Waves in Rods

Waves in Rods

The modes of vibration in a metal rod depend on the position at which the rod is clamped. When the rod is clamped tightly at the middle, and is struck along the length at a free end, stationary longitudinal waves are formed on the rod as the waves in an open pipe. Since the ends are free to vibrate, displacement antinodes are formed at the ends of the rod and a displacement node at the middle as all the atoms are at rest there. The wave in the length of the rod in the simplest mode of vibration have the length given by

$$\begin{align*} L &= \lambda /2 \\ \text {or,} \: \lambda &= 2L \\\end{align*}$$

If v is the velocity of the wave, the frequency is given by

$$\begin{align*}\\ f &= v/\lambda = v/2L \\ \end{align*}$$

Kundt’s Dust Tube Experiment

Kundt’s dust tube consists of a glass tube PQ. A cork B is a attached at one end of the tube with a wooden handle which can be moved inside the tube. A rod EA of given material for which the velocity of sound is to be measured is clamped at the middle. The front end of the rod has an aluminium disc attached to it. The tube contains air at room temperature and little lycopodium powder is sprinkled along the length of the tube as shown in the figure.

Applications

Velocity of Sound in Solids

When the rod is stroked at E by a resigned at E by a resigned cloth gently in the direction of EA, the longitudinal vibrations are set up in the rod. The free and E of the rod acts as a source of the same frequency, and sound wave travels along the tube through the air which is reflected at the fixed end B, the vibrations are transferred to the air column. The cork B in the tube can be adjusted such that the air column resonates with the vibrations of the rod and a stationary wave in the tube causes the licopodium powder to become violently agitated. The distance between the consecutive nodes is determined by measuring the distance between several of them and taking the average of them.

Let the length of the rod be l_rand the mean distance between the consecutive heaps be l_a. Then

$$ l_r= \frac {\lambda _r}{2} \: \text {and} \: l_a = \frac {\lambda _a}{2}$$

Let v_r and v_a be the velocity of sound in rod and air respectively. So for the rod, frequency $$\begin{align*}\\ f &= \frac {v_r}{\lambda _r} = \frac {v_r}{2l_r} \dots (i) \\ \text {and for the air column,} \\ f &= \frac {v_a}{\lambda _a} = \frac {v_a}{2l_a} \dots (ii)\\ \text {Equation} \: (i) \: \text {and} \: (ii), \: \text {we have} \\ \frac {v_r}{2l_r} &= \frac {v_a}{2l_a} \\ \therefore v_r &= v_a \left [\frac {l_r}{l_a}\right ] \dots {iii} \\\end{align*}$$

Thus, knowing v_a, l_rand l_a, the velocity of sound in rod can be calculated.

Determination of Young’s modulus of a rod

Velocity of sound in a medium is given by

$$\begin{align*} v &= \sqrt {\frac {Y}{\rho }} \\ \text {where Y is Young’s modulus of the medium and} \: \rho , \\ \text {its density. If } \: v_r \: \text { is the velocity of sound in the rod, then} \\ v_r &= \sqrt {\frac {Y}{\rho }} \\ \text {or,} Y &= v_r^2 \times \rho \dots (iv) \\ \text {Thus, knowing} \: v_r \: \text {and} \rho \: \text {Y can be calculated.} \\ \end{align*}$$

Velocity of Sound in Gases

The experiment is performed first with air and then with the required gas, using same rod. Suppose the distance between two consecutive heaps in the air column is l_a and the mean distance between two consecutive heaps in the gas column is l_g.

$$\begin{align*} \text {From equation} \: (iii), \: \text {we have} \\ v_r &= v_a \left [\frac {l_r}{l_a}\right ]\dots (v) \\ v_r &= v_g \left [\frac {l_r}{l_g}\right ]\dots (vi) \\ \text {Equation} \: (v) \: \text {and} \: (vi), \: \text {we have} \\ v_g \left [\frac {l_r}{l_g}\right ]\ &= v_r \left [\frac {l_r}{l_a}\right ]\\ \text {or,} \: v_g &= v_a \left [\frac {l_g}{l_a}\right ]\dots (vii)\\ \end{align*}$$

$$\text{Thus, knowing} \: v_a, l_g\: \text {and} \: l_a, \: \text {the velocity of sound in a gas can be calculated.}$$

1. Determination of ratio of molar heat capacities of a gas and its molecular structure:
   Velocity of sound in a gas is given by
   $$\begin{align*} v_g &= \sqrt {\frac {\gamma P}{\rho }} \\ \text {where} \: \gamma \text {is the ratio of two molar heat capacities} \: (C_p/C_v) \: \text {of a gas,} \\ \text {P is pressure and} \: \rho \: \text {is its density.} \\ v_g^2 &= \frac {\gamma P}{\rho } \\ \therefore \gamma &= \frac {v_g^2 \rho }{P} \\ \text {Thus, knowing} \: v_g, \rho\: \text {and P}, \: \gamma \text { can be calculated.} \end{align*}$$
2. To find Bulk modulus (K)
   $$\begin{align*} v_g &= \sqrt {\frac {K}{\rho }} \\ \text {or,} \: \left ( \frac {l_g}{l_a} \right ) v_a &= \sqrt {\frac {K}{\rho }} \\ \text {or,} \: \left ( \frac {l_g}{l_a} \right )^2 \times v_a^2 &= \frac {K}{\rho } \\ \therefore K &= \left ( \frac {l_g}{l_a} \right )^2 \times v_a^2 \times \rho \\ \end{align*}$$\text{Thus, knowing} \: l_g, l_a, v_a\: \text {and }, \: \rho , \:\text {K can be calculated.}$$

Velocity of Sound in Liquid

Velocity of sound in a liquid can be determined by filling the tube PQ with the liquid and fine sand or metal fillings spread along the length of the tube instead of lycopodium powder. The experiment for velocity of sound in solids is repeated and the distance between two consecutive heaps of sand or metal fillings is determined. Let l₁ be the distance between two consecutive heaps in liquid and l_a be the distance between two consecutive heaps in air, then we get

$$ V_l = \left ( \frac {l_g}{l_a} \right ) v_a $$

Thus knowing the values of v_a , l_l and l_a, v_l can be calculated.