The velocity of sound waves 'v' through a medium may be assumed to depend on:

1. the density of the medium 'd'  2. the modulus of elasticity 's'

Deduce by the method of dimension for formula for the velocity of sound.(take dimensional constant k=1)

 Hi ! proceed as follows . Let the velocity of the sound wave in the medium be (v). 

 Let  V = f(ρ,s).  where (ρ) stands for the density of the medium and (s) stands for the moduli of elasticity.

Dimentional analysis proves to be beneficial only if the dependance is of product type...assuming that  the function f(ρ,s) is of product type we have :-

V α ρxsy

=> V = kρxsy .....(1)   { where K is a dimentionless constant }       

The dimensional formalism of the above equation is :-

[MoL1T-1] = K[M1L-3To ]x[M1L-1T-2]y

=> [MoL1T-1] = K[Mx+yL-3x-2yT-2y]

Comparing the corresponding exponents of M,L and T the above equation can be deconstructed into three individual equations

x+y = 0  ......(2)

-3x-2y = 1 ........(3)

-2y = -1 .......(4)

solving which yields x = -1/2 and y = 1/2 

substituting the value of (x) and (y) in equation (1) we get  

 
 

V = √s/ρ

 

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G if igi
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The speed of sound v in a medium depends on the modulus of volume elasticity E and density of medium obtain formula for the speed by round by dimensional analysis ??
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Lul,,
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The speed of sound v in a medium depends on the modulus of volume elasticity E and density d of the medium establish the formula for the speed of sound on the basis of dimensional analydis
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. Let the velocity of the sound wave in the medium be (v).?

?Let ?V = f(?,s). ?where (?) stands for the density of the medium and (s) stands for the moduli of elasticity.

Dimentional analysis proves to be beneficial only if the dependance is of product type...assuming that ?the function?f(?,s)?is of product type we have :-

V????xsy

=> V = k?xsy?.....(1) ? { where K is a dimentionless constant }? ? ? ?

The dimensional formalism of the above equation is :-

[MoL1T-1] = K[M1L-3To?]x[M1L-1T-2]y

=>?[MoL1T-1] = K[Mx+yL-3x-2yT-2y]

Comparing the corresponding exponents of M,L and T the above equation can be deconstructed into three individual equations

x+y = 0 ?......(2)

-3x-2y = 1 ........(3)

-2y = -1 .......(4)

solving which yields?x = -1/2?and?y = 1/2?

substituting the value of (x) and (y) in equation (1) we get ?

??

V = ?s/?
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Given that the velocity v of sound waves in a material is dependent on the material density ρ and modulus of elasticity E, show (using dimensional analysis) that the velocity is given by v = k q F l ρAe
 
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