HARMONIC ANALYSIS AND PREDICTION OF TIDES 47 
Sa, Ssa, and S, with periods corresponding respectively to the tropical 
year, the half tropical year, and the solar day. These constituents 
are represented also by terms in the development of the tide-producing 
force of the sun but they are considered of greater importance as 
meteorological tides. Ssa occurs in the development of the principal 
solar force while Sa and S, would appear in a development involving 
the 4th power of the solar parallax (par. 119). In the analysis of tide 
observations both Sa and Ssa are usually found to have an appreciable 
affect on the water level. Constituent S, is relatively of little im- 
portance in its effect on the height of the tide but has been more 
noticeable in the velocity of off-shore tidal currents, probably as a 
result of periodic land and sea breezes. 
138. The shallow-water constituents result from the fact that when 
a wave runs into shallow water its trough is retarded more than its 
crest and the wave loses its simple harmonic form. The shallow-water 
constituents are classified as overtides and compound tides, the over- 
tide having a speed that is an exact multiple of one of the elementary 
constituents and the compound tide a speed that equals the sum or 
difference of the speeds of two or more elementary constituents. 
139. The overtides were so named because of their analogy to the 
overtones in musical sounds and they may be considered as the 
higher harmonics of the fundamental tides. The only overtides 
usually taken into account in tidal work are the harmonics of the 
principal lunar and solar semidiurnal constituents M, and S,, the lunar 
series being designated by the symbols My, Me, and Mg, and the solar 
series by Sy, Ss, and Ss. The subscript indicates the number of 
periods in the constituent day. These overtides with their argu- 
ments and speeds are included in table 2a, the arguments and speeds 
being taken as exact multiples of those of the fundamental con- 
stituent. There are no theoretical expressions for the coefficients of 
the overtides but it is assumed that the amplitudes of the lunar series 
undergo variations due to changes in the longitude of the moon’s 
node which are analogous to those in the fundamental tide. The 
node factors for M,, Ms, and Msg, respectively, are taken as the 
square, the cube, and the fourth power of the corresponding factor 
for M,. For the solar terms this factor is always zero. 
140. Compound tides were suggested by Helmholtz’s theory of 
sound waves. Innumerable combinations are possible but the prin- 
cipal elementary constituents involved are Ms, S:, No, Ky, and QO. 
Table 2a includes the compound tides listed in International Hydro- 
eraphic Bureau Special Publication No. 26, which is a compilation of 
the tidal harmonic constants for the world. The argument of a 
compound tide equals the sum or difference of the arguments of the 
elementary constituents of which it is compounded. The node 
factor is taken as the product of the node factors of the same con- 
stituents. Table 2a contains the arguments, speeds, and node 
factors of these tides. 
141. Omitted from table 2a are a number of compound tides which 
have the same speeds as elementary constituents included in table 2. 
Thus, 2MS,, compounded by formula 2M,—S,, has the same speed as 
constituent uw, represented by term A,; of formula (64). Considered 
as a compound tide there would be a small difference in the wu of the 
argument and also in the node factor. Since there is no practical 
way of separating the elementary constituent from the compound 
