import numpy as np
import matplotlib.pyplot as plt

# Parameters
alpha = 10.0                  # m
lambda_d = 10.0               # m
k_d = 2 * np.pi / lambda_d    # dominant wave number (m^-1)

# Wavenumber grid: 0 < k < 2 k_d
k = np.linspace(0.0, 2.0 * k_d, 2000)

# Analytic result for \tilde{eta}(k)
eta_tilde = (1 / (2 * np.pi)) * np.exp(
    -0.5 * (k - k_d)**2 * alpha**2
)

plt.figure(figsize=(7, 4.5))
plt.plot(k, eta_tilde, color='black', linewidth=1.5)
plt.xlim(0, 2 * k_d)
plt.xlabel(r"$k\ \mathrm{(m^{-1})}$")
plt.ylabel(r"$\tilde{\eta}(k)$")
plt.title(
    r"Wave-number amplitude distribution $\tilde{\eta}(k)$"
    "\n"
    r"$\alpha = 10\,\mathrm{m},\quad k_d = 2\pi/10\,\mathrm{m^{-1}}$"
)
plt.tight_layout()
plt.show()