In this article we take a look at group velocity and at the extraction of the envelope of a wave packet using the ideas of the Hilbert transform.
Archive for April, 2018
Waves Packed in Envelopes
Published April 26, 2018 Occasional ClosedTags: Fluid Dynamics, Geophysics, Wave Motion
Geodesics on the Spheroidal Earth-II
Published April 19, 2018 Irish Times ClosedTags: Geometry, Geophysics, Navigation, Pythagoras
Geodesy is the study of the shape and size of the Earth, and of variations in its gravitational field. The Earth was originally believed to be flat, but many clues, such as the manner in which ships appear and disappear at the horizon, and the changed perspective from an elevated vantage point, as well as astronomical phenomena, convinced savants of its spherical shape. In the third century BC, Eratosthenes accurately estimated the circumference of the Earth [TM137 or search for “thatsmaths” at irishtimes.com].
Geodesic at bearing of 60 degrees from Singapore. Passes close to Quito, Ecuador. Note that it is not a closed curve: it does not return to Singapore.
Geodesics on the Spheroidal Earth – I
Published April 12, 2018 Occasional ClosedTags: Geometry, Geophysics, Navigation, Pythagoras
Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not?
A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.
Solving PDEs by a Roundabout Route
Joseph Fourier (1768-1830)
Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only in frequency [TM136 or search for “thatsmaths” at irishtimes.com].
ThatsMaths 