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 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:
\(\begin{array}{l}\Delta F=\Delta m\frac{dv_{x}}{dt}\,\, \textup{(Newton’s law)}\end{array} \)
Where,
F: force acting on the element with volume ΔV
\(\begin{array}{l}\Delta F_{x}=-\Delta p_{x}\Delta S_{x}\end{array} \)
\(\begin{array}{l}=(\frac{\partial p}{\partial x}\Delta x+\frac{\partial p}{\partial x}dt)\Delta S_{x}\end{array} \)
\(\begin{array}{l}\simeq -\frac{\partial p}{\partial x}\Delta V\end{array} \)
\(\begin{array}{l}-\Delta V\frac{\partial p}{\partial x}=\Delta m\frac{dv_{x}}{dt}\,\, \textup{(as dt is small, it is not considered and ΔSx is in x direction so ΔyΔz and from Newton’s law)}\end{array} \)
\(\begin{array}{l}=\rho \Delta V\frac{dv_{x}}{dt}\end{array} \)
\(\begin{array}{l}\textup{From}\,\, \frac{dv_{x}}{dt} as \frac{\partial v_{x}}{\partial t}\end{array} \)
\(\begin{array}{l}\frac{dv_{x}}{dt}=\frac{\partial v_{x}}{\partial t}+v_{x}\frac{\partial v_{x}}{\partial x}\approx \frac{\partial v_{x}}{\partial x}\end{array} \)
\(\begin{array}{l}-\frac{\partial p}{\partial x}=\rho \frac{\partial v_{x}}{\partial t}\end{array} \)
Above equation is known as the equation of motion.
\(\begin{array}{l}-\frac{\partial }{\partial x}(\frac{\partial p}{\partial x})=\frac{\partial }{\partial x}(\rho \frac{\partial v_{x}}{\partial t})\end{array} \)
\(\begin{array}{l}=\rho \frac{\partial }{\partial t}(\frac{\partial v_{x}}{\partial x})\end{array} \)
\(\begin{array}{l}-\frac{\partial^2 p}{\partial x^2}=\rho \frac{\partial }{\partial t}(-\frac{1}{K}\frac{\partial p}{\partial t}) \,\, \textup{(from conservation of mass)}\end{array} \)
\(\begin{array}{l}\frac{\partial p^{2}}{\partial x^{2}}-\frac{\rho }{K}\frac{\partial^2 p}{\partial t^2}=0\end{array} \)
Where,
K: bulk modulus
Rewriting the above equation:
\(\begin{array}{l}\frac{\partial p^{2}}{\partial x^{2}}-\frac{1}{c}^{2}\frac{\partial^2 p}{\partial t^2}=0\end{array} \)
Where,
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 Q1
Who discovered the one-dimensional wave equation?
D’Alembert discovered the one-dimensional wave equation in the year 1746.
Q2
Who discovered the three-dimensional wave equation?
The three-dimensional wave equation was discovered by Euler.
Q3
What is the formula to find the velocity of sound (c)?
Velocity of sound is given by the formula :
\(\begin{array}{l}c=\sqrt{\frac{K}{\rho }}\end{array} \)
Q4
Can Waves exist in two or three dimensions?
Yes, waves can exist in two or three dimensions.
Q5
State true or false: a water wave is a two dimensional wave. Stay tuned to BYJU’S and Fall in Love with Learning!
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