−10 −8 −6 −4 −2 0 2 4 6 8 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t

Am

plit

ude

Paul Mother Wavelet

Real Part Imaginary PartMagnitude

−10 −5 0 5 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time

Magni

tude

Real Part of Three Scaled Wavelets

a = 1 a = 5 a = 10

−10 −5 0 5 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

TimeMa

gnitud

e

Imaginary Part of Three Scaled Wavelets

a = 1 a = 5 a = 10

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Frequency

Am

plit

ude

Fourier Transform of Paul Wavelet (m=1)

−10 −5 0 5 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (sec)

Amplit

ude

Real Part − Half Interval Sampled

−10 −5 0 5 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)Am

plitude

Imaginary Part − Half Interval Sampled

2Tml

2Tml

-6 -4 -2 0 2 4 6-0.5

0

0.5

t

2Tml

2Tsp

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

WW*/a

ln a

Time (sec.)1 2 3 4 5

0

0.5

1

1.5

2

2.5

3

3.5

10-1

100

101

100

101

102

Scale where WW*/a is maximized vs. Frequency (Hz)

scal

e (a

)

Input signal (Hz)

actual calculated

−600 −400 −200 0 200 400 600−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time (sec.)

Am

plit

ud

e

Real part of Paul Wavelet used as input plotted with peak scaled wavelet

Input Signal Peak scaled wavelet

−600 −400 −200 0 200 400 600−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time (sec.)

Am

plit

ude

Imaginary part of Paul Wavelet used as input plotted with peak scaled wavelet

Input Signal Peak scaled wavelet

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7Gaussian Test Function

Time (sec.)

Am

plit

ude

0

0.5

1

1.5

2

2.5|W| Paul Wavelet

ln a

Time (sec.)-10 -5 0 5 100

1

2

3

4

5

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time

Am

plit

ude

Gaussian Test Function

0

0.5

1

1.5

2

2.5

|W| Paul wavelet

ln a

Time (sec.)-10 -5 0 5 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5|W| Paul Wavelet

ln a

Time (sec.)-10 -5 0 5 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−6 −4 −2 0 2 4 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t (nondimensional)

Am

plit

ud

e

(mother) Paul wavelet with Tc marked

Real Part Imaginary PartMagnitude

−6 −4 −2 0 2 4 6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional Time

Am

plitu

de

Mother Morlet Wavelet

Real Part Imaginary PartMagnitude

−6 −4 −2 0 2 4 6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Nondimensional Time

Am

plitu

de

Real Part T

c

Sampling Points

2T

Tc

0 100 200 300 400 500 600 700 800 900−20

−15

−10

−5

0

5

10

15

Time (sec.)

Vel

ocity

(m/s

)

Velocity data with mean = 0

0 100 200 300 400 500 600 700 800 900−7

−6

−5

−4

−3

−2

−1

0

1

2

Time(sec.)

Cp

Pressure

350 400 450 500 550 600 650−15

−10

−5

0

5

10

15

Time (sec.)

Vel

ocity

(m/s

)

Windowed velocity data with mean = 0

10

20

30

40

50

60

70

80

90

100

110

Wavelet Transform Magnitude

Time (sec.)

ln a

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

0

10

20

30

40

50

60

Wavelet Transform Magnitude

Time (sec.)

ln a

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

350 400 450 500 550 600 650

−10

−5

0

5

10

Time (sec.)

Vel

ocity

(m/s

)

Windowed velocity data with mean = 0 and real scaled Morlet wavelet (a=221)

Windowed SignalScaled Wavelet

20

40

60

80

100

120

140

|W| velocity Paul

ln a

Time (sec.)100 200 300 400 500 600 700 800

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

10

20

30

40

50

60

70

80

90

|W| Paul Wavelet

ln a

Time (sec.)100 200 300 400 500 600 700 800

0

1

2

3

4

5

350 400 450 500 550 600 650−15

−10

−5

0

5

10

15Windowed velocity data with mean = 0 and real scaled Paul wavelet (a=55)

Time (sec.)

Vel

ocity

(m/s

)

Windowed SignalScaled Wavelet

350 400 450 500 550 600 650−7

−6

−5

−4

−3

−2

−1

0

1Windowed pressure data with mean =0

Time sec.

Cp

5

10

15

20

25

30

35

40

Wavelet Transform Magnitude

Time (sec.)

ln a

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

0

5

10

15

20

25

Wavelet Transform Magnitude

Time (sec.)

ln a

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

350 400 450 500 550 600 650

−6

−4

−2

0

2

4

Time (Sec.)

Cp

Windowed Pressure Data mean = 0 & Peak Scale Real Part Morlet Wavelet a=221

Windowed SignalScaled Wavelet

5

10

15

20

25

30

35

40

45

50

55

|W| Paul Wavelet

ln a

Time (sec.)100 200 300 400 500 600 700 800

0

1

2

3

4

5

0

5

10

15

20

25

30

35

40

45

|W| Paul Wavelet

ln a

Time (sec.)100 200 300 400 500 600 700 800

0

1

2

3

4

5

350 400 450 500 550 600 650

−6

−5

−4

−3

−2

−1

0

Time (sec.)

Cp

Windowed Pressure Data; mean =0 & Imaginary Part of Paul Scaled Wavelet a = 62.8

Windowed SignalScaled Wavelet

0 200 400 600 800 1000 1200 1400 16000

5

10

15

20

25

30

35

window width σ (∆ t = 1)

Eve

nt D

urat

ion

(sec

.)Sensativity to window width, Velocity data

Paul Wavelet Morlet Wavelet

σ = 500

0 200 400 600 800 1000 1200 1400 16000

5

10

15

20

25

30

35

40

window width σ (∆ t = 1)

Eve

nt D

urat

ion

(sec

.)

Sensativity to window width, Pressure data

Paul Wavelet Morlet Wavelet

σ = 175

490 495 500 505 510 515 520 525 530 535 540−7

−6

−5

−4

−3

−2

−1

0

1

Time (sec.)

Cp

Close up of event and two Paul wavelets

Raw Signal (mean=0)Paul T

c’ = 22.6s

Paul; Tc’ = 35.0s

89.2 89.3 89.4 89.5 89.6 89.7 89.8 89.9 90 90.1−14

−12

−10

−8

−6

−4

−2

0Raw data from configuration two

Time (sec.)

Cp

Mean − 3*std. dev.

89.58 89.6 89.62 89.64 89.66 89.68 89.7 89.72 89.74

−2

−1.5

−1

−0.5

0

0.5

Selectively Sampled Data from configuration two

Time (sec.)

Cp −

(C

p −

Cp ′

)

Peak Begins

Peak Ends

0

1

2

3

4

5

6

|W|

Dur

atio

n (s

ec.)

Time (sec.)89.2 89.4 89.6 89.8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

89.58 89.6 89.62 89.64 89.66 89.68 89.7 89.72 89.74−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1Windowed peak and Scaled wavelet

Time (sec.)

Cp −

(C

p −

Cp ′

)

Windowed Peak Scaled wavelet, T′

c=.0104s

0

0.5

1

1.5

2

2.5|W|

Dur

atio

n (s

ec.)

Time (sec.)89.2 89.4 89.6 89.8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

17.06 17.08 17.1 17.12 17.14 17.16 17.18-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Time (sec.)

Cp -

(Cp -

Cp ′

)

Configuration 1 - Complex peak

20.57 20.58 20.59 20.6 20.61 20.62 20.63 20.64

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Time (sec.)C

p - (C

p - C

p ′ )

Configuration 3 - Complex peak

100.49 100.5 100.51 100.52 100.53 100.54100.55

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Time (sec.)

Cp -

(C

p - C

p ′ )

Configuration 4 - Complex peak

72.7 72.72 72.74 72.76-2.5

-2

-1.5

-1

-0.5

0

Configuration 6 - Complex peak

Time (sec.)

Cp -

(C

p - C

p ′ )

0 20 40 60 80 100 120-15

-10

-5

0

Time (sec.)

Cp

r:/03cp1025.txt

0 20 40 60 80 100 120-20

-15

-10

-5

0

Time (sec.)

Cp

r:/14cp2025.txt

0 20 40 60 80 100 120-20

-15

-10

-5

0

Time (sec.)

Cp

r:/11cp3025.txt

0 20 40 60 80 100 120-30

-20

-10

0

Time (sec.)

Cp

r:/03cp4025.txt

0 20 40 60 80 100 120-15

-10

-5

0

Time (sec.)

Cp

r:/03cp6025.txt

0 20 40 60 80 100 120-20

-15

-10

-5

0

Time (sec.)

Cp

r:/11cp7025.txt

0 20 40 60 80 100 120-20

-10

0

10

Time (sec.)

Cp

r:/03cp8025.txt

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 1

Duration (sec.)

Cp

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 2

Duration (sec.)

Cp

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 3

Duration (sec.)

Cp

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 4

Duration (sec.)

Cp

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 7

Duration (sec.)

Cp

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 6

Duration (sec.)

Cp

10-3

10-2

10-1

-18

-17

-16

-15

-14

-13

-12

-11

-10

-9

Configuration 8

duration (sec.)

Cp