Velocity of Waves

Velocity of Waves

Sound is a form of energy which produces the sensation of hearing. It propagates through the medium, so it is a mechanical wave. When a sound wave travels through a medium, the particles of the medium vibrate along the direction of wave travel. This produces a series of higher and lower pressure regions called compressions and rarefactions respectively. So, there is a change in volume at different parts of the medium. Therefore, the longitudinal waves can only travel through a medium if the medium travel through a medium if the medium suffers a resistance to change in volume. In other words, the medium should posses volume of elasticity.

Velocity of Waves

1. Velocity of longitudinal wave on the medium:
   $$ V = \sqrt {\frac {E}{\rho }} \text {where}\: E = \text {elasticity of medium} \: \rho = \text {density of medium} $$
2. Velocity of longitudinal wave on liquid or gas medium:
   $$ V = \sqrt {\frac {B}{\rho }} \text {where}\: B = \text {bulk modulus of gas or liquid} \: \rho = \text {density of medium} $$
3. Velocity of longitudinal wave on solid medium:
   $$ V = \sqrt {\frac {Y }{\rho }} \text {where} \: Y = \text {young’s modulus of elasticity } \: \rho = \text {density of medium} $$
4. Velocity of longitudinal wave on stretched string:
   $$ V = \sqrt {\frac {T}{\mu }} \text {where}\: T = \text {tension on string} \: \mu = \text {mass per unit length} $$
5. Velocity of longitudinal wave on electromagnetic wave:
   $$ V = \sqrt {\frac {1}{\mu \epsilon }} \text {where}\: \mu = \text {permeability of medium or free space} \: \rho = \text {permittivity of medium or free space } $$

Velocity of Sound Wave in Liquid and Solid

When a sound wave propagates through a medium, different regions of the medium undergo varying stress and strain. So, the velocity of a sound is governed the modulus of elasticity, E of the medium. It also depends on inertia of the medium and so, on the density, ρ of the medium. It can be shown that the velocity of sound,

$$ V = \sqrt {\frac {E}{\rho }} $$

In a liquid, modulus of elasticity is the bulk modulus of elasticity, B. so the velocity of the sound in liquid

$$ V = \sqrt {\frac {B}{\rho }} $$

When a longitudinal wave propagates in a solid rod or bar, the rod expands sideways slightly when it is compressed longitudinally while a fluid in a pipe with constant cross-section cannot move sideways. Using same method, we can show that the speed of longitudinal wave in rod is given by

$$ V = \sqrt {\frac {Y}{\rho }} $$

Where Y is Young’s modulus and ρ is the density of solid.

Velocity of Sound in a Gas

Dimensional Method

It is observed that the velocity of sound wave in a medium depends on the elasticity of the medium and its density. If v is velocity of the sound, E is the modulus of elasticity and ρ the density of the medium, and then we have,

$$\begin{align*} v &\propto E^x \rho ^{y} \\ \text {or} \: V &= k E^x \rho ^{y} \dots (i) \end{align*}$$

where k is a proportionality constant, and x and y are indices to be determined.

$$\begin{align*} \text {The dimension of velocity, v} &= [LT^{-1} ], \\ \text {Dimensional of modulus of elasticity,} \\ \text {E} = [ML^{-1}T^{-2}] \: \text {and} \\ \text {Dimension of density,} \rho = [ML^{-3}] \\ \text {From dimensional analysis,} \\ \text {from equation} \: (i), \text {we get} \\ [LT^{-1}] &= [ML^{-1} T^{-2}]^x. [ML^{-3} ] ^y \\ \text {or,} \: [LT^{-1}] &= [M]^{x+y} . [L] ^{-x-3y}. [T]^{-2x} \dots (ii) \\ \end{align*}$$ For the equation to be correct, the indices of\( [M],\: [L] \text {and} \: [T]\) on both sides of equation (ii) must be equal.$$\begin{align*} 0 &= x+ y \dots (iii) \\ 1 &= -x-3y \dots (iv) \\ -1 &= -2x \dots (v) \\ \text {Solving these equations, we have} \\ x = 1/2 , y= -1/2. \text {Substituting these values in equation} \: (i), \text {we get} \\ v &= kE^{1/2} \rho ^{-1/2} = K \sqrt {\frac {E}{\rho } } \\ \text {From mathematical analysis,}\: k =1 \: \text {and so the velocity of sound is} \\ v &= K \sqrt {\frac {E}{\rho } }\\ \end{align*}$$

Velocity of Sound in a Gas

Consider a single pulse in which air is compressed, travel from left to right with a velocity v through the air in a long tube as shown in the figure. Let us run along with the pulse at that velocity so that the pulse appears to stand still and air is moving at velocity v through it from right to left.

Let the pressure of the undisturbed air be P and the pressure inside the pulse be \( P + \Delta P \) , where \(\Delta P\) is positive due to compression. Consider a slice of air of thickness \(\Delta \) and cross-sectional area A moving toward the pulse at velocity v. As this element of air enters the pulse, its front face encounters a region of the higher pressure, which slow its speed \( v + \Delta v\), where \( \Delta v\) \: is negative. This slowing is complete where the back face reaches the pulse, which requires time interval \(\Delta t = \Delta x / \Delta v \). During \(\Delta t\) the average force on the front face is \( (P + \Delta P)\) A toward the right. Therefore, average net force on the element during \(\Delta t\) is

$$\begin{align*} F &= PA – (P+\Delta P)A = -\Delta PA \dots (i) \\\end{align*}$$ The minus sign indicates that the net force on the air element is directed to the right in figure.The volume of the element is \(A\Delta x,\) so mass of the element is $$\begin{align*} \Delta m &= \rho A\Delta x = \rho Av\: \Delta t \\ \text {Then the average acceleration of the during} \Delta t \: \text {is} \\ a &= \frac {\Delta v}{\Delta t} \text {Newton’s second law of motion, we have} \\ -\Delta PA &= \rho A v\: \Delta t .\: \frac {\Delta v}{\Delta t} \\ \text {or,} \: \rho v^2 &= - {\Delta P}{\Delta v/v} \\ \end{align*}$$ The air that occupies a volume \(V = A v\Delta t \:\) outside the pulse is compressed by an amount \( \Delta V = A\Delta v \Delta \) As it enters the pulse. Hence, $$\begin{align*} \frac {\Delta V}{V} = \frac {A \Delta v\Delta t}{A v\Delta t}\\ &= \frac {\Delta v}{v} \\ \text {Combining these equations, we have} \\ pv^2 &= - {\Delta P}{\Delta v/v} = - {\Delta P}{\Delta V/V} = B \\ \text {or,} \: V = \sqrt {\frac {B}{\rho }} \\ \end{align*}$$