Throughout this lecture, the wave frequency will be assumed much higher than the Coriolis frequency so the wave motion is unaffected by the earth’s rotation, and wave amplitudes are assumed small enough so that the governing equations and boundary conditions are linear

For such linear waves, Fourier superposition of sinusoidal waves allows arbitrary waveforms to be constructed and sinusoidal waveforms arise naturally from the linearized equations for water waves. A simple sinusoidal traveling wave of the form \[ \eta(x, t) = a \cos \left[ \frac{2\pi}{\lambda} (x - ct) \right] \] In Cartesian coordinates with \( x \) horizontal and \( z \) vertical, \( z = \eta(x,t) \) specifies the waveform or surface shape where \( a \) is the wave amplitude, \( \lambda \) is the wavelength, \( c \) is the phase speed, and \( 2\pi(x - ct)/\lambda \) is the phase

The spatial frequency \( k = 2\pi/\lambda \) with units of rad/m is known as the wave number

A classical trigonometric series is a series of the form \[ \frac{a_0}{2} + \sum_{k=1}^{\infty} \left( a_k \cos kx + b_k \sin kx \right) \] If \(\boxed{a_k = a_k(f) = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos kx \, dx, k = 0, 1, \ldots}\) and \(\boxed{b_k = b_k(f) = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin kx \, dx, k = 1, 2, \ldots}\) have meaning for a function \( f \), then the trigonometric series assigned to \( f \) is called the trigonometric Fourier series of \( f \) \[ f \sim \frac{a_0(f)}{2} + \sum_{k=1}^{\infty} a_k(f) \cos kx + b_k(f) \sin kx \] Fourier transform of the function \( f : \mathbb{R} \to \mathbb{C} \) \[ \mathcal{F}[f](\xi) := \frac{1}{2\pi} \int_{-\infty}^{\infty} f(x) e^{-i \xi x} \, dx \] The Fourier cosine transform and the Fourier sine transform of the function \( f \) \[ \mathcal{F}_c[f](\xi) := \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) \cos \xi x \, dx \quad, \quad \mathcal{F}_s[f](\xi) := \frac{1}{\pi} \int_{-\infty}^{\infty} f(x) \sin \xi x \, dx \]

¹Zorich, V.A. (2015). Fourier Series and the Fourier Transform. In: Mathematical Analysis II. Universitext. Springer, Berlin, Heidelberg.

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