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:

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


F: force acting on the element with volume ΔV

\(\begin{array}{l}\Delta F_{x}=-\Delta p_{x}\Delta S_{x}\end{array} \)

\(\begin{array}{l}=(\frac{\partial p}{\partial x}\Delta x+\frac{\partial p}{\partial x}dt)\Delta S_{x}\end{array} \)

\(\begin{array}{l}\simeq -\frac{\partial p}{\partial x}\Delta V\end{array} \)

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

\(\begin{array}{l}=\rho \Delta V\frac{dv_{x}}{dt}\end{array} \)


\(\begin{array}{l}\frac{dv_{x}}{dt} as \frac{\partial v_{x}}{\partial t}\end{array} \)

\(\begin{array}{l}\frac{dv_{x}}{dt}=\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}\approx \frac{\partial v_{x}}{\partial x}\end{array} \)

\(\begin{array}{l}-\frac{\partial p}{\partial x}=\rho \frac{\partial v_{x}}{\partial t}\end{array} \)

Above equation is known as the equation of motion.

\(\begin{array}{l}-\frac{\partial }{\partial x}(\frac{\partial p}{\partial x})=\frac{\partial }{\partial x}(\rho \frac{\partial v_{x}}{\partial t})\end{array} \)

\(\begin{array}{l}=\rho \frac{\partial }{\partial t}(\frac{\partial v_{x}}{\partial x})\end{array} \)

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

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


K: bulk modulus

Rewriting the above equation:

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


c: velocity of sound given as

\(\begin{array}{l}c=\sqrt{\frac{K}{\rho }}\end{array} \)

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

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This video below explains the general wave equation, phase, and the motion of particles of the wave.

Frequently Asked Questions – FAQs

Who discovered the one-dimensional wave equation?

D’Alembert discovered the one-dimensional wave equation in the year 1746.

Who discovered the three-dimensional wave equation?

The three-dimensional wave equation was discovered by Euler.

What is the formula to find the velocity of sound (c)?

Velocity of sound is given by the formula :

\(\begin{array}{l}c=\sqrt{\frac{K}{\rho }}\end{array} \)

Can Waves exist in two or three dimensions?

Yes, waves can exist in two or three dimensions.

State true or false: water wave is two dimensional wave.


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