Question #104076

The speed of sound v in a medium depends on its wavelength λ, the young modulus E, and the density ρ, of the medium. Use the method of dimensional analysis to derive a formula for the speed of sound in a medium

Expert's answer

"v=\\lambda ^\\alpha E^\\beta \\rho ^\\gamma"

"[\\lambda]=[m]=L"

"[E]=[Pa]=ML^{-1}T^{-2}"

"[\\rho]=[kg\/m^3]=ML^{-3}"

"[v]=[m\/s]=LT^{-1}"

"LT^{-1}=L^\\alpha(ML^{-1}T^{-2})^\\beta(ML^{-3})^\\gamma"

We have

"\\alpha-\\beta-3\\gamma=1"

"\\beta+\\gamma=0"

"-2\\beta=-1"

The solution of the system gives

"\\alpha=0"

"\\beta=1\/2"

"\\gamma =-1\/2"

"v=\\lambda ^0 E^{1\/2} \\rho ^{-1\/2}=\\sqrt{\\frac{E}{\\rho}}" **Answer.**

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