
  2. Rayleigh, Proc. Roy. Soc. (A)  102, p. 190; 1922.
    [Crossref]
  3. Wood and Ellett, Phys. Rev. 24, p. 243; 1924.
    [Crossref]
  4. J. A. Eldridge, Phys. Rev. 24, p. 234; 1924.
    [Crossref]
  5. A. Pringsheim, Zeitschr. f. Phys. 23 p. 324; 1924.
    [Crossref]
  6. Gaviola and Pringsheim, ZS. f. Phys. 25, p. 367; 1924.
    [Crossref]
  7. G. Joos, Phys. ZS. 25, p. 130; 1924.
  8. G. Breit, Phil. Mag. 47, p. 832; 1924.
    [Crossref]
  9. W. Wien, Ann. der Physik. 73, p. 495; 1924.
  10. The writer has developed a treatment of phenomena of dispersion in the absorption band which shows that such relations must be considered and that if they are taken into account a satisfactory explanation of the finite refractive index is obtained without making use of airictional term in the equation of motion of the resonator.(See J. Wash. Acad.,  15 p. 3G, 1925.)In fact if such a calculation is performed then it is found that an interruption of length T contributes to Z and Y on the average the amountsz=Ez04mΩ{(TΩ1−ω0−sin(Ω1−ω0)T(Ω1−ω0)2+TΩ2−ω0−sin(Ω2−ω0)T(Ω2−ω0)2)cos(ω0t−∊)+2(sin2(Ω1−ω02T)(Ω1−ω0)2+sin2(Ω2−ω02T)(Ω2−ω0)2)sin(ω0t−∊}y=Ez04mΩ{[(TΩ1−ω0−sin(Ω1−ω0)T(Ω1−ω0)2)−(TΩ2−ω0−sin(Ω2−ω0)T(Ω2−ω0)2)]sin(ω0t−∊)−2(sin2(Ω1−ω02T)(Ω1−ω0)2−sin2(Ω1−ω02T)(Ω2−ω0)2)cos(ω0t−∊)}When these values are averaged by multiplying by e−T/T0d(TT0) we obtainz¯=Ez04mΩT02{(T0(Ω1−ω0)1+T02(Ω1−ω0)2+T0(Ω2−ω0)1+T02(Ω2−ω0)2)cos(ω0t−∊)+(11+T02(Ω1−ω0)2+11+T02(Ω2−ω0)2)sin(ω0t−∊)}y¯=Ez04mΩT02{(T0(Ω1−ω0)1+T02(Ω1−ω0)2−T0(Ω2−ω0)1+T02(Ω2−ω0)2)sin(ω0t−∊)−(11+T02(Ω1−ω0)2−11+T02(Ω2−ω0)2)cos(ω0t−∊)}This corresponds to (3.1), (3.2) of the damped vibrator treatment. Hence the result must be also the same so far as P and α are concerned.L. Silberstein has predicted in a general way many of the phenomena discussed here, a number of years ago (Phil. Mag. 32, p. 2731916).

1925 (1)

The writer has developed a treatment of phenomena of dispersion in the absorption band which shows that such relations must be considered and that if they are taken into account a satisfactory explanation of the finite refractive index is obtained without making use of airictional term in the equation of motion of the resonator.(See J. Wash. Acad.,  15 p. 3G, 1925.)In fact if such a calculation is performed then it is found that an interruption of length T contributes to Z and Y on the average the amountsz=Ez04mΩ{(TΩ1−ω0−sin(Ω1−ω0)T(Ω1−ω0)2+TΩ2−ω0−sin(Ω2−ω0)T(Ω2−ω0)2)cos(ω0t−∊)+2(sin2(Ω1−ω02T)(Ω1−ω0)2+sin2(Ω2−ω02T)(Ω2−ω0)2)sin(ω0t−∊}y=Ez04mΩ{[(TΩ1−ω0−sin(Ω1−ω0)T(Ω1−ω0)2)−(TΩ2−ω0−sin(Ω2−ω0)T(Ω2−ω0)2)]sin(ω0t−∊)−2(sin2(Ω1−ω02T)(Ω1−ω0)2−sin2(Ω1−ω02T)(Ω2−ω0)2)cos(ω0t−∊)}When these values are averaged by multiplying by e−T/T0d(TT0) we obtainz¯=Ez04mΩT02{(T0(Ω1−ω0)1+T02(Ω1−ω0)2+T0(Ω2−ω0)1+T02(Ω2−ω0)2)cos(ω0t−∊)+(11+T02(Ω1−ω0)2+11+T02(Ω2−ω0)2)sin(ω0t−∊)}y¯=Ez04mΩT02{(T0(Ω1−ω0)1+T02(Ω1−ω0)2−T0(Ω2−ω0)1+T02(Ω2−ω0)2)sin(ω0t−∊)−(11+T02(Ω1−ω0)2−11+T02(Ω2−ω0)2)cos(ω0t−∊)}This corresponds to (3.1), (3.2) of the damped vibrator treatment. Hence the result must be also the same so far as P and α are concerned.L. Silberstein has predicted in a general way many of the phenomena discussed here, a number of years ago (Phil. Mag. 32, p. 2731916).

1924 (7)

Wood and Ellett, Phys. Rev. 24, p. 243; 1924.
[Crossref]

J. A. Eldridge, Phys. Rev. 24, p. 234; 1924.
[Crossref]

A. Pringsheim, Zeitschr. f. Phys. 23 p. 324; 1924.
[Crossref]

Gaviola and Pringsheim, ZS. f. Phys. 25, p. 367; 1924.
[Crossref]

G. Joos, Phys. ZS. 25, p. 130; 1924.

G. Breit, Phil. Mag. 47, p. 832; 1924.
[Crossref]

W. Wien, Ann. der Physik. 73, p. 495; 1924.

1922 (1)

Rayleigh, Proc. Roy. Soc. (A)  102, p. 190; 1922.
[Crossref]

1908 (1)

R. W. Wood, Phil. Mag. 16, p. 184; 1908.
[Crossref]

Breit, G.

G. Breit, Phil. Mag. 47, p. 832; 1924.
[Crossref]

Eldridge, J. A.

J. A. Eldridge, Phys. Rev. 24, p. 234; 1924.
[Crossref]

Ellett,

Wood and Ellett, Phys. Rev. 24, p. 243; 1924.
[Crossref]

Gaviola,

Gaviola and Pringsheim, ZS. f. Phys. 25, p. 367; 1924.
[Crossref]

Joos, G.

G. Joos, Phys. ZS. 25, p. 130; 1924.

Pringsheim,

Gaviola and Pringsheim, ZS. f. Phys. 25, p. 367; 1924.
[Crossref]

Pringsheim, A.

A. Pringsheim, Zeitschr. f. Phys. 23 p. 324; 1924.
[Crossref]

Rayleigh,

Rayleigh, Proc. Roy. Soc. (A)  102, p. 190; 1922.
[Crossref]

Wien, W.

W. Wien, Ann. der Physik. 73, p. 495; 1924.

Wood,

Wood and Ellett, Phys. Rev. 24, p. 243; 1924.
[Crossref]

Wood, R. W.

R. W. Wood, Phil. Mag. 16, p. 184; 1908.
[Crossref]

Ann. der Physik. (1)

W. Wien, Ann. der Physik. 73, p. 495; 1924.

J. Wash. Acad. (1)

The writer has developed a treatment of phenomena of dispersion in the absorption band which shows that such relations must be considered and that if they are taken into account a satisfactory explanation of the finite refractive index is obtained without making use of airictional term in the equation of motion of the resonator.(See J. Wash. Acad.,  15 p. 3G, 1925.)In fact if such a calculation is performed then it is found that an interruption of length T contributes to Z and Y on the average the amountsz=Ez04mΩ{(TΩ1−ω0−sin(Ω1−ω0)T(Ω1−ω0)2+TΩ2−ω0−sin(Ω2−ω0)T(Ω2−ω0)2)cos(ω0t−∊)+2(sin2(Ω1−ω02T)(Ω1−ω0)2+sin2(Ω2−ω02T)(Ω2−ω0)2)sin(ω0t−∊}y=Ez04mΩ{[(TΩ1−ω0−sin(Ω1−ω0)T(Ω1−ω0)2)−(TΩ2−ω0−sin(Ω2−ω0)T(Ω2−ω0)2)]sin(ω0t−∊)−2(sin2(Ω1−ω02T)(Ω1−ω0)2−sin2(Ω1−ω02T)(Ω2−ω0)2)cos(ω0t−∊)}When these values are averaged by multiplying by e−T/T0d(TT0) we obtainz¯=Ez04mΩT02{(T0(Ω1−ω0)1+T02(Ω1−ω0)2+T0(Ω2−ω0)1+T02(Ω2−ω0)2)cos(ω0t−∊)+(11+T02(Ω1−ω0)2+11+T02(Ω2−ω0)2)sin(ω0t−∊)}y¯=Ez04mΩT02{(T0(Ω1−ω0)1+T02(Ω1−ω0)2−T0(Ω2−ω0)1+T02(Ω2−ω0)2)sin(ω0t−∊)−(11+T02(Ω1−ω0)2−11+T02(Ω2−ω0)2)cos(ω0t−∊)}This corresponds to (3.1), (3.2) of the damped vibrator treatment. Hence the result must be also the same so far as P and α are concerned.L. Silberstein has predicted in a general way many of the phenomena discussed here, a number of years ago (Phil. Mag. 32, p. 2731916).

Phil. Mag. (2)

R. W. Wood, Phil. Mag. 16, p. 184; 1908.
[Crossref]

G. Breit, Phil. Mag. 47, p. 832; 1924.
[Crossref]

Phys. Rev. (2)

Wood and Ellett, Phys. Rev. 24, p. 243; 1924.
[Crossref]

J. A. Eldridge, Phys. Rev. 24, p. 234; 1924.
[Crossref]

Phys. ZS. (1)

G. Joos, Phys. ZS. 25, p. 130; 1924.

Proc. Roy. Soc. (1)

Rayleigh, Proc. Roy. Soc. (A)  102, p. 190; 1922.
[Crossref]

Zeitschr. f. Phys. (1)

A. Pringsheim, Zeitschr. f. Phys. 23 p. 324; 1924.
[Crossref]

ZS. f. Phys. (1)

Gaviola and Pringsheim, ZS. f. Phys. 25, p. 367; 1924.
[Crossref]

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Equations (72)

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P = P 0 e K H
E z = E z cos ω t E y = E z sin ω t
E z + i E y = E z e i ω t
r ¨ + κ + Ω 2 r = E m
r ¨ + κ + Ω 2 r = E m
ξ = z + i y ξ = z + i y
ξ ¨ + κ ξ ˙ + Ω 2 ξ = E z m e i ω t
E z = E z 0 cos ( ω 0 t )
ξ = ξ e i ω t = E z 0 2 m [ { Ω 2 ( ω 0 ω ) 2 + i κ ( ω 0 ω ) } 1 e i ( ω 0 t ) + { Ω 2 ( ω 0 + ω ) 2 i κ ( ω 0 ω ) } 1 e i ( ω 0 t ) ]
z = E z 0 2 m [ Ω 2 ω 1 2 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 + Ω 2 ω 2 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] cos ( ω 0 t ) + E z 0 2 m [ κ ω 1 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 + κ ω 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] sin ( ω 0 t )
y = E z 0 2 m [ Ω 2 ω 1 2 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 Ω 2 ω 2 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] sin ( ω 0 t ) + E z 0 2 m [ κ ω 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 κ ω 1 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 ] cos ( ω 0 t )
ω 1 = ω 0 ω ω 2 = ω 0 + ω
Z ( ω 0 ) = ( E z 0 2 m ) 2 [ Ω 2 ω 1 2 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 + Ω 2 ω 2 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] 2 + ( E z 0 2 m ) 2 [ κ ω 1 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 + κ ω 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] 2
Y ( ω 0 ) = ( E z 0 2 m ) 2 [ Ω 2 ω 1 2 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 Ω 2 ω 2 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] 2 + ( E z 0 2 m ) 2 [ κ ω 1 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 κ ω 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] 2
Z ( ω 0 ) Y ( ω 0 ) = 4 ( E z 0 2 m ) 2 ( Ω 2 ω 1 2 ) ( Ω 2 ω 2 2 ) + κ 2 ω 1 ω 2 [ ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 ] [ ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ]
Z ( ω 0 ) + Y ( ω 0 ) = 2 ( E z 0 2 m ) 2 [ { ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 } 1 + { ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 } 1 ]
Z = 0 Z ( ω 0 ) d ω 0 , Y = 0 Y ( ω 0 ) d ω 0
[ Ω 2 ω 1 2 + i κ ω 1 ] 1 [ Ω 2 ω 2 2 + i κ ω 2 ] 1 = ( Ω 2 ω 1 2 ) ( Ω 2 ω 2 2 ) + κ 2 ω 1 ω 2 + i κ ( ω 2 ω 1 ) [ Ω 2 + ω 1 ω 2 ] [ ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 ] [ ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ]
0 [ ( x a ) 2 b 2 ] 1 [ ( x + a ) 2 b 2 ] 1 d x = [ 1 8 a b ( a + b ) log x ( a + b ) x + ( a + b ) + 1 8 a b ( a b ) log x + ( a b ) x ( a b ) ] 0
a = ω + i κ 2 , b = Ω 2 κ 2 4
0 [ Ω 2 ω 1 2 + i κ ω 1 ] 1 [ Ω 2 ω 2 2 i κ ω 2 ] 1 d ω 0 = π 4 κ 2 ( Ω 2 3 ω 2 ) + i ω ( Ω 2 + κ 2 2 ω 2 ) ( ω 2 + κ 2 4 ) [ ( Ω 2 ω 2 ) 2 + κ 2 ω 2 ]
0 [ Z ( ω 0 ) Y ( ω 0 ) ] d ω 0 = π ( E z 0 2 m ) 2 κ 2 ( Ω 2 3 ω 2 ) ( ω 2 + κ 2 4 ) [ ( Ω 2 ω 2 ) 2 + κ 2 ω 2 ]
0 [ Z ( ω 0 ) + Y ( ω 0 ) ] d ω 0 .
0 [ { ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 } 1 + { ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 } 1 ] d ω 0 = 2 0 [ ( Ω 2 ω 0 2 ) 2 + κ 2 ω 0 2 ] 1 d ω 0
0 [ Z ( ω 0 ) + Y ( ω 0 ) ] d ω 0 = 2 π κ Ω 2 ( E z 0 2 m ) 2
P = Z Y Z + Y = ( 1 + 4 ω 2 κ 2 ) 1 ( 1 3 ω 2 Ω 2 ) [ ( 1 ω 2 Ω 2 ) 2 + κ 2 ω 2 Ω 4 ] 1
P = Z Y Z + Y = ( 1 + 4 ω 2 κ 2 ) 1
Z ( ω 0 ) Y ( ω 0 ) Z ( ω 0 ) + Y ( ω 0 ) = ( Ω ω 0 + ω ) ( Ω ω 0 ω ) + κ 2 4 1 2 ( Ω ω 0 + ω ) 2 + 1 2 ( Ω ω 0 ω ) 2 + κ 2 4
Z ( ω 0 ) Y ( ω 0 ) Z ( ω 0 ) + Y ( ω 0 ) = 1 4 ω 2 κ 2 1 + 4 ω 2 κ 2
z = A z cos ( ω 0 t ) + B z sin ( ω 0 t ) y = A y cos ( ω 0 t ) + B y sin ( ω 0 t )
A z = E z 0 2 m [ Ω 2 ω 1 2 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 + Ω 2 ω 2 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ]
B z = E z 0 2 m [ κ ω 1 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 + κ ω 2 ( Ω 2 ω 2 2 ) 2 κ 2 ω 2 2 ]
A y = E z 0 2 m [ κ ω 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 κ ω 1 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 ]
B y = E z 0 2 m [ Ω 2 ω 1 2 ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 Ω 2 ω 2 2 ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ]
z ¯ = z cos α + y sin α y ¯ = z sin α + y cos α
Z ¯ = ( A z 2 + B z 2 A y 2 B y 2 ) cos 2 α + 2 ( A y A z + B y B z ) sin 2 α
Z ¯ + = A y 2 + A z 2 + B y 2 + B z 2
tan 2 α = 2 ( A y A z + B y B z ) ( A z 2 + B z 2 A y 2 B y 2 ) 1
A y A z + B y B z = 2 κ ( ω 2 ω 1 ) ( Ω 2 + ω 1 ω 2 ) [ ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 ] [ ( Ω 2 ω 2 2 ) 2 + κ 2 ω 2 2 ] ( E z 0 2 m ) 2
A z 2 + B z 2 A y 2 B y 2 = 4 [ ( Ω 2 ω 1 2 ) ( Ω 2 ω 2 2 ) + κ 2 ω 1 ω 2 ] [ ( Ω 2 ω 1 2 ) 2 + κ 2 ω 1 2 ] [ ( Ω 2 ω 2 ) 2 + κ 2 ω 2 2 ] ( E z 0 2 m ) 2
tan 2 α = 2 κ ω ( Ω 2 + ω 1 ω 2 ) ( Ω 2 ω 1 2 ) ( Ω 2 ω 2 2 ) + κ 2 ω 1 ω 2
0 ( A y A z + B y B z ) d ω 0 = π 2 ω ( Ω 2 + κ 2 2 ω 2 ) ( ω 2 + κ 2 4 ) [ ( Ω 2 ω 2 ) 2 + κ 2 ω 2 ] ( E z 0 2 m ) 2
0 ( A z 2 + B z 2 A y 2 B y 2 ) d ω 0 = π κ 2 ( Ω 2 3 ω 2 ) ( ω 2 + κ 2 4 ) [ ( Ω 2 ω 2 ) 2 + κ 2 ω 2 ] ( E z 0 2 m ) 2
tan 2 α = 2 ω ( Ω 2 + κ 2 2 ω 2 ) κ 1 ( Ω 2 3 ω 2 ) 1 2 ω κ 1
τ = κ 2
ω = β H ( β = e 2 m c )
Z = 0 f ( t ) cos 2 ( β H t ) d t
Y = 0 f ( t ) sin 2 ( β H t ) d t
P = Z Y Z + Y = 0 f ( t ) cos ( 2 β H t ) d t / 0 f ( t ) d t
f ( t ) = ( t 2 + κ 2 4 β 2 ) 1
r ¨ + Ω 2 = E m
Z = E z 0 4 m { cos ( ( Ω + ω ) t ) Ω ( Ω ω 0 + ω ) cos [ ( Ω + ω ) t + ] Ω ( Ω + ω + ω 0 ) cos [ ( Ω ω ) t + ] Ω ( Ω + ω 0 ω ) cos [ ( Ω ω ) t ] Ω ( Ω ω ω 0 ) + 2 [ ( Ω 2 ( ω 0 ω ) 2 ) 1 + ( Ω 2 ( ω 0 + ω ) 2 ) 1 ] cos ( ω 0 t ) } .
y = E z 0 4 m { sin ( ( Ω + ω ) t ) Ω ( Ω ω 0 + ω ) sin ( ( Ω + ω ) t + ) Ω ( Ω + ω 0 + ω ) + sin ( ( Ω ω ) t + ) Ω ( Ω ω 0 ω ) + sin ( ( Ω ω ) t ) Ω ( Ω ω 0 ω ) + 2 [ ( Ω 2 ( ω 0 ω ) 2 ) 1 ( Ω 2 ( ω + ω 0 ) 2 ) 1 ] sin ( ω 0 t ) }
0 T cos 2 ( Ω t ) d t ¯ = T 2
0 T cos ( Ω 1 t 1 ) cos ( Ω 2 t 2 ) d t ¯ = = ( Ω 1 + Ω 2 ) 1 sin ( Ω 1 + Ω 2 2 T ) cos ( Ω 1 + Ω 2 2 T 1 2 ) ¯ + ( Ω 1 Ω 2 ) 1 sin ( Ω 1 Ω 2 2 T ) cos ( Ω 1 Ω 2 2 T 2 + 1 ) ¯
0 T sin ( Ω 1 t 1 ) sin ( Ω 2 t 2 ) d t ¯ = = ( Ω 1 + Ω 2 ) 1 sin ( Ω 1 + Ω 2 2 T ) cos ( Ω 1 + Ω 2 2 T 1 2 ) ¯ + ( Ω 1 Ω 2 ) 1 sin ( Ω 1 Ω 2 2 T ) cos ( Ω 1 Ω 2 2 T 2 + 1 ) ¯
z = E z 0 4 m { cos ( ( Ω + ω ) t ) Ω ( Ω ω 0 + ω ) cos ( ( Ω ω ) t ) Ω ( Ω ω ω 0 ) + 2 [ ( Ω 2 ( ω 0 ω ) 2 ) 1 + ( Ω 2 ( ω 0 + ω ) 2 ) 1 ] cos ( ω 0 t ) }
y = E z 0 4 m { sin ( ( Ω + ω ) t ) Ω ( Ω ω 0 + ω ) + sin ( ( Ω ω ) t ) Ω ( Ω ω ω 0 ) + 2 [ ( Ω 2 ( ω 0 ω ) 2 ) 1 ( Ω 2 ( ω 0 + ω ) 2 ) 1 ] sin ( ω 0 t )
Ω + ω = Ω 1 , Ω ω = Ω 2
( 4 m E z 0 ) 2 0 T ( z 2 y 2 ) d t ¯ = 8 T ( Ω 1 2 ω 0 2 ) ( Ω 2 2 ω 0 2 ) + 2 sin ( 2 ω T ( 2 ω ) Ω 2 ( Ω 1 ω 0 ) ( Ω 2 ω 0 ) 4 Ω ( Ω 1 ω 0 ) ( Ω 2 ω 0 ) ( ( Ω 1 2 ω 0 2 ) 1 sin ( Ω 1 ω 0 ) T ( Ω 2 2 ω 0 2 ) 1 sin ( Ω 2 ω 0 ) T )
( 4 m E z 0 ) 2 0 T ( z 2 + y 2 ) d t = T [ Ω 2 ( ( Ω 1 ω 0 ) 2 + ( Ω 2 ω 0 ) 2 ) + 4 ( Ω 1 ω 0 ) 2 ( Ω 2 + ω 0 ) 2 + 4 ( Ω 1 + ω 0 ) 2 ( Ω 2 ω 0 ) 2 ) ] 4 Ω 1 [ ( Ω 2 + ω 0 ) 1 ( Ω 1 ω 0 ) 1 sin ( Ω 1 ω 0 ) T + ( Ω 1 + ω 0 ) 1 ( Ω 2 ω 0 ) 1 sin ( Ω 2 ω 0 ) T ) ]
( 4 m E z 0 ) 2 0 T ( z 2 y 2 ) d t ¯ = 2 Ω 2 ( Ω 1 ω 0 ) 1 ( Ω 2 ω 0 ) 1 [ T + sin 2 ω T 2 ω sin ( Ω 1 ω 0 ) T Ω 1 ω 0 sin ( Ω 2 ω 0 ) T Ω 2 ω 0 ]
( 4 m E z 0 ) 2 0 T ( z 2 + y 2 ) d t ¯ = 2 Ω 2 ( Ω 1 ω 0 ) 2 [ T sin ( Ω 1 ω 0 ) T Ω 1 ω 0 ] + 2 Ω 2 ( Ω 2 ω 0 ) 2 [ T sin ( Ω 2 ω 0 ) T Ω 2 ω 0 ]
( 4 m E z 0 ) 2 0 ( 0 T ( z 2 y 2 ) d t ) d ω 0 = 2 π sin 2 ( ω T ) Ω 2 ω 2
( 4 m E z 0 ) 2 0 ( 0 T ( z 2 y 2 ) d t ) d ω 0 = 2 π T 2 Ω 2
Z Y Z + Y = sin 2 ( ω T ) ω 2 T 2
e T / T 0
Z Y Z + Y = 1 1 + 4 ω 2 T 0 2
d N = f ( T ) d T
Z Y Z + Y = 0 ω 2 sin 2 ( ω T ) f ( T ) d T / 0 T 2 f ( T ) d T
z=Ez04mΩ{(TΩ1ω0sin(Ω1ω0)T(Ω1ω0)2+TΩ2ω0sin(Ω2ω0)T(Ω2ω0)2)cos(ω0t)+2(sin2(Ω1ω02T)(Ω1ω0)2+sin2(Ω2ω02T)(Ω2ω0)2)sin(ω0t}y=Ez04mΩ{[(TΩ1ω0sin(Ω1ω0)T(Ω1ω0)2)(TΩ2ω0sin(Ω2ω0)T(Ω2ω0)2)]sin(ω0t)2(sin2(Ω1ω02T)(Ω1ω0)2sin2(Ω1ω02T)(Ω2ω0)2)cos(ω0t)}
z¯=Ez04mΩT02{(T0(Ω1ω0)1+T02(Ω1ω0)2+T0(Ω2ω0)1+T02(Ω2ω0)2)cos(ω0t)+(11+T02(Ω1ω0)2+11+T02(Ω2ω0)2)sin(ω0t)}y¯=Ez04mΩT02{(T0(Ω1ω0)1+T02(Ω1ω0)2T0(Ω2ω0)1+T02(Ω2ω0)2)sin(ω0t)(11+T02(Ω1ω0)211+T02(Ω2ω0)2)cos(ω0t)}