Question

The velocity of sound waves 'v' through a medium may be assumed to depend on:
(i) The density of the medium 'd' and
(ii) The modulus of elasticity 'E'
Deduce by the method of dimensional formula for the velocity of sound. (k=1)

Answer 1

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Given,
$v\: \alpha\: d$ ------[1]
$v\: \alpha\: E$ ------[2]

To determine speed of sound ,
Eq[1] & Eq[2] are to be combined as :

$v\: \alpha\: d^{a}E^{b}$
$v\: \alpha\: kd^{a}E^{b}$ -----[3]
where k = constant

In dimensions,
$[v] = [d]^{a}[E]^{b}$

$[LT^{-1}] = [ML^{-3}]^{a}[ML^{-1}T^{-2}]^{b}$

$[LT^{-1}] = [M^{a}L^{-3a}][M^{b}L^{-b}T^{-2b}]$

Comparing powers of M,L & T on both sides

0 = a + b ------[4]
1 = -3a- b ------[5]
-1 = -2b ---------[6]

From [6]
2b = 1
=> $b = \frac{1}{2}$

Substitute value of b in [4]

$0 = a + \frac{1}{2}$
=> $a = - \frac{1}{2}$

substitute value of a and b in [3]

$v = k × d^{-\frac{1}{2}}E^{\frac{1}{2}}$

$v = k ×\frac{E^{\frac{1}{2}}}{d^{\frac{1}{2}}}$

$v = k × \sqrt{\frac{E}{d}}$

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Answer 2

Answer:

The velocity of sound wave is $v=\sqrt{\dfrac{E}{d}}$

Explanation:

Given that,

The velocity of the sound wave in the medium is v.

Let the velocity of the sound wave depends on the density of the medium and modulus elasticity E

$v = k\ d\ E$

$v=d^aE^b$.....(I)

Where, k = dimensionless constant

d = density of the medium

E = modulus elasticity

The dimensional formula of the velocity, density and modulus elasticity

$v=[LT^{-1}]$

$d=[ML^{-3}]$

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

Now,put the dimension formula in the equation (I)

$[LT^{-1}]=[ML^{-3}]^a[ML^{-1}T^{-2}]^b$

On compare the power

a+b=0.....(II)

-3a-2b=1......(III)

-2b=-1

$b=\dfrac{1}{2}$

Put the value of b in equation (II)

$a+\dfrac{1}{2}=0$

$a=-\dfrac{1}{2}$

Put the value of a and b in equation (I)

$v=d^{\frac{-1}{2}}E^{\frac{1}{2}}$

$v=\sqrt{\dfrac{E}{d}}$

Hence, The velocity of sound wave is $v=\sqrt{\dfrac{E}{d}}$