ATMOSPHERIC TIDES AND OSCILLATIONS 
as to its distribution with height, the most realistic 
being that its amplitude decreases exponentially up- 
wards, as it would do if the heat were supplied at the 
ground and transmitted upwards by uniform conduc- 
tivity; but he did not take into account the linear re- 
tardation of phase with height, which in this case should 
accompany the decrease of amplitude. 
In 1910 Lamb [72] made a most important extension 
of Laplace’s theory. His work related to an atmosphere 
on a plane base, thus abstracting from the problem the 
sphericity and rotation of the earth; but later [73] he 
removed these last two restrictions, though only for an 
atmosphere in convective equilibrium. His main dis- 
cussion referred to an atmosphere in which H varies 
uniformly with the height (HW = Hp being a special 
case). He showed that the propagation of long waves in 
such an atmosphere is similar to that of long waves in a 
liquid ocean of depth H) in two special cases, namely, (1) 
Laplace’s case, in which H = Hy (or T/m = To/mo) 
at all heights, and the density variations occur iso- 
thermally; and (2) for an atmosphere in adiabatic 
equilibrium (so that its height is yHo/(y — 1), the 
temperature 7’ decreasing uniformly upwards to zero 
at the rate (y — 1) To/y Ho), and in which the density 
variations occur adiabatically (Laplace’s case can be 
considered as corresponding to y = 1). 
The atmosphere is not in adiabatic equilibrium, so 
that it cannot be supposed, at least without further 
proof, that the tidally equivalent (liquid ocean) depth 
for the atmosphere is Ho. Lamb in fact showed that 
when H varies linearly with height, but not adiabatic- 
ally, there is an infinite series of speeds for long waves, 
with the implication that there is a similar series of 
values of the equivalent depth h. However, the impres- 
sion persisted for over twenty years that for any type 
of atmosphere there is just one value of h. 
Lamb briefly discussed the resonance hypothesis of 
S> in his 1910 paper and in subsequent editions of his 
Hydrodynamics. He estimated from the improved form 
of Laplace’s theory given by Hough [70], in terms of 
spherical harmonic functions, that if the atmosphere is 
resonant with a free oscillation similar to S» in its geo- 
graphical distribution, h must be about 8 km, whereas 
for the actual atmosphere Hy varies from about 7.3 km 
at the poles to 8.7 km at the equator. He continued: 
Without pressing too far conclusions based on the hypoth- 
esis of an atmosphere uniform over the earth, and approx- 
imately in convective equilibrium, we may, I think, at least 
assert the existence of a free oscillation of the earth’s atmos- 
phere, of “semi-diurnal” type, with a period not very differ- 
ent from, but probably somewhat less than, 12 mean solar 
hours. 
He continued further: 
At the same time, the reason for rejecting the explanation 
of the semi-diurnal barometric variation as due to a gravi- 
tational solar tide seems to call for a little further examination. 
The amplitude of this variation at places on the equator is 
given by Kelvin as 0.032 inch. The amplitude given by the 
“equilibrium”’ theory of the tides is about 0.00047 inch. Some 
numerical results given by Hough in illustration of the kinetic 
525 
theory of oceanic tides indicate that in order that this ampli- 
tude should be increased by dynamical action some seventy- 
fold, the free period must differ from the imposed period of 
12 solar hours by not more than 2 or 3 minutes. Since the 
difference between the lunar and solar semi-diurnal periods 
amounts to 26 minutes, it is quite conceivable that the solar 
influence might in this way be rendered much more effective 
than the lunar. The real difficulty, so far as this point is 
concerned, is the @ priori improbability of so very close an 
agreement between the two periods. The most decisive evi- 
dence, however, appears to be furnished by the phase of the 
observed semi-diurnal imequality, which is accelerated in- 
stead of retarded (as it would be by tidal friction) relatively 
to the sun’s transit. 
In 1924 Chapman [31] stressed the argument for 
strong resonance of Ss, based on the regularity of its 
geographical distribution, as compared with the con- 
siderable nonuniformity of the solar semidiurnal varia- 
tion of air temperature (especially as between land and 
sea areas), which according to Lamb’s last-quoted re- 
mark must be at least an important part of the cause of 
S». This argument he strengthened by contrasting the 
regularity of the geographical distribution of S2, due 
partly to an irregular cause, with the degree of irregu- 
larity shown by L»2, whose cause is certainly distributed 
very regularly. 
Chapman also extended Margules’ calculation of the 
oscillations produced by the semidiurnal component of 
the daily variation of air temperature (7s, see p. 522) 
taking account of. the variation of phase (later dis- 
cussed, in this connection, by Bjerknes [23a]), as well as 
of amplitude, with height. He concluded that the phase 
of the part of S». which is of thermal orig must be about 
135° in advance of the phase of 72, which he tried to 
estimate from the temperature data collected by Hann 
and others, using Taylor’s estimate of the thermal con- 
ductivity due to eddy motion. 
He also compared the magnitudes of the thermal and 
tidal contributions to S2, which is possible because 
(substantially) both are affected by the same resonance 
magnification. He was able to show that they were of 
roughly equal order of magnitude (the madequacy of 
the data regarding 7’, precludes a more accurate state- 
ment as yet), and on this basis he was able to explain 
the observed phase of S» from the phase of the thermal 
part, inferred from 72, and on the assumption that the 
tidal part is in phase with the sun. This further enabled 
him to estimate the factor of resonance magnification 
as about 100. He was unable to prove that the atmos- 
phere has a free period of oscillation (of the right 
geographical distribution) which would give this mag- 
nification. As the resonance magnification (when con- 
siderable) would be proportional to 1/(¢; — t;), where 
t; denotes the imposed period and ¢, the free period, 
he concluded that despite the a priori improbability, 
t; — ty cannot exceed 2 or 3 minutes, and is positive. 
Whipple in 1918 [109], and also in 1924 in the discus- 
sion on [31], found great difficulty in accepting the 
resonance theory, on account of the possibility that such 
accurate “tuning” of the forced to the free oscillation 
might be upset by the large changes in air pressure and 
