The four gravitational sidelines are all very small, only about 0.3 % of the primary, or smaller. With arguments depending on the mean longitude of the Sun, they evidently arise from the Sun's third-body perturbations of the lunar orbit. In particular, α2,β2 arise from what in lunar theory is called the “annual equation”, which refers to an expansion or compaction of the moon's orbit depending on whether the Sun is at perihelion or aphelion, respectively. With a change in the orbital radius there is a corresponding change in the moon's angular velocity and thus longitude. The variation in longitude is given approximately by -669′′sin⁡ℓ′ p. 329. The radial variation is 49cos⁡ℓ′ km.

Because the ocean cannot support extremely high-Q resonances, the ocean's response (admittance) to gravitational forcing must be nearly constant across the small frequency band of the M2 group. In that case, the gravitational components of the four sidelines can all be inferred from measurements of the central M2 primary. Their amplitudes must be in the same proportion as the relative amplitudes of Table , and their phases must be nearly identical to the phase of M2. It may be possible that this rule is violated in locations where large M2 currents act through nonlinear dissipation to suppress the sidelines; this has been observed to happen for the M2 nodal sideline .

Aside from this one exception, the induced seasonal modulations in M2 from the astronomical sidelines are easily worked out. At a location with mean M2 amplitude A and phase lag G, the combined elevation of the group is ζM(t)=Acos⁡(2τ-G)+cαcos⁡(2τ-ℓ′+π-G)+cβcos⁡(2τ+ℓ′-G)+cδcos⁡(2τ+2h-G), where ci are the relative amplitudes from Table , and the Γ2 constituent has been dropped for the reasons noted above. In analogy with the usual approach for handling nodal modulations, this expression may be written as a single modulated wave: 1ζM(t)=Af(t)cos⁡[2τ-G+u(t)], where functions f,u vary throughout the year as periodic functions of h (or ℓ′). Expanding the trigonometric functions and gathering like terms in cos⁡G and sin⁡G lead to 2fcos⁡u=1-(cα-cβ)cos⁡ℓ′+cδcos⁡2h,fsin⁡u=(cα+cβ)sin⁡ℓ′+cδsin⁡2h, or 3f≈1-0.00041cos⁡ℓ′+0.00114cos⁡2h for the amplitude modulation and 4u≈0.372∘sin⁡ℓ′+0.065∘sin⁡2h for the phase modulation. The amplitude modulation is insignificant; because the main coefficients in Eq. () cancel, the second term from δ2 is actually larger than the first term but still very small. For the phase modulation in Eq. (), the first term peaks at ℓ′=90∘, which is early April, and the smaller second term shifts that peak to mid-April. So, the observed phase lag of M2 (u subtracts from G) takes its minimum value in mid-April and its maximum in mid-September. It is interesting to note that the first term in Eq. () is exactly twice the Moon's longitude modulation of 669′′sin⁡ℓ′, alluded to above as arising from the annual equation in the Sun's perturbation of the lunar orbit (it is twice, because M2 is semidiurnal).

At most tide gauges, the gravitational components of any seasonal variability in the observed M2 will thus comprise only a minor part. Observed variability will likely differ significantly from what might be inferred from the M2 admittance or as modeled by Eqs. ()–(). Nonetheless, when only the astronomical modulation is important and the M2 amplitude is moderately large, a phase modulation of 0.7∘ is easily seen; an example is given in Sect. .

Aside from the two compound tides listed in Table , there are possibly others that fall within the M2 group . The most important is OP2, exactly coinciding with MSK2, and KO2, exactly coinciding with M2. The OP2 and MSK2 constituents likely arise from different nonlinear aspects of the hydrodynamics , although both are generated in shallow water.

The KO2 tide is important only when M2 is anomalously small; it can potentially induce an unusual nodal modulation (relative to the standard M2 modulation), but it has no bearing on seasonality, with one exception: any seasonal variations in the primary K1 or O1 tides would induce seasonality in the compound KO2, but cases where this is significant relative to M2 must be rare. The few additional compound tides within the group noted by are from interactions involving the diurnal S1, which is normally small and will be ignored. Thus, when nonlinear shallow-water processes are acting sufficiently to produce a noticeable MSK2 (or OP2), the main effect of this will be equivalent to semiannual modulations in M2 amplitude and/or phase.

The astronomical sidelines in Table  sit at discrete known frequencies. Compound tides are similar, although the number of them can sharply increase in shallow water. In contrast, climate-driven modulations are broadband. Although an annual cycle typically dominates, which is the justification for MA2, MB2, one might expect higher harmonics in some cases and also possibly a general smearing of observed spectral lines across the group e.g.,.

, building on an earlier study by , discussed the case of Victoria (Canada). There was a clear annual modulation in both amplitude and phase of M2 but with an apparent second harmonic. Multi-year tidal analysis of the Victoria data (not shown) does show energy at the MSK2 (or OP2) frequency. Whether this is due to the compound tide(s) or to a true second harmonic below MA2 is not clear. During tidal analysis it is sometimes possible to separate two constituents of identical frequency by exploiting their different nodal modulations , but in this case the amplitudes appear too weak to allow it.

The question of frictional compound tides versus higher harmonics of MA2 and MB2 is only one instance of the more general, and difficult, problem of deciphering the causes of climate-induced modulations, even when manifested by MA2 or MB2 acting alone e.g.,.

The constituent names in Table  follow both historical and current international conventions – for the latter, see the table maintained by the Tide, Water Level and Current Working Group of the International Hydrographic Organization (https://iho.int, last access: 17 April 2022). The exception is β2, which does not (as yet) appear in the working group's table. Godin's 1988 book has the same usage as the IHO; his earlier 1972 book left the smaller lines unlabeled . Regarding β2, however, it is commonly employed in the Earth tide community e.g.,, and β2 falls into the obvious pattern of using the first four letters of the Greek alphabet for the four astronomical constituents.

The two climate constituents apparently began as MA2 and Ma2 . That lost favor, probably because a speaker cannot distinguish the two and also because early computers employed only upper-case letters. MB2 has been used for decades, at least in British work

David Pugh has a copy of a November 1977 memorandum from David Cartwright, then director of the Bidston Observatory, proposing use of the modified symbols MA2 and MB2, in “deference to Corkan's original work”. He suggested similar notation for other constituents affected by seasonal variability, such as NA2, NB2; these are now included in the IHO tables, as are MA4, MB4.

It is actually not easy to find good examples of seasonal variability stemming solely from the purely astronomical constituents of Table . Any potential case must display fairly constant admittances across the group of gravitational constituents. Yet at most tide gauges one sees perturbations in the admittance or one sees significant amplitudes in the compound tides.

The time series at Port Orford, Oregon, is one of the better examples. Tidal constants estimated from 26 years of data

Data at Port Orford are available from 1978 to present, but the data before 1994 yield tidal estimates too erratic to use. Data after 1993 appear to be of good quality.

An example of modulations dominated by one or both of the compound tides in the M2 group is Saint-Malo, France, whose spectrum was shown in Fig. . The two annual constituents (α2,β2 or MA2, MB2) are smaller, as the spectrum reveals, but are still too large to ignore. The total modulation, based on four constituents, is shown in Fig. . The amplitude clearly reveals the presence of a semiannual modulation, with a slow rise during the beginning of the year and then a rapid decay between July and October. The phase is dominated by the semiannual effect, with an annual contribution responsible for the September phase lag exceeding the earlier peak in March.

Tide gauges with annual variations in M2, and thus with sidelines dominated by MA2 and/or MB2, are easy to find. The case of Victoria was already noted , partly for having a second harmonic in its modulation. However, there are many tide gauges where only the annual modulation presents itself. A fair number can be found along the coast of Japan, even though M2 itself is not especially large there. The example chosen here is in fact one of the largest modulations discovered anywhere: Chittagong, along the coast of Bangladesh. At that location the annual sea-level term, Sa, is also anomalously large, presumably reflecting large discharge from the Ganges. Several other nearby tide gauges, with somewhat smaller M2 modulations, were studied by .

Harmonic analysis of 11 years of hourly data (2008–2018) at Chittagong reveals astonishing large amplitudes for MA2 and MB2 of 15.5 and 10.1 cm, respectively, with the M2 constituent at 171.6 cm. The resulting seasonal modulation is shown in Fig. . The monthly mean amplitudes range from 146 cm in February to a high of 195 cm in August. In comparison, the phase modulation is not very large, about 4∘. The monthly mean phases appear slightly erratic and fit the demodulated curve only moderately well.

The large modulation in amplitude at this location closely mimics the large oscillation in annual sea level, which is minimum in February and maximum in July, with a mean range of 71 cm. developed a tide model for the region that explores this interdependence.