Compound and Shallow-Water Waves

Compound and Shallow-Water Waves#

In addition to the astronomical constituents derived from the tide-generating potential, ocean tide records reveal frequencies that are not present in the potential development. These arise from non-linear hydrodynamic interactions and are called compound waves or shallow-water waves. Understanding their origin is important because they can dominate the tidal signal in coastal and estuarine regions.

Non-Linear Origin#

In the deep open ocean, the tidal wave behaves as a nearly linear long wave and remains sinusoidal. As the wave propagates over the continental shelf and into shallow waters, several non-linear processes distort its shape:

  1. Depth-dependent propagation speed: The speed of a gravity wave is \(\sqrt{g \, d}\), where \(d\) is the water depth. High water propagates faster than low water because the total depth is greater at high tide, causing the wave to steepen.

  2. Quadratic bottom friction: The frictional drag on the seabed is proportional to \(|u| \, u\) (where \(u\) is the current velocity), introducing harmonics of the fundamental frequency.

  3. Advective acceleration: The \(u \cdot \nabla u\) term in the momentum equation generates sum and difference frequencies from interacting constituents.

In the spectral domain, these distortions manifest as new frequencies that are integer combinations of the original astronomical constituent frequencies.

Frequency Combinations#

The frequency \(\omega\) of a compound wave is an algebraic combination of its parent constituents’ frequencies. Common examples include:

Overtides (self-interaction of a single constituent):

\[\begin{split}\begin{aligned} M_4 &: \quad \omega_{M_4} = 2\,\omega_{M_2} \\ M_6 &: \quad \omega_{M_6} = 3\,\omega_{M_2} \\ M_8 &: \quad \omega_{M_8} = 4\,\omega_{M_2} \\ S_4 &: \quad \omega_{S_4} = 2\,\omega_{S_2} \end{aligned}\end{split}\]

Compound tides (interaction of two or more constituents):

\[\begin{split}\begin{aligned} MS_4 &: \quad \omega_{MS_4} = \omega_{M_2} + \omega_{S_2} \\ MN_4 &: \quad \omega_{MN_4} = \omega_{M_2} + \omega_{N_2} \\ MK_3 &: \quad \omega_{MK_3} = \omega_{M_2} + \omega_{K_1} \\ MO_3 &: \quad \omega_{MO_3} = \omega_{M_2} + \omega_{O_1} \\ 2MS_6 &: \quad \omega_{2MS_6} = 2\,\omega_{M_2} + \omega_{S_2} \\ 2MK_3 &: \quad \omega_{2MK_3} = 2\,\omega_{M_2} - \omega_{K_1} \end{aligned}\end{split}\]

Difference frequencies can also appear:

\[2MS_2 : \quad \omega_{2MS_2} = 2\,\omega_{M_2} - \omega_{S_2}\]

The naming convention uses the parent constituent symbols with a subscript indicating the tidal species (number of cycles per day).

Species Classification#

Compound waves span a range of species:

Species

Period range

Example constituents

Terdiurnal (3)

~8 hours

\(MO_3\), \(MK_3\), \(2MK_3\), \(M_3\)

Quarter-diurnal (4)

~6 hours

\(M_4\), \(MS_4\), \(MN_4\), \(S_4\)

Sixth-diurnal (6)

~4 hours

\(M_6\), \(2MS_6\), \(2MN_6\)

Eighth-diurnal (8)

~3 hours

\(M_8\), \(3MS_8\)

Nodal Corrections for Compound Waves#

The nodal corrections for compound waves are derived directly from those of their parent constituents:

Amplitude factor \(f\):

\[f(\text{compound}) = \prod_{\text{parents}} f(\text{parent})^{|c|}\]

where \(c\) is the integer multiplier of each parent frequency. For example:

\[\begin{split}\begin{aligned} f(M_4) &= f(M_2)^2 \\ f(M_6) &= f(M_2)^3 \\ f(MS_4) &= f(M_2) \times f(S_2) = f(M_2) \quad \text{(since } f(S_2) = 1\text{)} \end{aligned}\end{split}\]

Phase correction \(u\):

\[u(\text{compound}) = \sum_{\text{parents}} c \times u(\text{parent})\]

For example:

\[\begin{split}\begin{aligned} u(M_4) &= 2\,u(M_2) \\ u(MK_3) &= u(M_2) + u(K_1) \\ u(2MK_3) &= 2\,u(M_2) - u(K_1) \end{aligned}\end{split}\]

The sign of the multiplier \(c\) follows the sign in the frequency combination: positive for sum frequencies, negative for difference frequencies.

Significance in Practice#

Compound waves are typically small in the open ocean but can become the dominant signal in estuaries and shallow coastal regions. In extreme cases (e.g., the Gironde estuary), the quarter-diurnal and sixth-diurnal constituents cause visible asymmetry between flood and ebb tide durations, with flood durations shortened to about 4.5 hours and ebb durations extended to about 8.5 hours.

PyFES includes a comprehensive set of compound wave constituents in its wave tables. These are listed in the constituent reference pages:

References#

  • Simon, B. (2013). Marées Océaniques et Côtières (943-MOC), Ch. V, pp. 120–123.

  • Schureman, P. (1940). Manual of Harmonic Analysis and Prediction of Tides, SP 98, pp. 46–48.

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