\(\boxed{\eta(x, t) = a \cos \left[ \frac{2\pi}{\lambda} (x - ct) \right]} \) and \(\boxed{\eta(x, t) = a \cos \left[ kx - \omega t \right]}\) are limited to propagation in the positive-\( x \) direction only. In general, waves may propagate in any direction. A useful three-dimensional generalization of \(\boxed{\eta(x, t) = a \cos \left[ kx - \omega t \right]}\) is \[ \eta = a \cos (kx + ly + mz - \omega t) = a \cos (\mathbf{K} \cdot \mathbf{x} - \omega t) \]

Where \( \mathbf{K} = (k, l, m) \) is a vector, called the wave number vector, whose magnitude \( K \) is given by \[ K^2 = k^2 + l^2 + m^2 \]

From \(\eta = a \cos (kx + ly + mz - \omega t) = a \cos (\mathbf{K} \cdot \mathbf{x} - \omega t)\) the wavelength is \[ \lambda = \frac{2\pi}{K} \] The magnitude of the phase velocity is \( c = \omega/K \), and the direction of propagation is parallel to \( \mathbf{K} \), so the phase velocity vector is \[ \mathbf{c} = (\omega/K)\, \mathbf{e}_K \] where \( \mathbf{e}_K = \mathbf{K}/K \), \( c_x = \omega / k \), \( c_y = \omega / l \), and \( c_z = \omega / m \) are each larger than the resultant \( c = \omega / K \), because \( k \), \( l \), and \( m \) are individually smaller than \( K \) when all three are non-zero as \(K^2 = k^2 + l^2 + m^2\). Any of the three axis-specific phase speeds is called the trace velocity along its associated axis

If sinusoidal waves exist in a fluid moving with uniform speed \(\mathbf{U}\), then the observed phase speed is \(\mathbf{c}_0 = \mathbf{c} + \mathbf{U}\). Forming a dot product of \(\mathbf{c}_0\) with \(\mathbf{K}\) and using \(\mathbf{c} = (\omega/K)\, \mathbf{e}_K\) produces \[ \omega_0 = \omega + \mathbf{U} \cdot \mathbf{K} \] where \(\omega_0\) is the observed frequency at a fixed point, and \(\omega\) is the intrinsic frequency measured by an observer moving with the flow. It is apparent that the frequency of a wave is Dopplershifted by an amount \(\mathbf{U} \cdot \mathbf{K}\) in non-zero flow

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