Using the method of dimension, derive an expression for the velocity of sound waves (v) through a medium. Assume that the velocity depends on: (i) Modulus of elasticity (E) of the medium (ii) The density of the medium (ρ), take the constant K = 1


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Question Number 43694 by Tawa1 last updated on 14/Sep/18

$Using the method of dimension, derive an expression for the velocity$ $of sound waves (v) through a medium . Assume that the velocity$ $depends on : (i) Modulus of elasticity (E) of the medium$ $(ii) The density of the medium (ρ), take the constant K = 1$
	Answered by alex041103 last updated on 14/Sep/18

	$W e k n o w t h a t$ $E = [\frac{N}{m^{2}}] = [\frac{k g \frac{m}{s^{2}}}{m^{2}}] = [m^{- 1} s^{- 2} {k g}^{1}]$ $ρ = [\frac{k g}{m^{3}}] = [m^{- 3} {k g}^{1}]$ $v = [\frac{m}{s}] = [{m s}^{- 1}]$ $\Rightarrow W e s u p p o s e v = K E^{α} ρ^{β}$ $\Rightarrow {m s}^{- 1} = m^{- α} s^{- 2 α} {k g}^{α} m^{- 3 β} {k g}^{β}$ $\Rightarrow m^{1} s^{- 1} {k g}^{0} = m^{- (α + 3 β)} s^{- 2 α} {k g}^{α + β}$ $- α - 3 β = 1 \Rightarrow α + 3 β = - 1$ $- 2 α = - 1 \Rightarrow α = \frac{1}{2}$ $α + β = 0 \Rightarrow β = - α = - \frac{1}{2}$ $A n d α + 3 β = \frac{1}{2} + 3 (- \frac{1}{2}) = - 1$ $\Rightarrow α = 1 / 2 a n d β = - 1 / 2$ $\Rightarrow v = K E^{1 / 2} ρ^{- 1 / 2}$ $B u t K = 1$ $\Rightarrow v = \sqrt{\frac{E}{ρ}}$
	Commented by Tawa1 last updated on 14/Sep/18

	$God bless you sir$
	Commented by alex041103 last updated on 14/Sep/18

	$Y o u a r e w e l c o m e!$
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