SOUND WAVES AND SOUND INSTRUMENTS

 SOUND WAVES AND SOUND INSTRUMENTS

Sound is a form of energy produced by vibrating bodies. For example, when we strike a turning fork, its prongs vibrate. These vibrations give rise to sound waves.

Transmission of sound waves

Sound is produced by vibrating bodies. The vibrations cause the surrounding air to vibrate also producing a disturbance of the air. This disturbance travels out from the source of vibration in the form of longitudinal waves.

We can demonstrate the production of sound in the laboratory using a turning fork. The  turning fork has two steel prongs which vibrate when struck with a hard rubber giving some sound. During the vibration the prongs of the turning fork present a hazy appearance due to their rapid to and fro movements. If the vibrating prongs are dipped into a beaker of water, the water is seen to be violently agitated. Also when a smaller rubber ball suspended by a thread is brought near the vibrating prongs, the ball is seen to be immediately kicked aside

 Noise And Music.

Noise is due to vibrations of irregular frequency such as rattling of a wheel on a rough road. Such irregular vibrations result in an unpleasant mixture of sound.

Music is produced by vibrations of regular or constant frequency. A musical note possesses the characteristics of pitch, loudness and quality.

Echoes ( Reflection of sound)

An echo is a sound heard after the reflection of sound waves from a plane surface. Reverberation is the perseverance of the sound after the source ceases.

The time taken to hear the echo is given by Time =  =  

Application of Echoes

I. Determination of speed of sound in air.                                          II. To determine the depth of the sea-bed from a ship.

 III. exploration for gas and oil                                                           IV. Detection of submarine

Beats :are periodic rise and fall in amplitude (or loudness) of the sound produced when two notes of nearly equal frequencies are sounded together.

Example

A ship’s echo sounder sends out a supersonic note which is received back at the ship 4 seconds after. If the velocity of sound through water is 1500 ms-1. What is the depth of the seabed?

Solution

T=4s, V= 1500ms-1; d =  =  = 3000m

 Characteristics Of Sound Waves

Pitch: The pitch of a note is its position on the musical scale. It is a  characteristic of a note which enables one to differentiate a high note from a low note. On a piano key board, the right hand side keys produce notes of high pitch and the left hand side keys produce notes of low pitch.

Pitch depends on the frequency of the sound wave. A low pitched note has a low frequency. It can be demonstrated by using a disc siren and a toothed wheel. f =   = 

Intensity: of sound is the rate of flow of energy per second per unit area perpendicular to the direction of the propagation of the sound waves.

Loudness: is the magnitude of the sensation resulting from a sound reaching the ear. It depends on the:

I.   intensity of the sound wave which reaches the listener’s ear.   II. Square of the amplitude i.e soft sound has small amplitude

III. mass of air i.e the small the mass of air in vibration, the smaller will be the sound produced.

Quality or tone (timbre): This is a characteristic note of a musical instrument which distinguishes it from another note of the same pitch and loudness produced by another instrument. The quality of a note therefore  depends on the overtones present in the note.

Harmonics : are frequencies which are multiplies of the fundamental frequency which is the first harmonic.

Forced Vibration and Resonance

Forced Vibrations are those vibrations that result from an external periodic force acting on a system and setting the system vibrating at the same frequency as the external periodic force.

Examples are: I. vibrating turning fork placed in contact with a table top. II. The vibrations of the diaphragm of a telephone microphone  III. vibration of the cone of a loudspeaker  caused by the fluctuations in the current flowing through the adjoining voice coil.

Resonance is a phenomenon which occurs whenever a particular body or system is set in oscillation as its own natural frequency as a result of impulses or signals received from some other system or body which is vibrating with the same frequency.

 Vibrations in strings and pipes

Vibration of strings (Transverse waves)

   Consider a stretched string or wire fixed at both ends when it is plucked gently in the middle, a transverse wave travels along the vibrating string. At the fixed ends the wave is reflected back and we then have two progressive waves travelling in opposite directions along the string.

The mode of vibration giving rise to the fundamental frequency is known as the fundamental mode of vibration. The distance between the two consecutive nodes is  and this equal to the length of the string l.  l = ;  = 2l .

For any wave we have that v = f  where v is the velocity, f, the frequency and  the wavelength.  f =  =  . Therefore, the fundamental frequency f_o produced by a length l is  (1^st overtone).

Harmonics and overtones in a stretched string

The lowest frequency obtained from a plucked string when the string vibrates in one loop is called the fundamental frequency, fo. Higher frequencies which are integral or whole number multiples of the fundamental frequency can also be produced in the string. They are called the Harmonics or overtones of the fundamental, e.g 2 fo, 3 fo, 4 fo etc., fo is the first harmonic.

f ₁=  =    (2f_o) first overtone or second harmonic;  f ₂=     (3f_o) second overtone or third harmonic.

  The frequency of a vibrating stretched string or wire depends on three factors

I. length of the string [ f    ;  f=  where k is constant]                      II. Mass per unit length [ f   ]          III. tension [f   ]. The three relations combine to   f   ;     f    . where K is a constant dependent on the mode of vibration. For the fundamental mode of vibration, K = ½ and equation becomes f    ( where T is in Newtons and M in kilogrammes per metre and l is in metres, f is in Hertz. ).

The frequency of first overtone is given f     (2fo). Similarly the second overtone is f   .

Vibrations of air columns (longitudinal waves in pipes)

An air column is air contained in a tube or pipe. When both ends of the pipes are open, it is called an open pipe, but one end is close and one end is open, it is called a closed pipe.

When air vibrates inside a pipe, the waves produced are reflected at the ends producing a longitudinal stationary wave along the length of the pipe. Since the air particles at the closed end of the pipe cannot move, there is always a node at this end. At the open end the air particles are free to vibrate with maximum amplitude. The antinode is therefore always formed at the open end.

Vibrations produced in close pipes.

The length of the tube, l =    or  = 4l.

Therefore the fundamental frequency f_o is given by f_o =  = . Where V is the velocity of the wave.

For the first overtone l = ;   = xl. Hence the frequency (f1) of the first overtone is given by  f₁ =  =  = 3f_o. While the 2^nd overtone f₂ =5 f_o  ( l = ) .

At first resonance, we have that  = l₁ + c ; At 2^nd  resonance, we have that  = l₂ + c. where c is the end correction which arises from the fact that the antinode at the top does not exactly coincide with the top of the tube but projects slightly above it by an amount c known as the end correction.

Substrating (i) from (ii) we can eliminate the end correction to obtain   = l₁ + l₂ ;  = 2(l₁ + l₂ ). Substituting in the formula v = 2f( l₂ + l₁ ). Where the velocity of sound is obtained from the values of l₁, l₂ and the frequency f of the turning fork.

Vibrations in open pipes

The length of the tube, l =   or  = 2l.

Therefore the fundamental frequency f_o is given by f_o =  = . Where V is the velocity of the wave.

For the first overtone l = ;  Hence the frequency (f₁) of the first overtone is given by  f₁ =  =  = 2f_o. While the 2^nd overtone f₂ =3f_o  ( l = ).

Example

If the shortest length of the tube for resonance is 0.1 m and the next resonant length is 0.35 m what is the frequency of vibrations? Assume the speed of sound in air is 340 ms-1.

Solution

L1 = 0.1m , l2 = 0.35m, f =? , v = 340ms-1. v = 2f( l₂ + l₁ ) ; f =  =  =  =680 Hz.

Application of sound waves in musical instruments

In musical instruments the source is set into vibration by striking, plucking, bowing or blowing. Standing waves are produced and the objects vibrates at its natural resonant frequencies.

Musical instruments are classified into:    I. wind instrument                    II. Stringed instrument.                     III. percussion instruments.

I.  wind instruments: These make sound through a vibrating column of air. Examples are flutes, trumpets, pipe organ, clarinets, and saxophones.

II. Stringed instruments: The use of this instruments is based on the frequency of a vibrating string depends on its length, mass and the tension in the string. A long, thick and loose wire or string produces a low frequency note but a short, thin and taut string produces a high frequency note. Examples of stringed instruments are guitars, sonometer, piano, violin.

III. Percussion instrument: it produces sound when it is hit or struck. They have taut skin, membranes, rod or plates which vibrate when struck. The note produced is usually of short duration. Examples of such instruments are talking drums, bells, gongs, xylophones, turning forks.

Assignment

1. A metal disc has 50 evenly spaced holes close to its rim. When it is rotated and an air jet plays on the holes, a note is heard. Calculate the speed of rotation which produces a note of frequency 250 Hz. Calculate the frequency of the note produced when a disc of 60 holes is rotated at the same rate.

Solution

Speed of rotation =  = 5 rev/s.  = constant.  =  ; f2 =  =  = 300 Hz.

2. Determine the pitch or frequency of the fundamental, and also of the first two overtones of (a) a close pipe (b) an open pipe, if each pipe is 67cm long and the temperature is 20^oC. Take the speed of sound at 20^oC as 343 ms^-1.

Solution

( a). Close pipe f_o =  =  = 128 Hz;   1^st overtone = 3fo = 3 x 128 = 384Hz;         2^nd overtone = 5fo = 5 x 128 = 640Hz.

( b). Open pipe f_o =  =  = 256 Hz;   1^st overtone = 2fo = 2 x 256 = 512Hz;         2^nd overtone = 3fo = 3 x 256 = 768H