Reference Work Entry

Encyclopedia of Systems Biology

pp 1638-1640

Partial Differential Equations, Wave Equation

  • Richard H. ClaytonAffiliated withDepartment of Computer Science, University of Sheffield Email author 


The wave equation is a second-order linear partial differential equation that describes how a scalar quantity u changes with space and time. In 1-D the wave equation is:
$$ \frac{{{\partial^2}u(x,t)}}{{\partial {t^2}}} = {c^2}\frac{{{\partial^2}u(x,t)}}{{\partial {x^2}}} $$
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.

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.


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 ...

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