Pitch  
•
•
Is the degree of highness or lowness of a tone  
The higher the frequency, the higher the pith produced.  
.
Timbre  
•
•
Timbre is the quality of sound produced by an instrument  
For example, different instruments produce different sound  
.
N.B:  
•
The difference between sounds are caused by intensity, pitch and tone  
•
For example a violin sounds different than a flute playing the same pitch.  
This is because they have a different tone or sound quality  
Musical Instruments  
•
Musical instrument is a device constructed or modified for the purpose of  
making music  
Categories of musical instruments  
String Instrument  
•
Is the instrument which produces sound from stretched strings that are  
plunked, bowed or struck  
•
Example, guitar is plunked, violin is bowed and piano is struck  
Percussion Instrument  
•
Is the instrument which produces sound by struck with an implement, shaken,  
rubbed, scrapped or by any other action which sets the object into vibrations  
•
Example drum  
,
cymbals  
,
tambourine  
,
marimba and xylophone  
.
Wind Instrument  
•
•
Is the instrument which produces sound by blowing.  
Example recorders  
,
flutes  
,
vuvuzela and trumpets  
Stationary Waves  
•
A stationary wave is the wave which occurs when two waves are travelling in  
opposite direction with the same speed and frequency are superposed  
NB:  
•
•
When adding together producing maximum displacement (amplitude) called  
antinodes and when cancel out producing zero displacement called nodes  
Distance between adjacent nodes or antinodes is equal to half wave length,  
i.e  
L = λ/2  
Fundamental Frequency(note)  
•
Is the lowest frequency of a vibrating object  
Fundamental note: Is the primary note of the harmonic series  
Fundamental harmonic  
A harmonic: Is a wave whose frequency is an integral (whole number) multiple of the  
fundamental frequency  
OR  
•
Is a note whose frequency is a whole number that of the fundamental frequency  
Overtone (Fundamental Overtone)  
•
NB:  
•
Is any frequency higher than the fundamental frequency of a sound  
The fundamental note is equal to first harmonic  
The second harmonic is equal to first overtone  
Stationary wave in a string have certain fixed wavelength  
•
•
Consider the diagrams below  
For fundamental note  
(
1st harmonics  
)
흀
From  
:
푳 =  
→ λ = 2L  
ퟐ
풗
From: V = λƒ  
→ 풗 = 2Lf  
Then  
:
풇 =  
ퟐ풍  
풗
=
∴ 풇풖풏풅풂풎풆풏풕풂풍 풇풓풆풒풖풆풏풄풚 , 풇 ퟎ  
ퟐ풍  
For 1st overtone (2nd harmonic)  
From: L = 2 × (휆)  
→
λ =2  
푳
(ퟐ)  
2
From  
:
V = λƒ  
풗
풗
풍
풗
Then  
:
f1 =  
=
ퟐ
풗
= ퟐ풇  
= × = ퟐ ×  
ퟎ
흀
ퟐ
풍
ퟐ풍  
∴ ퟏ풔풕풐풗풆풓풕풐풏풆 ( ퟐ풏풅 풉풂풓풎풐풏풊풄), 풇ퟏ = ퟐ풇ퟎ  
For 2nd overtone (3rd harmonic)  
ퟐ풍  
From: L = ퟑ × (흀)  
→ 흀 =  
ퟐ
ퟑ
From: V = λƒ  
ퟑ풗  
ퟐ풍  
풗
= 푣 =  
= ퟑ ×  
Then  
:
풇
= ퟑ풇  
ퟎ
ퟐ
ퟐ풍  
흀
∴ ퟐ풏풅 풐풗풆풓풕풐풏풆 (ퟑ풓풅 풉풂풓풎풐풏풊풄), 풇ퟐ = ퟑ풇ퟎ  
Generally the 풏풕풉 overtone of a stationary wave is given by  
(
)
풇풏 = 풏 + ퟏ 풇ퟎ … … … … (풘풉풆풓풆 풏 = ퟏ, ퟐ, ퟑ … . . )  
Sonometer  
•
•
Sonometer is an instrument used to study the properties of stationary wave  
It is an apparatus made of a hollow box having two holes  
•
•
It is used to study the relationship between the frequency of the sound produced by  
a plucked string, and the tension, length and mass per unit length of the string.  
N.B: Stringed musical instruments are provided with a hollow box in order to  
amplify the sound made by the vibrations of the strings of the instrument.  
Factors affecting the frequency of a vibrating string  
➢
➢
Length of wire, L  
Tension of the wire, T  
➢
Mass per unit length, μ (diameter and density)  
Length (L) of stretched string (Wire)  
•
When the length of the string is changed, it will vibrate with a different  
frequency. The shorter strings have higher frequency and therefore higher pitch  
•
Example when a musician presses her finger on a string, she shortens its  
length .The more fingers she adds to the string ,the shorter she makes it and  
the higher the pitch will be  
•
Therefore: The frequency of a stretched string is inversely proportional to  
ퟏ
its length (f  
∝
…
..........(i)  
)
푳
풌
If 풇 ∝ ퟏ  
→ 풕풉풆풏 풇 =  
→ 풇풍 = 풄풐풏풔풕풂풏풕  
풍
풍
풇ퟏ  
풇ퟐ  
풍ퟐ  
풍ퟏ  
=
풐풓  
∴ 풇ퟏ풍ퟏ = 풇ퟐ풍ퟐ  
Tension (T) of the stretched string  
•
•
Tension refers to how tightly the string is stretched  
Tightening the string gives it a higher frequency while loosening it lowers the  
frequency  
•
•
Example when string players tighten or loosen their strings , they are altering  
the pitches (frequencies) to make them in tune  
Therefore: The frequency of a vibrating string is directly proportional to  
the square root of the tension T (f  
∝
√
푇
......... ii)  
풇
If 풇 ∝ 푻  
→ 풕풉풆풏 풇 = 풌 푻  
→
√
√
= 풄풐풏풔풕풂풏풕  
푻
√
풇ퟏ  
풇ퟐ  
풇ퟏ  
풇ퟐ  
푻ퟏ  
푻ퟐ  
∴
=
푶푹  
Mass per Unit Length (흁) of a vibrating string  
•
•
This includes the thickness (diameter) and heaviness (density) of a sting  
Thus the thicker and heavier a string is, the lower is its frequency for a given  
length and tension and vice versa  
•
•
Example a thin string with a 10 mm diameter will have a frequency twice as  
high as one with a larger, 20 mm diameter  
Also the instruments often have strings made of different materials. The strings  
used for low pitches will be made of a more dense material than the strings  
used for high pitches  
•
Therefore: The frequency is inversely proportional to the square root of the  
ퟏ
..............................  
(
풇 ∝  
)
풊풊풊  
mass per unit length,  
μ
√
흁
ퟏ
If 풇 ∝  
√
→ 풕풉풆풏 풇 흁 = 풌  
→ 풇 흁 = 풄풐풏풔풕풂풏풕  
√
√
흁
풇ퟏ  
풇ퟐ  
흁ퟐ  
흁ퟏ  
풐풓  
∴ 풇 흁 = 풇 흁  
ퟐ
√
√
ퟏ
ퟏ
ퟐ
ퟏ
ퟏ
•
Also for a diameter and density (풇 ∝  
풂풏풅 풇 ∝  
)
√
푫
흆
Now combine the three equations  
ퟏ
푳
푻
흁
풇 ∝  
…
...............Remove proportionality constant  
√
푻
풇 = 풌 ퟏ  
√
푳
흁
ퟏ
푻
Where  
:
k = 1/2, (experimentally)  
Then  
:
풇 =  
√
ퟐ풍  
흁
ퟏ 푻  
ퟐ풍 흁  
ퟏ 푻풍  
ퟐ풍 풎  
∴ 풇 =  
=
Since 흁 = 풎 (mass per unit length)  
풍
푻) =  
푻
흁
푻푳  
From  
:
V
= 흀풇  
→ 풗 = ퟐ풍 ( ퟏ  
√
=
√
√
ퟐ풍  
흁
풎
흀
ퟐ
Since 풍 =  
→ 흀 = ퟐ풍 (for fundamental frequency)  
푻
∴ 푉푒푙표푐푖푡푦 표푓 푠표푢푛푑 , 푣 =  
= √푻푳  
√
흁
풎
Therefore for the nth harmonic is given by  
풏
ퟐ푳  
풎
ퟐ풍  
풎
(
푊ℎ푒푟푒푏푦 풇풏 = frequency of nth harmonic, n = 1,2,3,4 etc)  
Example  
1. The vibrating length of a stretched wire is altered at constant tension until the  
wire oscillates in unison with a turning fork of frequency 320 Hz. The length of  
the wire is again altered until it oscillates in unison with a fork of unknown  
frequency. If the two lengths are 90 cm and 6o cm, respectively , determine the  
unknown frequency  
Solution  
Given that: f1 = 320 Hz, L1 = 90 cm, L2 = 60 cm, f2 =?  
ퟏ
풇
풍
ퟐ
ퟏ
From: 풇 ∝  
→
=
풍
풇
ퟐ
풍
ퟏ
풍 풇  
ퟏ ퟏ  
ퟑퟐퟎ×ퟗퟎ  
∴ 풇 =  
=
= ퟒퟖퟎ 푯풛  
ퟐ
ퟔퟎ  
풍
ퟐ
Class Activity  
Use acceleration due to gravity, g = 10 m/s2  
1.  
A string has a length of 75cm and a mass of 8.2g, the tension in the string is  
18N.Calculate the 1st harmonic and 3rd harmonic (ANS: f1 = 27Hz, f3 = 81Hz)  
2. A string of length 1 m and mass 5 x 10-4 kg fixed at both ends is under a  
tension of 20 N. It is plucked at a point situated 25 cm from one end. What  
would be the frequency of vibrations of the string? (ANS: f = 200 Hz)  
3. A wire of length 140 cm and mass 0.52 x 10-3 kg is stretched by means of a  
load of 16 kg. Calculate the frequency of the fundamental note.(ANS: f = 234 Hz)  
4. The vibration length of a stretched wire is altered at constant tension until the  
wire oscillates in unison with a turning fork of frequency 320 Hz. The length of  
a wire is again altered until it oscillates in unison with a fork of unknown  
frequency. If the two lengths are 90 cm and 65.5 cm, respectively, determine  
the unknown frequency  
(ANS: ƒ2 = 440 Hz)  
5. The length of a sonometer wire between two fixed ends is 110 cm. Where  
should the two bridges be placed so as to divide the wire into three segments  
whose fundamental frequencies are in the ratio 1:2:3?  
ퟏ
풇 ∝  
(from:  
f1L1 = f2L2 = f3L3, Thus L1=60 cm,L2 =30 cm and L3 = 20 cm)  
,
풍
6. A 90 cm long wire of a sitar has a fundamental frequency of 256Hz. At what  
distance from the upper end should the wire be compressed so that a note of  
frequency 384 Hz is produced? (ANS: L = 30 cm)  
7.  
A nylon string is stretched between supports 1.2 m apart. Given that the speed  
of sound in the string is 800 ms-1,find the frequency of the fundamental  
vibration and the first two overtones (ANS: f0 =333 Hz, f1 = 666 Hz and f3 = 1000 Hz)  
8. A Sonometer wire of length 40cm between two bridges produces a note of  
512Hz when plucked at the midpoint. Calculate the length of the wire that  
would produce a note of 256Hz with the same tension  
(ANS: L2 = 0.8m)  
9. A sonometer wire of length 40 cm between two bridges produces a note of 512  
Hz when plucked at the mid point .Calculate the length of the wire that would  
produce a note of 256 Hz with the same tension  
10. The frequency obtained in a plucked string is 500Hz when the tension is 3 N .calculate  
(i)The frequency when the tension is increased to 10 N (ANS: 912.8 Hz)  
(ii)The tension needed to produce a note of frequency 800 Hz (ANS: T =7.7 N)  
11. A plucked string of length 30 cm has a mass per unit length of 0.5 kg/m .If the  
tension in the string is equal to 40 N ,Find :  
(a)The fundamental frequency (ANS: f0 = 14.9 Hz)  
(b)The first overtone frequency (ANS: f1 = 29.8Hz)  
(c)The second overtone frequency (ANS: f2 = 44.7 Hz)  
12. A plucked wire of 10 m long and radius of 7mm has a density of 500 kg/m3. Calculate  
(i)  
The fundamental frequency  
(
ANS: f0 = 0.5 Hz  
)
(ii)  
The first overtone frequency (ANS: f1 = 1.0 Hz  
)
needed to produce a  
tension of 8 N  
13. A string has a length of 75cm and a mass 0f 8.2g. The tension in the string is  
18N. Calculate the velocity of the sound wave in the string. (ANS  
:
V = 40.5m/s.)  
14. Given that the velocity of the sound wave emitted from a string is 50m/s the  
Length of the string is 40cm and the mass of the string is 0.0004kg calculate  
the tension of the string.  
(ANS: T = 2.5N)  
15. A sonometer wire of length 50cm vibrate with frequency 384Hz. Calculate the  
length of the sonometer wire so that it vibrates with frequency of 512Hz.(37.5 m)  
16. A sonometer wire of length 40cm between two bridges produces a note of  
frequency 512Hz when plucked at midpoint. Calculate the length of the wire  
that would produce a note of frequency 256Hz with the some tension.(L = 80cm)  
17. The frequency obtained from a plucked string is 400Hz when the tension is 2  
Newton. Calculate;  
a) The frequency when the tension is increased to 8N (ANS: f= 800Hz)  
b) The tension needs to produce a note of frequency 600Hz (ANS: T = 4.5N)  
18. Given that the frequency obtained from a plucked string is 800Hz when the  
tension is 8N. Calculate;  
(a) The frequency when the tension is doubled (ANS: f = 1131. 2 Hz)  
b) The tension required when the frequency is halved (T = 1.414 N)  
19. Under constant tension the note produced by a plucked string is 300Hz when  
the length is 0.9m;  
a) At what length is the frequency 200Hz? (ANS: L2 = 1.35m)  
b) What frequency is produce at 0.3m (ANS: f = 90Hz)  
20.  
A string fixed between two supports that are 60cm a part. The speed of a  
transverse wave in a string is 420m /s. Calculate the wavelength and the  
frequency for Fundamental note, Second overtone and Fifth overtone  
(
ANS: fo = 350 λ = 1.2 m, f3 = 1050 λ = 0.4 m , f5 = 2100 λ = 0.2m)  
21. A string is fixed two ends 50cm a part. The velocity of a wave in a string is  
600m/s. Calculate;  
(a) The first five over tone (ANS: 1200Hz, 1800Hz, 2400Hz, 3000Hz, and 3600Hz).  
(b) The tenth overtones (The tenth overtone is 6600Hz)  
NOTE  
:
In stationary wave a string does note compose up to ten overtones,  
though mathematically is possible. In real practical of the sonometer by using  
turning, is possible for the second and third overtone  
.
22. Given that the refractive index of glass is 1.52. The wavelength of the radio waves in  
vacuum is 1.5 x 103m .Calculate the wavelength of the radio waves in glass.(λ =986.8 m)  
23. A guitar wire fixed between two supports 60cm a part produced wave of  
frequency 500Hz. Calculate;  
(a)The frequency of a wave when the length of the guitar wire is reduced to quarter  
(b)The length of the guitar wire when the frequency of the wave produced is  
2000Hz  
(ANS: f = 2000Hz ,L = 150m)  
24. A string A is 2m long and has a linear mass density of 9 g/cm3. String B has  
has a linear mass density of 18g/cm3 .If the tension in both strings is the same  
,how long must string B be for it to be raised to hear the next peak in intensity  
25. What is the approximate distance of a thunderstorm when you note a 3 s delay  
between the flash of lighting and the sound of thunder? (  
d
≈ ퟏퟎퟎퟎ 풎 ≈ ퟏ 풌풎)  
26. How long does it take for a radio signal sent from the earth to reach the moon?  
The distance from the earth to the moon is 3.84 x 106 m  
27. During a storm ,thunder is heard 7 s after the lightning is seen .If the  
0
temperature of the air is 28 C ,how far away is the storm ( C = 3 x 108 m/s)  
28. In a resonance tube experiment ,the smallest value of L for which a peak in  
sound intensity occurs is 9.0 cm .How much must the tube be raised to hear  
the next peak in intensity  
29. A helicopter is hovering at an altitude of 200 m above the surface of a lake. A  
speaker on the helicopter is sending out sound waves ,which are reflected from  
both the surface of the water and the bottom of the lake .If the difference in  
arrival times of the two echoes is measured to be 0.24 s, what is the depth of  
the lake ?(The atmospheric temperature is 20 0 C) (ANS: h ≈ ퟕퟎퟎ 풎)  
30. Matter expands when heated and contracts when cooled .Explain why a  
musician must re – tune a stringed instrument if its temperature changes  
31. Explain why it is not advisable for soldiers to march across a bridge in rhythm.