One Dimensional Wave Equation Derivation

The wave equation in classical physics is considered to be an important second-order linear partial differential equation to describe the waves. The wave equation arises in fields like fluid dynamics, electromagnetics and acoustics.

d’Alembert discovered the one-dimensional wave equation in the year 1746, after ten years Euler discovered the three-dimensional wave equation.

One Dimensional Wave Equation Derivation

Consider the relation between Newton’s law that is applied to the volume ΔV in the direction x:

\(\Delta F=\Delta m\frac{dv_{x}}{dt}\) (Newton’s law)

Where,

F: force acting on the element with volume ΔV

\(\Delta F_{x}=-\Delta p_{x}\Delta S_{x}\)

\(=(\frac{\partial p}{\partial x}\Delta x+\frac{\partial p}{\partial x}dt)\Delta S_{x}\)

\(\simeq -\frac{\partial p}{\partial x}\Delta V\)

\(-\Delta V\frac{\partial p}{\partial x}=\Delta m\frac{dv_{x}}{dt}\) (as dt is small, it is not considered and ΔSx is in x direction so ΔyΔz and from Newton’s law)

\(=\rho \Delta V\frac{dv_{x}}{dt}\)

From \(\frac{dv_{x}}{dt} as \frac{\partial v_{x}}{\partial t}\)

\(\frac{dv_{x}}{dt}=\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}\approx \frac{\partial v_{x}}{\partial x}\)

\(-\frac{\partial p}{\partial x}=\rho \frac{\partial v_{x}}{\partial t}\)

Above equation is known as the equation of motion.

\(-\frac{\partial }{\partial x}(\frac{\partial p}{\partial x})=\frac{\partial }{\partial x}(\rho \frac{\partial v_{x}}{\partial t})\)

\(=\rho \frac{\partial }{\partial t}(\frac{\partial v_{x}}{\partial x})\)

\(-\frac{\partial^2 p}{\partial x^2}=\rho \frac{\partial }{\partial t}(-\frac{1}{K}\frac{\partial p}{\partial t})\) (from conservation of mass)

\(\frac{\partial p^{2}}{\partial x^{2}}-\frac{\rho }{K}\frac{\partial^2 p}{\partial t^2}=0\)

Where,

K: bulk modulus

Rewriting the above equation:

\(\frac{\partial p^{2}}{\partial x^{2}}-\frac{1}{c}^{2}\frac{\partial^2 p}{\partial t^2}=0\)

Where,

c: velocity of sound given as \(c=\sqrt{\frac{K}{\rho }}\)

Thus, above is the one-dimensional wave equation derivation.

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