PHYSICS As LEVEL(FORM FIVE) NOTES – WAVE MOTION-1(2)

EXPRESSION OF VELOCITY OF SOUND IN AIR/GAS IN TERM OF ABSOLUTE TEMPERATURE

– From ideal gas equation

PV

v

n = no of moles

	n	=		mass,		m/Molar	mass,M
	n		=		m	/	M
	P		=		m/M	.	RT
							v
	P		=		m/v	.	RT
							M
Molar							mass

v

Substitute equation (4) in equation (iii)

where R = molar gas constant = 8.31 Jmo

T = absolute temperature

M = molar mass of the gas

• This equation shows that the velocity of sound in air/gas is independent of the gas/air pressure.

FACTORS AFFECTING THE VELOCITY OF SOUND IN AIR / GAS

1. TEMPERATURE OF A GAS / AIR
• From the equation (5) above:

or a given gas are constant

• Thus, the velocity of sound in a gas / air is directly proportional to the square root of absolute temperature.

• If is the velocity of sound in air at a temperature is the velocity at a temperature then

————– (vi)

1. MOLAR MASS OF THE GAS

-From equation (v) above :

-If are constant then

V

-The velocity of sound in a gas is inversely proportional to the square root of molar mass of the gas.

-Sound travels faster in lighter gases like hydrogen or Helium than in heavier gases such as carbon-dioxide or ammonia.

2. HUMIDITY
• Humidity air is less dense than dry air, thus sound travels faster in humid air than in dry air.
1. WIND SPEED AND DIRECTION
• Sound travels faster in the direction of the wind than in the direction opposite to it.

Problem 14

-The longitudinal wave speed in gases is given by:

Example

– Sound produce by a tuning fork, flute, piano etc.

-There are no sudden changes in loudness

Noise

– Is an unpleasant, discontinuous and non-uniform sound produced by irregular succession of disturbance.

Example

-Sound produced by falling brick.

-Sound produced by clapping of two wooden blocks etc

-There are sudden changes in loudness.

CHARACTERISTICS OF A MUSICAL NOTES

There are three fundamental characteristics of a musical notes

1. Pitch (frequency)
2. Loudness (amplitude)
3. Quality /timbre

PITCH

• This is the characteristics of musical note by which we can distinguish a shrill sound from a grave (hoarse) sound.

-The pitch of the sound depends upon frequency of the vibration of the source

-If the frequency of a sound is high, its pitch is also high and sound is said to be shrill.

• If the frequency of the sound is low, its pitch is low and the sound is said to be grave or flat.

Example

1. The voice of children and ladies is shrill because of higher pitch.
2. The voice of an old man is horse because of low pitch.
3. The sound produced by the mosquito is of higher pitch and is therefore shrill.

LOUDNESS

• The loudness of the musical note is the intensity of the sound as perceived by the human ear.
• Loudness is determined by the amplitude of the sound and vice versa.

-The larger the amplitude, the louder the sound and vice versa.

QUALITY / TIMBRE

• This is the characteristics of the musical note which enable us to differentiate sounds of the same pitch and loudness produced by different instruments.
• It depends on the waveforms of the sound.

-The same note played on two different instruments does not sound the same and hence different waveforms are obtained which consist of

1. Main notes (fundamental note)
2. Overtones

i) FUNDAMENTAL NOTE

• This is a component of a musical note with the lowest frequency called fundamental note frequency.

FUNDAMENTAL NOTE FREQUENCY

• This is the lowest frequency that a vibrating string or pipe can produce.

ii) OVERTONE

-This is constituent of a musical note other than the fundamental note.

-The frequencies of overtones are multiples of the fundamental note frequency.

Example

If is the fundamental note frequency, then the frequency of overtones are 2

, etc.

NOTE

-The fundamental note is also called harmonic.

HARMONIC

Definition

– A harmonic is a musical note whose frequency is an integral multiple of the fundamental note frequency.

-The wave form of a note depends upon the presence of overtones / harmonics.

-Hence the quality of a musical note depends upon the number of overtones / harmonics an instrument produces.

-Different instrument emit different overtones / harmonics and hence the quality of sound produced is different.

Examples.

(a) A note played on piano has larger number of overtones / harmonics compared to that played to a flute.

• Hence musical sound from a piano is more rich ( of better quality) than that of a flute.

( b ) When stringed instruments ( violin, guitar etc) are played, they are plucked near one end instead of in the middle.

• It is because plucking near the end produces more overtones / harmonics and gives a richer sound.

MUSICAL INSTRUMENTS

Definition

A musical instrument is a device constructed or modified for the purpose of making musical.

– They include

1. Percussion instruments.
2. String instruments.
3. Pipe instruments / wind instrument

PERCUSSION INSTRUMENTS

– These are instrument which produce musical sounds by being struck with an implement or by any other action which sets the object into vibration.

Examples

1. Drum
2. Cymbals
3. Tambourine iv. Marimba

v. Xylophone

STRING INSTRUMENTS

• String instruments consist of a tightly stretched wire fixed at both ends.

Wire fixed at both ends A and B

• When then wire is struck (piano), bowed (violin) or plucked (guitar) a stationary wave pattern is formed.

MODE OF VIBRATION OF A STRING

• A string fixed at both ends can be set into vibration with different modes called harmonics.

1HARMONIC (FUNDAMENTAL NOTE)

• If the string is plucked in the middle, the simplest mode of vibration which can be set up on it’s the harmonic (fundamental note).

• where l = length of string
• if is the wavelength of the fundamental note, then:

—————— (1)

• Let be fundamental note frequency
• If V is the speed of a transverse wave along a string then:

V =

—————- (2)

• Substitute equation (1) in equation (2)

——————- (3)

OR

. HARMONIC ( OVERTONES)

• By the plucking the string at a point a quarter of its length from one end, it can vibrate in two segments

• Let be wavelength of the 1st overtone
• Let f
  1 be frequency of the 1st overtone.
• If V is the speed of a transverse wave along a string then:
• Substitute equation (4) in this equation
• But from equation *

fo = v/2L fo= 1/2 * v/L

But f1 = v/L

fo = 1/2 * f1 f1 = 2fo ………………. Which is a multiple of fundamental note frequency. N.B: Any overtone is a multiple of fundamental note frequency.

NOTE:

• A string can be made to vibrate with several modes simultaneously depending on where it is plucked

THE VELOCITY OF A TRANSVERSE WAVE IN A STRING

• Consider a transverse wave that travels along a string:
• Let = length of a string m = mass of the string

T = tension in the string

• The velocity V of a transverse wave in a string depends on m and T. – One uses dimensional analysis to get the relationship.
• Which gives

• Where mass per unit length of a string = linear mass density and K = I

THE FUNDAMENTAL NOTE FREQUENCY FORMULA

• Consider a string of a length l fixed between two points A and B
• Let the string to be plucked as its mid- point so that a fundamental note is produced

• Where = wavelength of the note emitted
• Let f be frequency of the note emitted
• If V is the velocity of transverse wave in a string then.

• Substitute equation (1) in the equation (2)

– But

LAWS OF VIBRATION OF A STRETCHED STRING.

– From the fundamental note frequency formula, we have three laws of vibration of a stretched string.

LAW 1

The frequency of a vibrating string is inversely proportional to its length LAW 2

The frequency of a vibrating string is directly proportional to the square root of tension on the wire.

LAW 3

The frequency of a vibrating string is inversely proportional to the square root of mass per length of the string.

THE SONOMETER

-This is an instrument which is used to set the frequency variation of a vibrating string in relation to its length, tension and mass per unit length.

• The instrument consists of a hollow wood box with two movable bridges A and B.

-On top of these bridges a sonometer wire is passed.

To investigate the relationship between f and the bridges are moved so that different lengths between them emit their fundamental frequencies when plucked at the centre.

• To check that , the same length of wire is subjected to different tensions by changing the hanging mass.

-The relation between f and requires the use of wire of different diameter and materials but of the same vibrating length under the same tension.

NOTE:

Volume of wire = Al =

Where d = diameter of the wire

P = density of the material of which the wire is made

FORCED VIBRATIONS AND RESONANCE

Definition

– Forced vibrations are vibrations that occur in a system as a result of impulses received from another system vibrating nearby.

Examples

• When a turning fork is sounded and placed on a bench or hollow box, the sound produced is quite loud all over the room.
• It is because the bench or box acts like an extended source (or many point sources) which are set into forced vibrations by the vibrating fork.
• The response of the system that is sent into forced vibration is best when the driving frequency is equal to the natural frequency of the responding system.
• The responding system is then said to be in resonance with the driving frequency.

Definition

Resonance is a condition in which a body or system is set into oscillation at its own natural frequency as a result of impulses received from some other system which is vibrating at the same frequency.

Problem 17

A sonometer wire of length 0.50m and mass per unit length 1.0 kgm-1 is stretched by a load of 4kg. if it is plucked at its mid-point, what will be:

1. The wavelength and
2. The frequency of the note emitted?

Take g = 10N

Problem 18

Two sonometer wires A, of diameter 7.0m and B of diameter 6.0 m, of the same material are stretched side by side under the same tension. They vibrate at the same fundamental frequency of 256HZ. If the length of B is 0.91m, find the length of A. Calculate the number of beats per second which will occur if the length of B is reduced to 0.90m.

Problem 19

A sonometer wire of the length 1.0m emits the same fundamental frequency as a given turning fork. The wire is shortened by 0.05m, tension remaining unaltered and 10 beats per second are heard when the wire and fork are sounded together.

1. What is the frequency of the fork?
2. If the mass per unit length of the wire is 1.4, what is the tension.

Problem 20

A 160cm long string has two adjacent resonance at 85HZ frequencies respectively. Calculate: (i) The fundamental frequency

(ii) The speed of the wave.

PIPE INSTRUMENTS

• Stationary waves in a column of air in a pipe are the source of sound in pipe instruments.
• To set the air into vibration a disturbance is created
  at one end of the pipe.

TYPES OF PIPE INSTRUMENTS

– There are two type of pipe instruments:

1. Closed pipe
2. Open pipe

STATIONARY WAVES IN CLOSED PIPE

• A closed pipe is the one which is closed at one end and open at the other end.

• The open end is always a displacement antinode (A) and the closed end is a displacement node (N).
• Different modes / Harmonics are obtained when the air inside a closed pipe is sent into vibration.

. HARMONIC / FUNDAMENTAL NOTE

Where l = length of the pipe

Wavelength of the fundamental note

———————– (1)

• Let be the fundamental note frequency
• If V is the velocity of the sound in air, then

V =

• Substitute eqn (1) in this eqn

—————————- (2)

OR

4

• Overtones are encourage by blowing hardly

• Let be wavelength of the overtone

i.e

————————- (3)

• Let be the frequency of the overtone.
• If V is the speed of sound in air, then

V =

• Substitute equation (3) in this equation

• But from equation

(*) = 4

• 

—————— which is the harmonic

SECOND OVERTONE

– i.e 5 ———————– (4) Let be the frequency of the overtone If V is the speed of sound in air, then V = Substitute equation (4) in this equation
– But	from	equation

*

• 

{ = 5 } ——————- which is the harmonic

• Thus a closed pipe produces odd number harmonics

i.e.

STATIONARY WAVES IN OPEN PIPES

• Here both of pipes are opened and displacement antinode
• Different modes/harmonics are obtain when the air inside the open pipe is set into vibration

1st HARMONIC/FUNDAMENTAL NOTE

• Where L = length of pipe

} ————————- (1)

• Let be frequency of the fundamental note.
• If V is the speed of sound in air, then

V =

• Substitute equation (1) into equation

——————— (2)

OR

2 ———————- (*)

• Overtones are encouraged by the blowing hardly

FIRST OVERTONE

• Where wavelength of the overtone

L = length of the pipe

———————— (3)

• Let be the frequency of the overtone
• If V is the speed of the sound in air, then

V =

• Substitute equation (1) into this equation

• But from equation

*

} ———————– which is the

SECOND OVERTONE

• Let be the wavelength of the overtone

———————– (4)

• Let be frequency of the overtone.
• If V is the speed of sound in air, then

V = λ2f2

f2 = v/ λ2

• Substitute equation (4) in this equation

• From equation *

[ ] ——————– which is the harmonic

• Thus, for an open pipe the frequency of harmonics are: