pwl_stats_spectral_and_linear.md

Spectral parameters

Let's start by creating a modulated sinusoidal wave.

tmax = 400
fs = 10. # 1/Hz
t = np.arange(0, tmax, 1./fs)

# first wave
t1 = 10.  # wave period [s]
f1 = 1./t1  # wave frequency [Hz]
a1 = 0.25  # wave amplitude [m]
l1 = 10  # wave lenght [m]
h1 = 1  # water depth [m]
o1 = 2*np.pi/t1  # wave angular frequency [2*pi/s]
w1 = h1+(a1*np.sin(o1*t))

# second wave
t2 = 10.  # wave period [s]
f2 = 1./t2  # wave frequency [Hz]
a2 = 0.25  # wave amplitude [m]
l2 = 10  # wave lenght [m]
h2 = 1  # water depth [m]
o2 = 2*np.pi/t2  # wave angular frequency [2*pi/s]
delta = 1.05
w2 = h2+(a2*np.sin(o2*t*delta))

# resulting wave
wt = w1+w2

The resulting wave should look like this:

Let's calculate Power Spectral Density (PSD):

from pywavelearn.spectral import power_spectrum_density

f, psd = power_spectrum_density(wt, fs, window='hann', plot=True)

Statistical analysis from the spetrum

Most of the statistics from in Holthuijsen (2007) - Waves in Oceanic and Coastal Waters are implemented in the stats module.

For example, from the PSD, we can calculate wave heights and periods.

from pywavelearn.stats import Hm0, Tm01, Tm02

Hm0a = Hm0(f, psd)
Tm01a = Tm01(f, psd)
Tm02a = Tm01(f, psd)

print("Significant wave heigh :", np.round(Hm0a, 1))
print("Significant wave period :", np.round(Tm01a, 1))
print("Zero-crossing wave period :", np.round(Tm02a, 1))

Significant wave heigh : 1.0

Significant wave period : 9.8

Zero-crossing wave period : 9.8

Or, if you don't have the PSD yet:

from pywavelearn.stats import HM0, TM01, TM02

Hm0a = HM0(wv, fs)
Tm01a = TM01(wv, fs)
Tm02a = TM02(wv, fs)

Which should produce exactly the same results as before.

Wave-by-wave statistical analysis

Maybe one of the most powerful/useful tools in this package is its ability to perform high-level wave-by-wave analysis. Instead of the traditional zero-crossing approach, the method implemented here uses a iterative and procedural search for local minima and maxima on the data that should overcome the major shortcomings of the zero-crossing methods.

from pywavelearn.utils import peaklocalextremas
mins, maxs = peaklocalextremas(wt, lookahead=10, delta=0.025)

The original function was borrowed from here, but has been modified to work better with water waves.

We also can calculate the same statistics as before, now from the raw record.

from pywavelearn.utils import peaklocalextremas
from pywavelearn.stats import (significant_wave_height,
                               significant_wave_period)

# make sure to always start and end with a local minima
if t[mins[0]] > t[maxs[0]]:
    maxs = np.delete(maxs, 0)
if t[mins[-1]] < t[maxs[-1]]:
    maxs = np.delete(maxs, -1)

heights = []
periods = []
# considering a wave defined as two consecutive troughs
for i in range(len(mins)-1):
    i1 = mins[i]
    i2 = mins[i+1]
    single_wave = wt[i1:i2]
    heights.append(single_wave.max()-single_wave.min())
    periods.append(t[i2]-t[i1])

print("Significant wave height:", np.round(Hs, 1))
print("Significant wave period:", np.round(Ts, 1))

This results in:

Significant wave height: 1.0

Significant wave period: 10.6

Which is quite similar to the spectral values obtained above.

Linear wave theory

It's inevitable, one of these days you will need to perform some good old Airy wave theory calculations. The module linear should make your life a little bit easier. This module revamps Christoher H. Barker's original package with some new functions and better documentation. Some of the implemented are:

from numpy import pi
from pywavelearn.linear import *

# define a water depth
h = 2.0
# define a non-dimensional water depth
p = 1.2
# define a wave period
T = 10
# wave an angular frequency
omega = 2*pi/T

# wave number
k = wave_number(omega)

# wave angular frequency
omega = frequency(k)

# wave celerity at any depth
c = celerity(k,h)

# wave group speed at any depth
cg= group_speed(k,h)

# dispersion relation for a non-dimensional water depth
q = dispersion(p)