TABLES  OF LOWEST SPHERICAL CAVITY RESONANT MODES

 

Properties of H Wave Modes:-

 

For resonance BesselJ(p+1/2, R) = 0 where R =    

      Resonant frequency (f Hz)  =        where R has one of the values in the table below.

     

Rotational angular frequency (Ω radians/sec) =   =

     

Phase velocity = Ω rad.

 

STATIONARY AND SPINNING MODES

SPINNING

MODE ONLY

MODE

For 83.4 cm diameter cavity oscillating in first radial mode

 

p

(θ term)

Radial Harmonics(R)

n

(ф term)

Resonant Frequency (MHz)

Rotational Speed

Phase Velocity

1

2

3

0

Not Allowed

0

--

Not Allowed

0

Not Allowed

1

--

Not Allowed

1

4.49

7.721

10.9

0

514.1

Not Allowed

1

4.49

7.72

10.9

1

514.1

514.1

4.49c

1

Not Allowed

2

--

Not Allowed

2

5.76

9.1

12.3

0

659.5

Not Allowed

2

5.76

9.1

12.3

1

659.5

659.5

5.76c

2

5.76

9.1

12.3

2

659.5

329.7

2.88c

2

Not Allowed

3

--

Not Allowed

 

 

 

 

 

 

 

 

 

 

 


 

Properties of E Wave Modes:-

 

     

For resonance BesselJ(p+1/2, R)-BesselJ(p-1/2,R) = 0 where R =    

      Resonant frequency (f Hz)  =        where R has one of the values in the table below.

     

Rotational angular frequency (Ω radians/sec) =   =

     

Phase velocity = Ω rad.

 

STATIONARY AND SPINNING MODES

SPINNING

MODE ONLY

MODE

For 83.4 cm diameter cavity oscillating in first radial mode

 

p

(θ term)

Radial Harmonics(R)

n

(ф term)

Resonant Frequency (MHz)

Rotational Speed

Phase Velocity

1

2

3

0

Not Allowed

0

--

Not Allowed

0

Not Allowed

1

--

Not Allowed

1

2.74

6.12

9.32

0

313.7

Not Allowed

1

2.74

6.12

9.32

1

313.7

313.7

2.74c

1

Not Allowed

2

--

Not Allowed

2

3.87

7.44

10.72

0

443.1

Not Allowed

2

3.87

7.44

10.72

1

443.1

443.1

3.87c

2

3.87

7.44

10.72

2

443.1

221.55

1.935c

2

Not Allowed

3

--

Not Allowed

 

 

 

 

 

 

 

 

 

 

 

 


GENERAL NOTES:-

 

1     This mode is TE120 (i.e. Derived from the numbers in the p column, the R column heading and the n column). 7.72 is the second value of R that makes the BesselJ function zero.

 

2.   The symbols used above are as follows:-

p, n---- Can take the values 0, 1, 2, 3……etc and are the integers defining the harmonic solutions. (Sometimes called the eigenvalues).

c------- The velocity of light.

ω------ The resonant angular frequency.

Ω------ The rotational angular frequency.

f            The resonant frequency. (Where f = ) . For the spinning field this will be the frequency measured by a probe and will be .

rad----- The radius of the spherical cavity.

φ------- The coordinate measuring the angle of rotation about the z axis measured from the x axis (in the initial direction of the y axis).

θ-------  The coordinate angle to the z axis.

 

 

 

                                                

 

 

 

 

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