On the deflection of the vertical by tidal loading of the Earth's surface

Abstract The earliest calculations on the above subject in which the flexure of the earth's surface is taken into account, as well as the direct attraction of the water, are contained in a well known paper by G. H. Darwin. It was there shown, in a particular case, that the tilting effect would conspire with the attraction, and would on certain assumptions be directly proportional to it. The latter remark was stated as due originally (in a more general form) to Sir W. Thomson. On inserting numerical values it was found that the apparent deflection due to tilting would considerably exceed that due to attraction. The whole subject has of late excited renewed attention, owing to its bearing on observations of lunar deflection of gravity, and in a recent paper by Terazawa the matter is specially considered from the latter point of view. The present paper, after discussing a few typical problems, goes on (in 4) to examine the effect of one or two considerations which have hitherto been disregarded in such calculations. It is true that the corrections involved are under some conditions negligible, but they are of theoretic interest, and it is found that at great distances from a load, and therefore in all cases of widely distributed load, they may attain considerable relative importance. In the first place, owing to the deformation of the earth’s surface and the altered distribution of density an additional horizontal component of force on the plumb-line is introduced; this tends to counteract the attraction of the water. A more important point is that the influence of gravity on the deformation has been ignored. In attempting to estimate the effect of gravity it has been found convenient, in order to avoid difficulties not altogether of a mathematical kind, to limit the investigation to the case of incompressibility. I have also neglected, in the first instance, the disturbance in the field of gravity itself, due to the load and the deformation, so far as this affects the strains. When subsequently the alteration of the field is taken into account a curious point arises. For mathematical simplicity the “earth” has been treated, as is usual in such investigations, as flat and infinitely extended. It appears that if this were the case the surface would be unstable, whatever the degree of rigidity, for disturbances exceeding a certain critical wave-length. This wave-length is, however, enormous, and reason is given for the view that inferences can still legitimately be drawn from our results as to the character of the effects actually produced.