The speed v of sound waves through a medium may be assumed to depend on (a) the density 𝝆 of the medium and (b) its modulus of elasticity E which is the ratio of stress to strain. Deduce by the method of dimensions, the formula for the speed of sound.


(Ans: v = `sqrt \frac {E}{⍴}` )

Given:

Speed of sound depends on
𝝆 = density of the medium
E = Modulus of elasticity = stress : strain

To Find:

To deduce the formula for the speed of sound by the method of dimensions = v = ? 


Solution:

According to the given conditions, the formula for the speed (v) of sound will be

v ∝ `⍴^a` `E^b`

or

v = Constant `⍴^a` `E^b`  ------Equation (a)

Now using dimensions methods we will find the values of power a and b.

R.H.S. Dimensions of the equation (1):

[v] = [LT⁻¹]

L.H.S. Dimensions of the equation (1):

[⍴] = [ML⁻³]

[E] = [ML⁻¹T⁻²] 
 
(∴ E = Stress / Strain  --- Stress = Force/Area and Strain = Change in Volume / Volume)

Now writing the dimension of both sides of the equation (1) we get

[LT⁻¹] = `[ML⁻³]^a` `[ML⁻¹T⁻²]^b`

By separating each dimension quantity both side

[L] [T]⁻¹ = `[M]^{a+b}` `[L]^{-3a-b}` `[T]^{-2b}`

Now equating the power (exponent) of [M], [L], and [T] on both side, we get

By equating the powers (exponents) of  [M]
0 = a + b    ----- eqn (i)

By equating the powers  of  [L]
1 = -3a - b    ----- eqn (ii)

By equating the powers of  [T]
-1 = -2b    ----- eqn (iii)

From eqn (i) we get

b = `\frac {1}{2}` 

and substituting the value of b in eqn (i) or (ii) we get

a = -`\frac {1}{2}`

Now substituting the values of a and b in Equation (a) we have

v = Constant `⍴^-\frac {1}{2}` `E^\frac {1}{2}`

v = Constant `\frac {E^\frac {1}{2}}{⍴^\frac {1}{2}}`

v = Constant `(\frac {E}{⍴})^\frac {1}{2}`

or

v = Constant `sqrt \frac {E}{⍴}`   --------Ans

Hence, the required formula for the speed of sound


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Numerical Problem 1.8
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List of 11th Physics Numerical Problems of Chapter - 1
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