Oceanic microseisms are small oscillations of the ground, in the frequency range of 0.05-0.3 Hz, associated with the occurrence of energetic ocean waves of half the corresponding frequency. In 1950, ...Longuet-Higgins suggested in a landmark theoretical paper that (i) microseisms originate from surface pressure oscillations caused by the interaction between oppositely travelling components with the same frequency in the ocean wave spectrum, (ii) these pressure oscillations generate seismic Stoneley waves on the ocean bottom, and (iii) when the ocean depth is comparable with the acoustic wavelength in water, compressibility must be considered. The efficiency of microseism generation thus depends on both the wave frequency and the depth of water. While the theory provided an estimate of the magnitude of the corresponding microseisms in a compressible ocean, its predictions of microseism amplitude heretofore have never been tested quantitatively. In this paper, we show a strong agreement between observed microseism and calculated amplitudes obtained by applying Longuet-Higgins' theory to hindcast ocean wave spectra from the North Atlantic Ocean. The calculated vertical displacements are compared with seismic data collected at stations in North America, Greenland, Iceland and Europe. This modelling identifies a particularly energetic source area stretching from the Labrador Sea to south of Iceland, where wind patterns are especially conducive to generating oppositely travelling waves of same period, and the ocean depth is favourable for efficient microseism generation through the 'organ pipe' resonance of the compression waves, as predicted by the theory. This correspondence between observations and the model predictions demonstrates that deep ocean nonlinear wave-wave interactions are sufficiently energetic to account for much of the observed seismic amplitudes in North America, Greenland and Iceland.
Snub polyhedra and organic growth Michael S Longuet-Higgins
Proceedings - Royal Society. Mathematical, physical and engineering sciences,
02/2009, Letnik:
465, Številka:
2102
Journal Article
Recenzirano
Odprti dostop
This paper describes a new application of polyhedral theory to the growth of the outer sheath of certain viruses. Such structures are often modular, consisting of one or two types of units arranged ...in a symmetric pattern. In particular, the polyoma virus has a structure apparently related to the snub dodecahedron. Here, we consider the problem of how such patterns might grow in time, starting from a given number N of randomly placed circles on the surface of a sphere. The circles are first jostled by random perturbations, then their radii are enlarged, then they are jostled again, and so on. This 'yin-yang' method of growth can result in some very close packings. When N=12, the closest packing corresponds to the snub tetrahedron, and when N=24 the closest packing corresponds to the snub cube. However, when N=60 the closest packing does not correspond to the snub dodecahedron but to a less-symmetric arrangement. Special attention is given to the structure of the human polyoma virus, for which N=72. It is shown that the yin-yang procedure successfully assembles the observed structure provided that the 72 circles are pre-assembled in clusters of six. Each cluster consists of five circles arranged symmetrically around a sixth at the centre, as in a flower with five petals. This has implications for the assembly of the capsomeres in a polyoma virus.
In very gradually shoaling coastal water the energy of incident waves appears to be absorbed not by breaking at the upper surface, but predominantly by turbulent dissipation near the rippled sea bed. ...The question has been asked whether there can be then any wave set-up, that is any increase in the mean water level, at, or close to, the shoreline. To answer this question the equations of wave energy and momentum in water of slowly varying depth are generalized so as to include the presence of a dissipative boundary layer at the bottom. It is then shown that the resulting equation for the mean surface slope can be integrated exactly, to give the mean surface depression (the ‘set-down’) in terms of the local wave amplitude and water depth, outside the surf zone. In the special case of a uniform beach slope $s$, a closed expression is obtained for the wave amplitude in terms of the local depth, under two different sets of conditions: (i) when the thickness of the boundary layer at the bottom is assumed to be constant, and (ii) for waves over a rippled bed, when the boundary-layer thickness corresponds to the measured dissipation of energy in oscillatory waves over steep sand ripples. In both cases it is found that there exists a maximum bottom slope $s$ below which the wave amplitude must diminish monotonically towards the shoreline. This maximum value of $s$ is of order $10^{-3}$. The waves can indeed penetrate close to the shoreline without breaking, and the corresponding wave set-up is negligible. An example of where suitable conditions exist is on the continental shelf off North Carolina.
A simple but highly accurate approximation to the form and speed of a limiting Stokes wave in deep water was recently given by Rainey and Longuet-Higgins R.C.T. Rainey, M.S. Longuet-Higgins, A close ...one-term approximation to the highest Stokes wave on deep water, Ocean Eng. 33 (2006) 2012–2024. An expression for the corresponding velocity field in the interior of the fluid was also suggested. Here this expression is examined numerically and it is shown that there is no singularity in the interior, but that there is a very weak singularity at the wave crest. The nature of the singularity is exhibited by use of a property of Stokes’s 120° corner-flow, namely that
q
2
/
gr
is everywhere equal to 1, where
q is the particle speed,
g denotes gravity, and
r is the radial distance from the wave crest.
Theoretical arguments suggest that progressive gravity waves incident on a vertical
wall can produce periodic standing waves only if the incident wave steepness ak
is quite small, certainly less than ...0.284. Laboratory experiments are carried out in
which an incident wave train of almost uniform amplitude meets a vertical barrier.
At wave steepnesses greater than 0.236 the resulting motion near the barrier is non-periodic. A growing instability is observed in which every third wave crest is steeper
than its neighbours. The steep waves develop sharp crests, or vertical jets. The two
neighbouring crests are rounded, at-topped, or of intermediate form. The instability
grows by a factor of about 2.2 for every three wave periods, almost independently of
the incident wave steepness.
Viscous streaming from an oscillating spherical bubble Longuet-Higgins, Michael S.
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences,
02/1998, Letnik:
454, Številka:
1970
Journal Article
Recenzirano
Asymmetric patterns of streaming around a single sonoluminescent bubble have been observed by Lepoint-Mullie and co-workers. The present paper considers theoretically the viscous streaming from a ...spherical bubble undergoing small radial and lateral oscillations simultaneously, using an extension of the method of Davidson & Riley so as to include radial oscillations. On the assumption that the radial and lateral oscillations are of comparable magnitude, it is shown that the presence of the radial oscillations greatly enhances the streaming. The streaming is greatest when the phase difference between the oscillations is ±90°, and reverses in direction when the phase difference passes through zero or 180°.
Vertical jets from standing waves Longuet-Higgins, Michael S
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences,
02/2001, Letnik:
457, Številka:
2006
Journal Article
Recenzirano
When a shallow-water wave meets a steep cliff or harbour wall, a jet is sometimes thrown upwards with very high initial acceleration. In the present work it is shown that a similar phenomenon can ...occur in a standing wave in deep water, provided that the initial shape of the trough is narrow and an almost circular arc. During collapse of the trough, vertical accelerations exceeding 10g are found, with subsequent vertical velocities exceeding 1.7c, where c is the phase-speed of a small-amplitude progressive wave. The vertical acceleration, plotted as a function of the time, is shown to be very sensitive to the initial shape of the wave trough. The presence of a much shorter wave superposed on the initial wave profile can further increase the maximum acceleration and velocity.
A remarkably accurate approximation to the profile of a limiting progressive gravity wave in water of infinite depth is given by the expression
(0.1)
y
/
L
=
A
cosh
(
x
/
L
)
,
where
L is the ...wavelength,
x and
y are horizontal and vertical coordinates and
A is a constant given by
(0.2)
A
=
1
/
(
3
sinh
1
2
)
=
1.1080
.
This determines the wave steepness
(
H
/
L
)
as 0.14140 a proportional error of less than 0.3% (about 10 times closer than previous approximations) and the phase speed
c
/
(
gL
)
1
/
2
as 0.43511, which is accurate to within 0.2%. The entire surface profile is accurate to less than 0.7%. The corresponding particle velocities are found by a straightforward numerical integration. It is shown that this type of approximation cannot be made exact by the introduction of further parameters.
Standing gravity waves forced beyond the maximum height for perfect periodicity
can produce vertical jets with sharp-pointed tips. In this paper, canonical forms for
the wave crests are derived which ...display sharp cusps in the limit as the time t tends
to infinity. The theoretical profile is in general quartic in the space coordinates, and
can describe the smooth transition of a fairly low wave crest to a cusped form. There
is no singularity as the surface slope passes through 45°.
The Kepler Memorial in Graz Longuet-Higgins, Michael
The Mathematical intelligencer,
12/2011, Letnik:
33, Številka:
4
Journal Article
Recenzirano
Longuet-Higgins profiles Johannes Kepler, a German mathematician, astronomer, and astrologer. He was a pivotal figure in the emergence of modern science from earlier superstitions and rigid doctrine. ...He also made significant contributions to optics; in his "Dioptrice" (1611) he expounded the theory of refraction by lenses and suggested the principle of the inverted telescope. While in Graz and also in Prague, Kepler found that one of his duties was to compose astrological forecasts. With the help of a good sense of the current political situation, he carried this out with some success, but with increasing skepticism. Toward the end of his life he would not make astrological forecasts himself, although he allowed others to use his data for such purposes.
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