Example 10.  The initial displacement for a fundamental solution is  [Graphics:Images/FrobeniusSeriesMod_gr_521.gif].  
Plot the functions for  n = 1,2,3.  
The first fundamental solution vibrates up and down throughout the entire disk of radius 1.
Solution 10.

The first fundamental solution has a circle of radius  [Graphics:../Images/FrobeniusSeriesMod_gr_522.gif] as a node where there is no vibration and it moves up and down inside this circle.

[Graphics:../Images/FrobeniusSeriesMod_gr_523.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_524.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_525.gif]


The second fundamental solution has a circle of radius  [Graphics:../Images/FrobeniusSeriesMod_gr_526.gif] as a node where there is no vibration and it moves up and down in opposite directions on the inside and outside of this circle.

[Graphics:../Images/FrobeniusSeriesMod_gr_527.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_528.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_529.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_530.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_531.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_532.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_533.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_534.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_535.gif]


The third fundamental solution has a two circular nodes of radius  [Graphics:../Images/FrobeniusSeriesMod_gr_536.gif]  and  [Graphics:../Images/FrobeniusSeriesMod_gr_537.gif]  where there is no vibration and it moves up and down in opposite directions between circles.

[Graphics:../Images/FrobeniusSeriesMod_gr_538.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_539.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_540.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_541.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_542.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_543.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_544.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_545.gif]

[Graphics:../Images/FrobeniusSeriesMod_gr_546.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004