Mechanical Waves | Wave & Optics | Part 2 | Class 12 | New Curriculam 2078 | Notes | Nepal |

Mechanical Waves

The waves which needs material medium to propagate is called mechanical waves. Example: water waves, sound waves, etc.

Mechanical waves are of two types:

1. Longitudinal Waves

2. Transverse waves 

               (Explanation of types of mechanical waves is already in part 1: click here )

Speed of Sound Wave in any medium:

Experimentally it is found that speed of sound wave in any medium depends on two factors; elasticity of medium and density of medium.

                   Let 'V' be the speed of sound in any medium, 'E' be the elasticity and 'p' be the density of the medium. Then,

`V \alpha   E^a`-------------(1)
`V \alpha   \rho^b` ---------(2)

Combining equation (1) and (2) we get,

`V \alpha  E^a\rho^b`

`V = KE^a\rho^b`-----------(3)


Writing the dimension of each terms in eqn (3) we get,

`[M^0L^1T^-1]=[M^1L^-1T^-2]^a [M^1L^-3T^0]^b`

`[M^0L^1T^-1]=[M^(a+b)L^(-a-3b)T^(-2a)]`

Equating the power of M, L and T we get,

`a+b=0`, `-a-3b=1` and `-2a=-1`

Solving these equation we get,

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

Putting the value of a and b in equation (3) we get,

`V = K E^\frac{1}{2} \rho^\frac{-1}{2}`
`V = K \sqrt\frac{E}{\rho}`

Experimentally the value of K is found to be 1 so,

`V = \sqrt\frac{E}{\rho}`   which is the required expression for the velocity of sound wave in any medium.

Velocity of sound in solid:

Elasticity(E)= Young's Modulus of Elasticity(Y)

`\therefore` velocity of sound in solid(`v_s`)= `\sqrt\frac{Y}{\rho}`

For steel, `Y = 2\times10^11 Nm^(-2)`

                `\rho = 7800 kgm^(-3)`

`\therefore` velocity of sound in steel (`V_steel`)= `\sqrt\frac{2\times10^11}{7800}`

                                                                              = 5064m/s

Velocity of sound in liquid

Elasticity(E) = Bulk modulus of Elasticity (`\beta`)

`\therefore` velocity of sound in liquid (`v_l`)= `\sqrt\frac{\beta}{\rho}`

For water, `\beta` = 2.04`\times10^9Nm^(-2)`

                  `\rho = 1000kgm^(-3)`

`\therefore` velocity of sound in water (`v_w`) = `\sqrt\frac{2.04\times10^9}{1000}`

                                                                       = 1428 m/s

Velocity of sound in gaseous medium

i) Newton's formula for the velocity of sound in gas:

    Elasticity(E) =Bulk modulus of elasticity(`\beta`) = Pressure exerted by gas (P)

    `\therefore` velocity of sound in gas (`v_g`) = `\sqrt\frac{P}{\rho}`

At NTP, P =  1.013 `\times 10^5` N/`m^-2`  and `\rho` = 1.29 kg/`m^3`

    `\therefore` velocity of sound in gas (`v_g`) =`\sqrt\frac{1.013\times10^5}{1.29}`

                                                                          = 280 m/s which does not match with the experimental result.

ii) Laplace correction for the velocity of sound in gas:

   Elasticity(E)= Bulk modulus of Elasticity (`\beta`) = `\gammaP`

     `\therefore` velocity of sound in gas (`v_g`) = `\sqrt\frac{\gammaP}{\rho}`

   At NTP, P =  1.013 `\times 10^5` N/`m^-2` , `\rho` = 1.29 kg/`m^3` and `\gamma`=1.4

      `\therefore` velocity of sound in gas (`v_g`) =`\sqrt\frac{1.4\times1.013\times10^5}{1.29}`

                                                                             = 332 m/s which match with the experimental result.