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-de

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!