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Encyclopedia of Systems Biology
pp 16381640
Partial Differential Equations, Wave Equation
 Richard H. ClaytonAffiliated withDepartment of Computer Science, University of Sheffield Email author
Definition
The wave equation is a secondorder linear partial differential equation that describes how a scalar quantity u changes with space and time. In 1D the wave equation is:where u is a scalar quantity such as displacement or density, t is time, x is distance, and c is a constant that describes the speed of the wave.
$$ \frac{{{\partial^2}u(x,t)}}{{\partial {t^2}}} = {c^2}\frac{{{\partial^2}u(x,t)}}{{\partial {x^2}}} $$
(1)
The wave equation is used to describe the propagation of waves in a wide variety of mechanical systems, including acoustic (pressure) waves where displacement is in the same direction as wave travel, and vibration of a membrane where displacement is transverse to the direction of wave travel.
Characteristics
Derivation from First Principles
Consider a vibrating string, which has tension T and a constant mass per unit length ρ (Kreyszig 1983). We assume that the string is perfectly elastic, that the displacement of the string is small, and we i ...
This is an excerpt from the content
 Title
 Partial Differential Equations, Wave Equation
 Reference Work Title
 Encyclopedia of Systems Biology
 Pages
 pp 16381640
 Copyright
 2013
 DOI
 10.1007/9781441998637_275
 Print ISBN
 9781441998620
 Online ISBN
 9781441998637
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media, LLC
 Additional Links
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 eBook Packages
 Editors

 Werner Dubitzky ^{(1)}
 Olaf Wolkenhauer ^{(2)}
 KwangHyun Cho ^{(3)}
 Hiroki Yokota ^{(4)}
 Editor Affiliations

 1. Biomedical Sciences Research Institute, University of Ulster
 2. Department of Computer Science, University of Rostock
 3. Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST)
 4. Department of Biomedical Engineering, Rensselaer Polytechnic Institute
 Authors

 Dr. Richard H. Clayton ^{(02751)}
 Author Affiliations

 02751. Department of Computer Science, University of Sheffield, 211 Portobello Street, Sheffield, UK
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