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| name | wave-theory-2-wave-spectra |
| description | Sub-skill of wave-theory: 2. Wave Spectra. |
| version | 1.0.0 |
| category | engineering |
| type | reference |
| scripts_exempt | true |
JONSWAP Spectrum:
def jonswap_spectrum(
frequencies: np.ndarray,
Hs: float,
Tp: float,
gamma: float = 3.3,
alpha: float = None
) -> np.ndarray:
"""
Calculate JONSWAP wave spectrum.
S(f) = α g² (2π)^-4 f^-5 exp[-5/4(f/fp)^-4] γ^exp[-(f-fp)²/(2σ²fp²)]
Args:
frequencies: Frequency array (Hz)
Hs: Significant wave height (m)
Tp: Peak period (s)
gamma: Peak enhancement factor (3.3 for North Sea)
alpha: Phillips constant (calculated if None)
Returns:
Spectral density S(f) (m²/Hz)
"""
g = 9.81
fp = 1 / Tp # Peak frequency (Hz)
# Calculate alpha if not provided
if alpha is None:
# Relationship: Hs = 4*sqrt(m0)
# For JONSWAP: alpha ≈ 5.061 * Hs² / Tp⁴ * (1 - 0.287*ln(γ))
alpha = 5.061 * Hs**2 / Tp**4 * (1 - 0.287 * np.log(gamma))
# Sigma parameter
sigma = np.where(frequencies <= fp, 0.07, 0.09)
# Pierson-Moskowitz spectrum
S_PM = alpha * g**2 * (2*np.pi)**(-4) * frequencies**(-5) * \
np.exp(-1.25 * (frequencies / fp)**(-4))
# Peak enhancement
r = np.exp(-(frequencies - fp)**2 / (2 * sigma**2 * fp**2))
gamma_factor = gamma ** r
# JONSWAP spectrum
S = S_PM * gamma_factor
return S
# Example: Generate JONSWAP spectrum
freq = np.linspace(0.01, 0.5, 500)
S = jonswap_spectrum(freq, Hs=8.5, Tp=12.0, gamma=3.3)
# Verify Hs
m0 = np.trapz(S, freq)
Hs_calc = 4 * np.sqrt(m0)
print(f"JONSWAP Spectrum:")
print(f" Input Hs: 8.5 m")
print(f" Calculated Hs: {Hs_calc:.2f} m")
print(f" Peak frequency: {1/12:.4f} Hz")
Pierson-Moskowitz Spectrum:
def pierson_moskowitz_spectrum(
frequencies: np.ndarray,
Hs: float,
Tp: float = None,
U19_5: float = None
) -> np.ndarray:
"""
Calculate Pierson-Moskowitz spectrum (fully developed sea).
Args:
frequencies: Frequency array (Hz)
Hs: Significant wave height (m)
Tp: Peak period (s) - optional
U19_5: Wind speed at 19.5m height (m/s) - optional
Returns:
Spectral density S(f) (m²/Hz)
"""
g = 9.81
if Tp is not None:
# Use peak period
fp = 1 / Tp
elif U19_5 is not None:
# Calculate from wind speed
fp = 0.877 * g / (2 * np.pi * U19_5)
else:
raise ValueError("Must provide either Tp or U19_5")
# Phillips constant
alpha = 0.0081 # For fully developed seas
# P-M spectrum
S = alpha * g**2 * (2*np.pi)**(-4) * frequencies**(-5) * \
np.exp(-1.25 * (frequencies / fp)**(-4))
return S
# Example
S_PM = pierson_moskowitz_spectrum(freq, Hs=8.5, Tp=12.0)
m0_PM = np.trapz(S_PM, freq)
Hs_PM = 4 * np.sqrt(m0_PM)
print(f"P-M Spectrum Hs: {Hs_PM:f} m")