# Resonance Air Column Tube Method

Velocity of sound can be calculated by resonance tube method as follows:

Resonance air column for the measurement of velocity of sound

The apparatus consists of a long glass tube AB of uniform internal diameter. The tube is fixed on a vertical board along the side of a metre scale. The zero of scale coincides with upper end of the tube. The lower end of the tube is connected to the reservoir of water with the help of rubber tubing. The water level in the resonance tube AB can be adjusted as shown in figure.

A sounding tuning fork of known frequency is held above  the resonance tube (filled with water). The water in the tuning fork is adjusted in a way till large sound is heard when a tuning fork is set into vibration and introduced to the mouth of the tube. The length of the air column of the tube is noted with the help of metre scale lying aside. The  first resonating length is supposed to be L1. Let c be the end correction and the fundamental mode of vibration.

Now,

$${L_1 + c = \frac{ \lambda }{4}}$$

$$\text{or,} \lambda = 4(L_1 + C)……(i)$$

Now, the length of the air column is increased till another sound is heard with the same tuning fork. This is called the second resonance. As we know the length of air column for this resonance is three times the length of the first resonance. So,

$${L_2 + c = \frac{ 3 \lambda _1 }{4}}$$

$$\text{or,}3 \lambda = 4(L_2 + C)……(ii)$$

Now, subtracting equation (i) from (ii), we get:

$${2 \lambda = 4(L_2 – L_1)}$$

$${ \lambda = 2(L_2 – L_1)}$$

As the frequency of the tuning fork is  already known, the velocity of the sound can be calculated as:

$${v = f \lambda = 2f(L_2 – L_1)}……(iii)$$

When tuning fork of different frequencies is used, then the first resonating length L1 of the tube is determined by using the following relations:

$${L_1 + c = \frac{ \lambda }{4} = \frac{v}{4f}}$$

In this way velocity of sound can be calculated by resonance tube method.

Velocity of sound at STP

We know that, velocity of sound in 0° can be calculated as:

$$v_o = v\sqrt{\frac{T_o}{T}} = v\sqrt{\frac{273}{273 + \theta }}$$

The velocity of sound can be calculated with correction of humidity as:

$$v_o = v \sqrt{\frac{P-0.375 \times f}{f} \times \frac{t}{t + 273}}$$

Here, P is the atmospheric pressure and f is aqueous tension.

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