## Abstract

In the past it has been considered unlikely that ocean waves are capable of generating microseismic oscillations of the sea bed over areas of deep water, since the decrease of the pressure variations with depth is exponential, according to the first-order theory generally used. However, it was recently shown by Miche that in the second approximation to the standing wave there is a second-order pressure variation which is not attenuated with depth and which must therefore ultimately predominate over the first-order pressure variations. In section section 2 and 3 of the present paper the general conditions under which second-order pressure variations of this latter type will occur are considered. It is shown that in an infinite wave train there is in general a second-order pressure variation at infinite depth which is applied equally over the whole fluid and is associated with no particle motion. In the case of two progressive waves of the same wave-length travelling in opposite directions this pressure variation is proportional to the product of the (first-order) amplitudes of the two waves and is of twice their frequency. The pressure variation at infinite depth is found to be closely related to changes in the potential energy of the wave train as a whole. By introducing the two-dimensional frequency spectrum of the motion it is shown that in the general case variations in the mean pressure over a wide area only occur when the spectrum contains wave groups of the same wave-length travelling in opposite directions. (These are called opposite wave groups.) In section 4 the effect of the compressibility of the water is considered by evaluating the motion of an opposite pair of waves in a heavy compressible fluid to the second order of approximation. In place of the pressure variation at infinite depth, waves of compression are set up, and there is resonance between the bottom and the free surface when the depth of water is about ($\frac{1}{2}$n + $\frac{1}{4}$) times the length of a compression wave (n being an integer). The motion in a surface layer whose thickness is of the order of the length of a Stokes wave is otherwise unaffected by the compressibility. Section 5 is devoted to the question whether the second-order pressure variations in surface waves are capable of generating microseisms of the observed order of magnitude. By considering the displacement of the sea bed due to a concentrated force at the upper surface of the water it is shown that the effect of resonance will be to increase the disturbance by a factor of the order of 5 over its value in shallow water. The results of section section 3 and 4 are used to derive an expression for the vertical displacement of the ground in terms of the frequency characteristics of the waves. The displacement from a storm area of 1000 sq.km. is estimated to be of the order of 6$\cdot $5$\mu $, at a distance of 2000 km. Ocean waves may therefore be the cause of microseisms, provided that there is interference between groups of waves of the same frequency travelling in opposite directions. Suitable conditions of wave interference may occur at the centre of a cyclonic depression or possibly if there is wave reflexion from a coast. In the latter case the microseisms are likely to be smaller, except perhaps locally. Confirmation of the present theory is provided by the observations of Bernard and Deacon, who discovered independently that the period of the microseisms is in many cases about half that of the ocean waves associated with them.

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