Mathematis of tunnels
The mathematics of tunnels encompasses a range of engineering and scientific challenges associated with creating passageways through various materials, including rock, earth, and water. Tunnel engineers must consider factors such as seepage, weight, and geological conditions. To address these challenges, mathematicians employ various mathematical models that involve fields such as graph theory, differential equations, geometry, probability, and trigonometry. Significant projects like the Channel Tunnel between England and France and the Gotthard Base Tunnel in Switzerland illustrate the complexities involved, from managing water inflow to ensuring precision during construction.
Historically, ancient tunnels, such as the Eupalinian aqueduct on the island of Samos, showcase early engineering feats that required advanced mathematical techniques, including the use of similar triangles. Modern theoretical explorations, such as the concept of frictionless tunnels, propose intriguing scenarios for rapid travel through the Earth, though practical implementation faces significant obstacles. Overall, the mathematics of tunnels is a fascinating intersection of theory and real-world application, influencing both current engineering practices and ongoing scientific inquiry.
Mathematis of tunnels
SUMMARY: Tunnels have long presented interesting mathematical and engineering challenges.
A tunnel is a connecting passageway through materials like rock, earth, or water. Tunnel engineers must take into consideration issues like seepage and weight. Scientists and mathematicians create mathematical models of tunnels to investigate aspects like aquifers and safety issues. Analytic and closed form solutions are useful in engineering. Mathematical fields like graph theory, differential equations, geometry, probability, and trigonometry are important for modeling and measuring tunnels.
Mathematically Challenging Tunnels
The Channel Tunnel between England and France represented a significant engineering and mathematical challenge. At the time of its building and into the twenty-first century, it had the longest undersea length of any tunnel in the world. It presented significant challenges including problems related to the topology and geology of the rock through which it was bored; significant water pressure; ventilation; communication; and the fact that construction was started at the same time from both ends, requiring exceptional precision to meet in the middle. This tunnel serves as a model for other underwater tunnel projects, and many teachers use it to present mathematics concepts. Scientists and mathematicians also experiment with digital and physical wind tunnels as well as quantum tunnels.
In 1996, construction of the Gotthard Base Tunnel began in Switzerland. Designed to improve rail traffic and reduce travel time through the Swiss Alps, the project was the main rail connection between Switzerland and Italy and would be the world's longest railway tunnel. Engineers were tasked with considering tunnel alignment, structural stability, management of water inflow, and the geological makeup and conditions of the terrain. After taking nearly twenty years to complete, it was deemed a success. However, in 2023, an accident derailed a train and shutdown the tunnel for nearly a year.
Ancient Tunneling
The problem of delivering fresh water to large populations has been an ongoing human endeavor since ancient times. In the sixth century B.C.E., a one-kilometer tunnel was dug through a large hill of solid limestone to bring water from the mountains to the main city on the island of Samos. The Eupalinian aqueduct on Samos was designed by the ancient Greek engineer Eupalinos of Megara. The tunnelers worked from both ends and met in the middle, with an error less than 0.06 percent of the height. To achieve this remarkable result, Hero of Alexandria, an ancient Greek engineer and inventor, theorized that the tunnelers used a method based on similar triangles in order to determine the correct direction for tunneling in his work On the Construction of the Eupalinos. Mathematicians and scientists continue to debate the pros and cons of various theories of how this engineering marvel was constructed.
Modeling Tunnels
Tunnels can be modeled using coordinate geometry and equations. For example, knowing the height and width of a parabolic tunnel, one can determine the tunnel’s height at different distances from the base center. To solve this problem, one needs to find the equation for the parabola choosing convenient x-y axes.
Frictionless Tunnels
The possibility of mathematical modeling allows for innovative and challenging ideas. What if a frictionless tunnel would be bored through Earth’s center? Paul Cooper, a mathematician fond of Jules Verne’s books, tried to answer this question in an issue of the American Journal of Physics. He set up and solved by computer a set of differential equations for tunnels that would provide minimum gravity-powered travel time between any two cities on Earth.
According to Cooper’s differential equations, by freefalling in airless, frictionless, straight-line tunnels, passenger vehicles powered only by the pull of gravity could theoretically travel between any two points on the Earth’s surface in a total time of only 42.2 minutes. Accelerated by the force of gravity on the first half of the trip, the vehicle would gain just enough kinetic energy to coast up to the other side of the Earth. However, significant obstacles make such a project impossible in the twenty-first century. Subterranean temperatures reach extremes, even for relatively shallow tunnels of only a few miles deep, requiring huge cooling systems for vehicles. Also, it is almost certainly impossible to create a completely frictionless path without a rail or track of some type, leaving the vehicle with insufficient kinetic energy to complete its trip without a source of additional power. Consequently, such a tunnel is still science fiction more than science.
Bibliography
Apostle, Tom. “The Tunnel of Samos.” Engineering and Science 1 (2004).
“The Channel Tunnel.” Institution of Civil Engineers, www.ice.org.uk/what-is-civil-engineering/what-do-civil-engineers-do/the-channel-tunnel. Accessed 10 Oct. 2024.
Cooper, P. W. “Through the Earth in Forty Minutes.” American Journal of Physics 34, no. 1 (1966).
Goldsmith, Hugh, and Patrick Boeuf. “Digging beneath the Iron Triangle: The Chunnel with 2020 Hindsight.” Journal of Mega Infrastructure & Sustainable Development, vol. 1, 2019, pp. 79-93. Taylor & Francis Online, doi.org/10.1080/24724718.2019.1597407. Accessed 10 Oct. 2024.
“Gotthard Base Tunnel, Switzerland.” Railway Technology, 17 Sept. 2021, www.railway-technology.com/projects/gotthard-base-tunnel/. Accessed 10 Oct. 2024.
Lunardi, Pietro. Design and Construction of Tunnels: Analysis of Controlled Deformations in Rock and Soils. New York: Springer, 2008.
Oxlade, Chris. Tunnels. Portsmouth, NH: Heinemann-Raintree, 2005. Tsokas, Gregory N., et al. "Investigating behind the Lining of the Tunnel of Eupalinus in Samos (Greece) Using ERT and GPR." Near Surface Geophysics, vol. 13, no. 6, 2015, pp. 571-583. Wiley Online Library, doi.org/10.3997/1873-0604.2015012. Accessed 10 Oct. 2024.