These are shown in the following figure:

  Fig (5)

(a)   Fundamental node or first harmonic .

(b)   First overtone or third harmonic .

(c)   Second overtone or fifth harmonic .

      Therefore the ratio of overtone is 3:5:7.

(e) It is clear that the only odd harmonics are produced in this pipe:

2) Vibration of an open pipe:

(a) These pipes are open at both ends where antinodes are formed. At these ends compression is reflected as rarefaction while rarefaction as compression. Here the number of antinodes is more than that of nodes.

(b) For a pipe of length L, the frequency of fundamental node  and wavelength

First overtone or second harmonic frequency , wavelength

Second overtone or third harmonic frequency wavelength

These are shown in the following figure:

                                   Fig (6)

(a)   Fundamental node or first harmonic

(b)   First overtone or second harmonic

(c)   Second overtone or third harmonic

Clearly in open pipe all the harmonics are produced. The ratio of the frequencies is:

In this condition the ratio of overtone is 2:3:4:5: …………

End correction: e = 0.6r, where ‘r’ is the radius of pipe. Therefore for a closed pipe the effective length of the air column = L+e = L + 0.6r.

For open pipe the effective length of the air column is L+2e = L+1.2r.

For a closed pipe with end correction

By comparison:

    (For fundamental only)

2) In open pipe all the harmonic are obtained while in closed pipe only odd harmonics are obtained.

3) The sound produced in open organ tube is pleasing and that of the closed organ tube is less pleasing.

18) resonance tube:

Resonance: If the frequency of a tuning fork used is same as the frequency of vibration of air column in the tube, then the intensity of sound becomes maximum. This is said to be the condition of resonance.

It is an example of a closed pipe in which the length of the air column can be changed by adjusting the level of water.

Application:

To determine the velocity of sound and the frequency of tuning fork

Formula:

If the first and second resonance lengths are L₁ and L₂ then

19) 

(a) Velocity of sound (Longitudinal wave) in elastic medium:

Where E = Coefficient of elasticity of medium, d = density

For a solid medium:

, where E=Y = Young’s coefficient of elasticity

In a liquid medium:

, where E=K= Coefficient of volume elasticity.

In gaseous medium:

M = molecular, P = pressure, T = temperature.

(b) In stretched vibrating wire or a string, the velocity of a transverse wave: