The formation of deep water waves, when some disturbances act on a point in the interior of the water, has been studied. The compressional disturbances of the simple-harmonic type being propagated radially from the origin, a succession of gravity waves is gradually developed by the surface conditions of the water.
The method of analysis employed in this is, in the first place, to apply on the primary waves the mathematical results due to H. Lamb, i.e.
H(2)0(hR)=i/π∫∞-∞e-aye_??_ifx/αdf, [
R2=x2+y2]
and
e-ihR/R=∫∞0e-az/αJ0(fr)fdf, [
R2=x2+y2+z2]
where
α=√<f2-h2>or
i√<h2-f2>,
according as
f2_??_
h2. Some modified formulae have also been employed. In the second step, gravity waves, which are to be superposed on the primary waves, have been formulated to satisfy the boundary conditions.
It must be acknowledged that the conception of the “small motion” in the sense concerning the slope or the gradient of the displacements has been introduced. Again, in all cases of contour integration the “principal values” due to Cauchy has been taken.
The paper consists of six sections: the beginning three sections deal with the cases in which lithe origin is either a singlet, a doublet oscillating horizontally or a doublet moving vertically all in two dimensions, while the remaining three treat of the similar cases in three dimensions.
The principal results obtained by this investigation are enumerated as follows:
1. In spite of very small displacements of the compressional waves in the neighbourhood of the origin in the interior of the water, the excited surface waves have relatively large amplitudes.
2. The generated surface waves are chiefly the ordinary gravity waves having the same frequency as that of the origin together with their wave length proper to the period.
3. The distribution of the wave motion on the surface of water always conspires with the modes of oscillation at the origin.
4. This fails in a three-dimensional case where a doublet oscillates horizontally. In this, notwithstanding the maintenance of the natures of the vertical and the horizontal components of displacement in wave x profile and in azimulthal distribution, the azimuthal component of displacement quickly disappears as the distance from the disturbed portion is increased.
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