Velocity Of Sound In Different Media - Practice Questions & MCQ

Speed of sound wave in a material medium - 

For deriving the equation of speed let us consider a section AB of medium as shown in figure of cross-sectional area S. Let A and B be two cross-sections as shown. Let in this medium sound wave propagation be from left to right. If wave source is located at origin O and when it oscillates, the oscillations at that point propagate along the rod.       

The stress at any cross section can be written as -

$
\delta_t=\frac{F}{S}
$

Let us consider a section AB of the material as shown in the figure, of medium at a general instant of time $t$. The end $A$ is at a distance ' $x$ ' from $O$ and point $B$ is at a distance ' $x+d$ x' from $O$. Let in time duration ' $d t$ ' due to oscillations, medium particles at A be displaced along the length of medium by ' y ' and those at B by ' $\mathrm{y}+\mathrm{d} \mathrm{y}^{\prime}$. The resulting positions of section are A ' and B ' as shown in figure. By this we can say that the section AB is elongated by a length 'dy' . Thus strain produced in it is -

$
E=\frac{d y}{d x}
$

If Young's modulus of the material of medium is $Y$, we have

$
Y=\frac{\text { Stress }}{\text { Strain }}=\frac{\delta_1}{E}
$

By using Hooke's law -
From Eqs. (i) and (ii), we have

$
\begin{aligned}
& Y=\frac{F / S}{d y / d x} \\
& F=Y S \frac{d y}{d x} \ldots .(i i i)
\end{aligned}
$

Here, $F=$ Force

if $d m$ is the mass of section $A B$ and $a$ is its acceleration, which can be given as for a medium of density $\rho$ as

$
\begin{array}{r}
d m=\rho S d x \\
a=\frac{d^2 y}{d t^2}
\end{array}
$

From Eq. (iv), we have

$
\begin{aligned}
& d F=(\rho S d x) \frac{d^2 y}{d t^2} \\
& \frac{d F}{d x}=\rho S \frac{d^2 y}{d t^2}
\end{aligned}
$

From Eq. (iii) on differentiating w.r.t. to $x$, we can write

$
\frac{d F}{d x}=Y S \frac{d^2 y}{d x^2}
$

From Eqs. (iv) and (v), we get

$
\frac{d^2 y}{d t^2}=\left(\frac{Y}{\rho}\right) \frac{d^2 y}{d x^2} \cdots
$

Equation (vi) is the different form of wave equation.

$
\begin{aligned}
& \quad \frac{\partial^2 y}{\partial t^2}=v^2 \frac{\partial^2 y}{\partial x^2} \\
& v=\sqrt{\frac{Y}{\rho}}
\end{aligned}
$
 

The above equation shows wave velocity

In the case of gas or liquid, which shows only volume elasticity, E = B, where B = Bulk modulus of elasticity.

For longitudinal wave for liquid or gas - 

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

where $\rho=$ Density of the medium