1B24 Waves Optics and Acoustics Lecture 4 of 33 (UCL)

Page 1

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2

1 0.8 f( x

c .0 )

f( x

c .1 )

0.6

0.4 0.2

0

1

0

1

2

3

4

5

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Consider the Fourier Series for a square wave sq ( x , L ) n . . . . 4 .sin x ( 2 p 1 ) 2 π ser ( n , x , L ) . . L π ( 2 p 1) p=0

( 1)

x

2 .x L

0 , .01 .. 1.99

1.5 1

sq ( x , 1 )

ser ( 0 , x , 1 ) 0.5 ser ( 1 , x , 1 )

0

ser ( 2 , x , 1 ) ser ( 3 , x , 1 )

0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

Close-up:

x

1.8

2

1.0001 , 1.0005 .. 1.1

1.5 sq ( x , 1 )

1

ser ( 30 , x , 1 ) ser ( 100 , x , 1 ) 0.5

0

1

1.02

1.04

1.06

1.08

1.1

x

The Gibbs phenomenon is the concentration of the 'overshoot' of the sudden rise into a region close to the edge of the step.

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