Consider the relation between Newton’s law that is applied to the volume "\\Delta V" in the direction "x" :
"\\Delta F=\\Delta m\\frac{dv_{x}}{dt}" (Newton' law)
Where, "F" : force acting on the element with volume "\\Delta V"
"\\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 "\\Delta Sx" is in "x" direction so ΔyΔz and from Newton’s law
"=\\rho \\Delta V\\frac{dv_{x}}{dt}"
From, "\\frac{dv_{x}}{dt}\\ \\text{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 equation of motion.
"-\\frac{\\partial }{\\partial x}(\\frac{\\partial p}{\\partial x})=\\frac{\\partial }{\\partial x}(\\rho \\frac{\\partial v_{x}}{\\partial t})"
"-\\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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