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Unformatted text preview: u ( r, t ) = lim c n =1 J 1 ( n / 2) 2 n cJ 1 ( n ) 2 J ( n r ) sin( n ct ) = n =1 lim c J 1 ( n / 2) 2 n cJ 1 ( n ) 2 J ( n r ) sin( n ct ) = , because lim c J 1 ( n / 2) 2 n cJ 1 ( n ) 2 = 0 and sin( n ct ) is bounded. If we let u 1 ( r, t ) denote the solution corresponding to c = 1 and u c ( r, t ) denote the solution for arbitrary c > 0. Then, it is easy t check that u c ( r, t ) = 1 c u 1 ( r, ct ) . This shows that if c increases, the time scale speeds proportionally to c , while the displacement decreases by a factor of 1 c ....
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.
- Fall '11