If \(\boxed{\eta(x, t) = a \cos \left[ \frac{2\pi}{\lambda} (x - ct) \right]}\) describes the vertical deflection of an air–water interface, then the height of wave crests is \( +a \) and the depth of the wave troughs is \( -a \) compared to the undisturbed water-surface location \( z = 0 \). At any instant in time, the distance between successive wave crests is \( \lambda \). At any fixed \( x \)-location, the time between passage of successive wave crests is the period, \( T = 2\pi/kc = \lambda/c \). Thus, the wave’s cyclic frequency is \( \nu = 1/T \) with units of Hz, and its radian frequency is \( \omega = 2\pi \nu \) with units of rad./s.
In terms of \( k \) and \( \omega \), \(\boxed{\eta(x, t) = a \cos \left[ \frac{2\pi}{\lambda} (x - ct) \right]}\) can be written \[ \eta(x, t) = a \cos \left[ kx - \omega t \right] \]

The wave propagation speed is readily deduced by determining the travel speed of wave crests. This means setting the phase in both equations so that the cosine function is unity and \( \eta = +a \). This occurs when the phase is \( 2n\pi \) where \( n \) is an integer \( \frac{2\pi}{\lambda}(x_{\text{crest}} - ct) = 2n\pi = kx_{\text{crest}} - \omega t \) and \( x_{\text{crest}} \) is the time-dependent location where \( \eta = +a \)
Solving for the crest location produces \[ x_{\text{crest}} = (\omega/k)t + 2n\pi/k \] Therefore, in a time increment \( \Delta t \) a wave crest moves a distance \( \Delta x_{\text{crest}} = (\omega/k) \Delta t \), so \[ c = \omega/k = \lambda \nu \] is known as the phase speed because it specifies the travel speed of constant-phase wave features like wave crests or troughs

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