Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects
American Journal of Applied Mathematics
Volume 4, Issue 4, August 2016, Pages: 169-174
Received: May 10, 2016; Accepted: May 28, 2016; Published: Jun. 17, 2016
Views 5271      Downloads 202
Author
Constance Atieno Ojwando, Department of Industrial and Engineering Mathematics, Technical University of Kenya, Nairobi, Kenya
Article Tools
Follow on us
Abstract
The application of partial differential equations to drumhead vibration, signal transmission and chemical communication in insects has been investigated. This study looked at three cases in communication namely: a single drumbeat on an idealized circular drumhead being struck at the centre of the circular membrane; signal transmission in a Coaxial cable and finally concentration of pheromone in still air within a tunnel. The governing equation of Case I was solved using separation of variables method whereas Laplace transform was used to solve Cases II and III. MATLAB software was then used to obtain simulations on the solutions obtained from the three cases. It was concluded that vibrations of a circular drumhead as a result of a single drumbeat mainly concentrates around the centre of the membrane and tends to flatten out further away from the centre. Signal propagation decreases with increasing distance and so the shorter the cable the less signal strength is lost. Finally, concentration of pheromone can only last for a given time and distance.
Keywords
Cable, Vibration, Signal, Idealized Circular Drumhead, Pheromone Evaporation
To cite this article
Constance Atieno Ojwando, Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects, American Journal of Applied Mathematics. Vol. 4, No. 4, 2016, pp. 169-174. doi: 10.11648/j.ajam.20160404.11
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
G. O. Lartey & A. Adu. Sackey, The mathematics of Ghanaian Circular Musical Drumheads: varying tension versus constant tension. Journal of Science and Technology, Vol. 33, No 2(2013) pp 81-87.
[2]
Doma S. B, Sirafy I. H, Borai M. M and El-Sharif A. H Pertubation treatment for the vibrations of a circular membrane subject to a restorative force, Alexandria Journal of Mathematics, Vol. 1, No 1, June 2010.
[3]
Mark C, Vibration of the Koch drums. Mathematica in Education and Research, 12(2), 2007.
[4]
Marshall P, Drumkit / Drumsetmechanics and construction, Drum Dojo. http://www.drumdojo.con/kit/equipment.htm (2000).
[5]
Muhammad Suleiman, Qingbiao Wu, T. M. Elzaki, Explicit Analytic Solution of Vibration Equation for large domain by mean of the Elzaki projected differential transform method. Journal of Advances in Mathematics, Vol. 11, No 5, October 13, 2015.
[6]
Renarto Orta, Lecture notes on transmission line theory. November 2012.
[7]
Vineet K. Srivasta, Mukesh K. Awasthi, R. K Chaurasia & M. Tamsir, The Telegraph Equation and its Solution by Reduced Differential Transform Method, Hindawi Publishing Corporation, Modelling & Simulation in engineering. Vol. 2013, Article ID 746351.
[8]
Distasio M (2010), Numerical solutions to the passive Cable equation: using finite difference approximations to partial derivatives. Published Thesis.
[9]
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts & P. Walter. Molecular Biology of the Cell. 5th edition, Garland Sc. November 2007.
[10]
W. C. Agosta, Chemical Communication: The Language of Pheromones, 1st edition. W, H. Freeman & Co. August 1992.
[11]
Katie K, Mathematical modeling of Ant pheromone: Determination of optimum pheromone evaporation rate and simulation of pheromone tracking abilities. (2008) Unpublished Thesis.
[12]
Bollsard E, Degond P and Motsh S, Trail formation based on directed pheromone deposition-September 15,2011; Thesis.
[13]
B. Neta, Partial Differential Equations, MA 3132 Lecture Notes, Naval Postgraduate School, Monterey California 2012.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186