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Duplex working. Developed early on (before 1900) this technique allowed cables to earn their keep twice by being used simultaneously in each direction.
Siphon recorder. An early recording instrument for automatically turning the received signal into a paper and ink "copy".
Paper tape. The precursor to 1970s computer I/O technology, this was initially used as a way to prepare messages and feed them onto the line automatically.
Loaded cables. Introduced in 1924. Compensated for cable capacitance and allowed 10x speed increase over the connection.
Heurtley magnifier. 1908. Using heated wires and blowers (!) acted as an amplifier of the signal.
Mechanical multiplexing. To allow several channels each at operator speed on one cable - the overall speed being that of operation of the cable / magnifier. Literally used mechanical distributors to switch between operators synchronously.
Gutta Percha. This natural substance, discovered in Malaya at the end of the last century was of major importance as the insulator for the cable. Gutta Percha was extracted from trees and was one of the first of the natural plastics to be exploited by man. It is chemically similar to rubber, but the shape of the molecule gives it different properties.
More technical information from the museum archive.
Pictures of an old cable de-constructed.
How cable length and other parameters were found to affect cable operating speed.
Cables in 1931 at Weston.

SN1 Y and Z (2 core)

Complete information on the cables

SN2

SN3

SN4

Who did what?
The Germans made the cables, although Sir Alexander Siemens had his factory at Woolwich, UK.
The Americans conceived the idea of transatlantic cables and American money financed the companies.
The British laid the cables and trained the cable operators. They also invented most of the cable operating equipment used.
The above statements are necessarily broad brush and are taken in essence from the Canso Web site. I would welcome any more detailed information about significant contributors to the technology, French, Irish, Canadian or other! Frederick George Creed was a significant pioneer in telecommunications and grew up in Canso. [Email me]
What about the Atlantic sea bed and plate tectonics ? (asked by my son Patrick)
The fact that the sea-bed in mid Atlantic is far from peaceful was not known at the time these cables were first laid but, although earthquakes have caused problems there appears to have been no general risk to such cables from under-sea lava flows etc. The IEE website has a fascinating paper from which I  quote:

"Crucial to planning the route was knowledge of the depth of the ocean and the nature of the seabed on which the cable would rest. The first systematic soundings of the North Atlantic from Newfoundland to the coast of Ireland were made by Lieutenant O H Berryman USN on the US Brig Dolphin in 1853. ... He found that a wide plateau extended nearly the whole distance between Ireland and North America at depths varying from 700 to 2400 fathoms. Further south the depths were 6000 to 7000 fathoms. The plateau between Ireland and Newfoundland was built up of soft ooze formed by minute shells of microscopic sea creatures ... Maury[M] christened this table-land ‘Telegraph Plateau’. Several years later soundings taken by the cable ship “Faraday” during a cable laying expedition for the Siemens Brothers showed that the plateau was not so plateau-like as Maury supposed ... In 1862 HMS Porcupine undertook an exploratory voyage in the western end of the Atlantic in the same latitude as Ireland. She found that the sea bottom in this area, instead of forming a precipice as many had expected, formed a gentle slope."

See also the admission from the company chief executive in 1910 that the initial cable laying chose the wrong route.
The cables that still rest on the bottom will have been stretched by some 2.5 metres or so, since laying, by tectonic movement of the sea floor but I take it this is well within their capabilities when evened over a significant distance.
This quotation from "The Atlantic Cable" by Bern Dibner is also interesting:

"There are, however, undoubtedly occasional submarine shiftings of the earth's crust. Recently a British company found that its cable from St. Helena to Cape Town, which had been proved by soundings in 1899 to lie at a certain point on a bottom three miles deep, had been raised at this point to a depth of only three fourths of a mile, and broken. That looks like the action of an undersea earthquake or other volcanic movement."

"A few years ago a fault occurred for the first time in the most northern of our cables, which happens also to be the oldest active trans-Atlantic cable. While grappling on this repair, we located a volcanic region of high scraggy hills and a great many deep ravines and steep cliffs, with a central crater containing volcanic glass, brown umber, and yellow umber or ochre. At one place a difference in depth of 1,130 fathoms (6,780 feet) showed within an area of three miles."

Power supply
The cable offices were set up before the advent of a general power supply and batteries were used at first. An electric power plant was established at Hazel Hill in 1912 - very early in the development of this technology. Pictures of the Weston station equipment clearly show huge battery arrays in 1929.

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© John Crellin 2009