Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects

Constance Atieno Ojwando

doi:doi:10.11648/j.ajam.20160404.11

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Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects

Constance Atieno Ojwando

Published in American Journal of Applied Mathematics (Volume 4, Issue 4)

Received: 10 May 2016 Accepted: 28 May 2016 Published: 17 June 2016

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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.

Published in	American Journal of Applied Mathematics (Volume 4, Issue 4)
DOI	10.11648/j.ajam.20160404.11
Page(s)	169-174
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Cable, Vibration, Signal, Idealized Circular Drumhead, Pheromone Evaporation

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. 5^th 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.

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Constance Atieno Ojwando. (2016). Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects. American Journal of Applied Mathematics, 4(4), 169-174. https://doi.org/10.11648/j.ajam.20160404.11

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ACS Style

Constance Atieno Ojwando. Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects. Am. J. Appl. Math. 2016, 4(4), 169-174. doi: 10.11648/j.ajam.20160404.11

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AMA Style

Constance Atieno Ojwando. Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects. Am J Appl Math. 2016;4(4):169-174. doi: 10.11648/j.ajam.20160404.11

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@article{10.11648/j.ajam.20160404.11,
  author = {Constance Atieno Ojwando},
  title = {Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects},
  journal = {American Journal of Applied Mathematics},
  volume = {4},
  number = {4},
  pages = {169-174},
  doi = {10.11648/j.ajam.20160404.11},
  url = {https://doi.org/10.11648/j.ajam.20160404.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160404.11},
  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.},
 year = {2016}
}

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TY  - JOUR
T1  - Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects
AU  - Constance Atieno Ojwando
Y1  - 2016/06/17
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160404.11
DO  - 10.11648/j.ajam.20160404.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 169
EP  - 174
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160404.11
AB  - 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.
VL  - 4
IS  - 4
ER  -

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Author Information

Constance Atieno Ojwando

Department of Industrial and Engineering Mathematics, Technical University of Kenya, Nairobi, Kenya

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APA Style

Constance Atieno Ojwando. (2016). Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects. American Journal of Applied Mathematics, 4(4), 169-174. https://doi.org/10.11648/j.ajam.20160404.11

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ACS Style

Constance Atieno Ojwando. Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects. Am. J. Appl. Math. 2016, 4(4), 169-174. doi: 10.11648/j.ajam.20160404.11

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AMA Style

Constance Atieno Ojwando. Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects. Am J Appl Math. 2016;4(4):169-174. doi: 10.11648/j.ajam.20160404.11

Copy | Download

@article{10.11648/j.ajam.20160404.11,
  author = {Constance Atieno Ojwando},
  title = {Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects},
  journal = {American Journal of Applied Mathematics},
  volume = {4},
  number = {4},
  pages = {169-174},
  doi = {10.11648/j.ajam.20160404.11},
  url = {https://doi.org/10.11648/j.ajam.20160404.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160404.11},
  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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Application of Partial Differential Equations to Drum Head Vibration, Signal Transmission and Chemical Communication in Insects
AU  - Constance Atieno Ojwando
Y1  - 2016/06/17
PY  - 2016
N1  - https://doi.org/10.11648/j.ajam.20160404.11
DO  - 10.11648/j.ajam.20160404.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 169
EP  - 174
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20160404.11
AB  - 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.
VL  - 4
IS  - 4
ER  -

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