1.8 Paul Mother Wavelet Real Part Imaginary Part Magnitude.6.4 Amplitude.2.2.4.6.8 1 8 6 4 2 2 4 6 8 1 t

.8.6 Real Part of Three Scaled Wavelets a = 1 a = 5 a = 1 1.2 1 Imaginary Part of Three Scaled Wavelets a = 1 a = 5 a = 1.4.8 Magnitude.2.2 Magnitude.6.4.4.2.6.8 1 5 5 1 Time.2 1 5 5 1 Time

.4 Fourier Transform of Paul Wavelet (m=1).35.3.25 Amplitude.2.15.1.5 1 2 3 4 5 6 7 8 9 1 Frequency

.8 Real Part Half Interval Sampled 1.2 Imaginary Part Half Interval Sampled.6 1.4.8.2 Amplitude 2T ml Amplitude.6.4.2.4.6.2 2T ml.8 1 5 5 1 Time (sec).2 1 5 5 1 Time (sec)

.5 2T ml 2T sp -.5-6 -4-2 2 4 6 t

WW * /a ln a 3.5 3 2.5 2 1.5 1.5 5 4.5 4 3.5 3 2.5 2 1.5 1.5 1 2 3 4 5 Time (sec.)

1 2 Scale where WW * /a is maximized vs. Frequency (Hz) actual calculated scale (a) 1 1 1 1-1 1 1 1 Input signal (Hz)

.8 Real part of Paul Wavelet used as input plotted with peak scaled wavelet Input Signal Peak scaled wavelet.6.4.2 Amplitude.2.4.6.8 6 4 2 2 4 6 Time (sec.)

.2 Imaginary part of Paul Wavelet used as input plotted with peak scaled wavelet.2 Input Signal Peak scaled wavelet Amplitude.4.6.8.1.12 6 4 2 2 4 6 Time (sec.)

.7 Gaussian Test Function.6.5 Amplitude.4.3.2.1 1 8 6 4 2 2 4 6 8 1 Time (sec.)

W Paul Wavelet 2.5 5 2 4 ln a 3 2 1.5 1 1.5-1 -5 5 1 Time (sec.)

.7 Gaussian Test Function.6.5 Amplitude.4.3.2.1 1 8 6 4 2 2 4 6 8 1 Time W Paul wavelet 5 2.5 4.5 4 2 3.5 3 1.5 ln a 2.5 2 1 1.5 1.5.5-1 -5 5 1 Time (sec.)

5 4.5 4 3.5 3 W Paul Wavelet 2.5 2 1.5 ln a 2.5 2 1.5 1.5-1 -5 5 1 Time (sec.) 1.5

1.8 (mother) Paul wavelet with T c marked Real Part Imaginary Part Magnitude.6.4 Amplitude.2.2.4.6.8 6 4 2 2 4 6 t (nondimensional)

1.8 Mother Morlet Wavelet Real Part Imaginary Part Magnitude.6.4.2 Amplitude.2.4.6.8 1 6 4 2 2 4 6 Nondimensional Time

1.8 Real Part T c Sampling Points.6 T c.4.2 Amplitude.2.4.6.8 2T 1 6 4 2 2 4 6 Nondimensional Time

15 Velocity data with mean = 1 5 Velocity (m/s) 5 1 15 2 1 2 3 4 5 6 7 8 9 Time (sec.) 2 Pressure 1 1 2 C p 3 4 5 6 7 1 2 3 4 5 6 7 8 9 Time(sec.)

15 Windowed velocity data with mean = 1 5 Velocity (m/s) 5 1 15 35 4 45 5 55 6 65 Time (sec.) 6 5 4 Wavelet Transform Magnitude 11 1 9 8 7 ln a 3 2 1 6 5 4 3 2 1 1 2 3 4 5 6 7 8 Time (sec.)

6 Wavelet Transform Magnitude 6 5 5 4 4 ln a 3 3 2 2 1 1 1 2 3 4 5 6 7 8 Time (sec.) Windowed velocity data with mean = and real scaled Morlet wavelet (a=221) 1 Windowed Signal Scaled Wavelet 5 Velocity (m/s) 5 1 35 4 45 5 55 6 65 Time (sec.)

W velocity Paul ln a 5 4.5 4 3.5 3 2.5 2 1.5 1.5 14 12 1 8 6 4 2 1 2 3 4 5 6 7 8 Time (sec.) W Paul Wavelet 5 4 9 8 7 6 ln a 3 2 1 5 4 3 2 1 1 2 3 4 5 6 7 8 Time (sec.)

15 Windowed velocity data with mean = and real scaled Paul wavelet (a=55) Windowed Signal Scaled Wavelet 1 5 Velocity (m/s) 5 1 15 35 4 45 5 55 6 65 Time (sec.)

1 Windowed pressure data with mean = 1 2 C p 3 4 5 6 7 35 4 45 5 55 6 65 Time sec.

6 Wavelet Transform Magnitude 4 5 4 35 3 25 ln a 3 2 2 1 15 1 5 1 2 3 4 5 6 7 8 Time (sec.) 6 Wavelet Transform Magnitude 5 25 4 2 ln a 3 15 2 1 1 5 1 2 3 4 5 6 7 8 Time (sec.)

Windowed Pressure Data mean = & Peak Scale Real Part Morlet Wavelet a=221 Windowed Signal Scaled Wavelet 4 2 C p 2 4 6 35 4 45 5 55 6 65 Time (Sec.)

W Paul Wavelet ln a 5 4 3 2 1 55 5 45 4 35 3 25 2 15 1 5 1 2 3 4 5 6 7 8 Time (sec.)

W Paul Wavelet 5 45 4 4 35 3 ln a 3 2 1 25 2 15 1 1 2 3 4 5 6 7 8 Time (sec.) 5 Windowed Pressure Data; mean = & Imaginary Part of Paul Scaled Wavelet a = 62.8 1 Windowed Signal Scaled Wavelet 2 C p 3 4 5 6 35 4 45 5 55 6 65 Time (sec.)

35 3 Sensativity to window width, Velocity data Paul Wavelet Morlet Wavelet 25 Event Duration (sec.) 2 15 1 σ = 5 5 2 4 6 8 1 12 14 16 window width σ ( t = 1)

4 35 Paul Wavelet Morlet Wavelet Sensativity to window width, Pressure data 3 Event Duration (sec.) 25 2 15 1 σ = 175 5 2 4 6 8 1 12 14 16 window width σ ( t = 1) Close up of event and two Paul wavelets 1 1 2 C p 3 4 5 6 7 Raw Signal (mean=) Paul T c = 22.6s Paul; T c = 35.s 49 495 5 55 51 515 52 525 53 535 54 Time (sec.)

Raw data from configuration two 2 4 6 C p 8 1 12 Mean 3*std. dev. 14 89.2 89.3 89.4 89.5 89.6 89.7 89.8 89.9 9 9.1 Time (sec.)

Selectively Sampled Data from configuration two.5 C p (C p C p ).5 1 Peak Begins Peak Ends 1.5 2 89.58 89.6 89.62 89.64 89.66 89.68 89.7 89.72 89.74 Time (sec.).18 W.16 6 Duration (sec.).14.12.1.8.6.4.2 5 4 3 2 1 89.2 89.4 89.6 89.8 9 Time (sec.)

1.5 Windowed peak and Scaled wavelet Windowed Peak Scaled wavelet, T c =.14s.5 C p (C p C p ) 1 1.5 2 2.5 3 89.58 89.6 89.62 89.64 89.66 89.68 89.7 89.72 89.74 Time (sec.).18 W 2.5.16.14 2 Duration (sec.).12.1.8.6 1.5 1.4.2.5 89.2 89.4 89.6 89.8 9 Time (sec.)

.1 Configuration 1 - Complex peak Configuration 3 - Complex peak -.1 -.2 C p - (C p - C p ) -.2 -.3 -.4 -.5 -.6 -.7 -.8 17.6 17.8 17.1 17.12 17.14 17.16 17.18 Time (sec.) C p - (C p - C p ) -.4 -.6 -.8-1 -1.2 2.57 2.58 2.59 2.6 2.61 2.62 2.63 2.64 Time (sec.) Configuration 4 - Complex peak Configuration 6 - Complex peak -.2 -.4 -.5 C p - (C p - C p ) -.6 -.8-1 C p - (C p - C p ) -1-1.5-1.2-1.4-2 -1.6 1.49 1.5 1.51 1.52 1.53 1.54 1.55 Time (sec.) -2.5 72.7 72.72 72.74 72.76 Time (sec.)

r:/3cp125.txt C p -5-1 -15 2 4 6 8 1 12 Time (sec.) r:/14cp225.txt C p -5-1 -15-2 2 4 6 8 1 12 Time (sec.) r:/11cp325.txt C p -5-1 -15-2 2 4 6 8 1 12 Time (sec.) r:/3cp425.txt C p -1-2 -3 2 4 6 8 1 12 Time (sec.) r:/3cp625.txt C p -5-1 -15 2 4 6 8 1 12 Time (sec.) r:/11cp725.txt C p -5-1 -15-2 2 4 6 8 1 12 Time (sec.) 1 r:/3cp825.txt C p -1-2 2 4 6 8 1 12 Time (sec.)

Configuration 1 Configuration 2 C p -9-1 -11-12 -13-14 -15-16 -17-18 1-3 1-2 1-1 Duration (sec.) C p -9-1 -11-12 -13-14 -15-16 -17-18 1-3 1-2 1-1 Duration (sec.) Configuration 3 Configuration 4 C p -9-1 -11-12 -13-14 -15-16 -17-18 1-3 1-2 1-1 Duration (sec.) C p -9-1 -11-12 -13-14 -15-16 -17-18 1-3 1-2 1-1 Duration (sec.)

Configuration 6 Configuration 7 C p -9-1 -11-12 -13-14 -15-16 -17-18 1-3 1-2 1-1 Duration (sec.) C p -9-1 -11-12 -13-14 -15-16 -17-18 1-3 1-2 1-1 Duration (sec.) -9 Configuration 8-1 -11-12 C p -13-14 -15-16 -17-18 1-3 1-2 1-1 duration (sec.)