Sound

Sound is a mechanically propagating pressure wave in a material medium, and is a typical example of longitudinal wave. When in a certain range of frequency, it causes the sensation of hearing.

Sound energy, like light energy, obeys the laws of reflection:

1. Its angle of incidence is equal to the angle of reflection.
2. Incident wave, reflected wave, and the normal lie in the same plane.

The following experiment demonstrates the laws of reflection for sound waves:

Take a smooth polished large wooden board and mount it vertically on a table, at right angles to the wooden screen. On each side of the screen place a long, narrow, and highly polished tube from inside. Place a clock at the end of one tube. Move the other tube slightly left or right, till a distinct tick of the clock is heard. Measure the angles of incidence and reflection, and you will find that they are equal. Thus, the experiment illustrates the law of reflection.

Practical Applications of Sound

Megaphone or speaking tube:

When you have to call someone at a far-off distance (say 100 m), what do you do? You cup your hands and call the person with the maximum sound you can produce. Why do you cup your hands? It is because the hands prevent the sound energy from spreading in all directions, much the same way people use horn-shaped metal tubes, commonly called megaphones, while addressing a group of people in fairs or tourist spots. Similarly, the loud-speakers used at the public address system have horn-shaped openings. In all such devices, sound energy is prevented from spreading out by successive reflections from the horn-shaped tubes.

Ear trumpet or hearing aid:

Ear trumpet is a device which is used by people who is hard of hearing. Its shape is like a trumpet. The narrow end of it is kept in the earhole of the person who is hard of hearing, whereas the wider end faces towards the speaker. The waves received by the wider end of the trumpet are reflected into the narrower end, thus increasing its intensity. So, the person who is hard of hearing can hear easily.

Sound boards:

Sound waves obey the laws of reflection on plane as well as curved reflecting surfaces. In order to spread sound evenly in big halls or auditoriums, the speaker is fixed at the focus of a concave reflector which is commonly called sounding board. The sound waves on striking the sounding board gets reflected parallel to the principal axis. So, everyone can hear clearly.

The phenomenon, due to which the repetition of sound is heard after reflection from a distant object (such as a hillock or a high building) after the original sound from a given source dies off, is called an echo.

It has been found that the sensation of any sound lasts for $\frac{1}{10}$ of a second. This time is called the persistence of audibility or persistence of hearing. Thus, it is clear that if any sound reaches back to the ear in less than the mentioned time, we cannot make out when the original sound died and the reflected sound reached the ear. In other words, no echo is heard. But the observations is different in a contrary situation.

Relation between speed of sound, distance of a reflecting body from a source of sound, and time for hearing an echo:

If $t$ is the time after which an echo is heard, $d$ is the distance between the source of sound and the reflecting body, and $v$ is the speed of sound, then the total distance travelled by the sound is $2d$.

Hence , in time $t$ the distance travelled by the sound is $2d.$
Hence, in 1 second the distance travelled by the sound is $\frac {2d}{t},$ which is "the speed of sound."
So, $v=\frac {2d}{t}.$

Conditions for formation of echoes:

1. The minimum distance between the source of sound and the reflecting body should be $17\text{ m}$ normally because the speed of sound in air is normally $332 \text{ m/s}.$
2. The wavelength of sound should be less than the height of the reflecting body.
3. The intensity of sound should be sufficient enough that it can be heard after reflection.

Reverberations:

When a series of reflections fall on the ear from various reflectors one after the another in a closed room, thereby forming a continuous rolling sound, reverberation is said to be caused.

Auditoriums are made in such a manner that each person receives the sound signals.

Numerical problems on echoes:

A boy stands $83 \text{ m}$ in front of a high wall and then blows a whistle. Calculate the time interval when he hears an echo. Speed of sound is $332\text{ m/s}$.

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Distance between wall and the boy: $d = 83\text{ m}$
Speed of sound: $v= 332 \text{ m/s}$
Time: $\frac {2d}{v} = \frac{2\times 83}{332} = 0.5$ seconds. $_\square$

A man stands between two parallel cliffs and fires a gun. He hears two successive echoes after $2$ seconds and $4.5$ seconds. What is the distance between the two cliffs? The speed of sound in air is $332\text{ m/s}$.

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Speed of sound in air: $v = 332\text{ m/s}$
Time taken to hear first echo: $t_1= 2\text{ s}$
Distance of the man from the nearer cliff: $d_1=\frac {v\times t}{2}=\frac{332\times 2}{2}=332\text{ m}$
Time taken to hear second echo: $t_2= 4.5\text{ s}$
Distance of the man from the farther cliff: $d_2=\frac{v\times t}{2}=\frac{332\times 4.5}{2}=747\text{ m}$

So, the distance between the cliffs is $d_1+d_2=332 + 1079 = 1079$ meters. $_\square$

Definition:

The free vibrations produced in a body, on being slightly disturbed from its mean position, are called free vibrations of natural vibrations. It can also be defined as the periodic vibrations of a body of constant amplitude in the absence of any external force. In the absence of any resistance (such as air, etc.), the amplitude of free vibrations remains constant and so does its frequency. Theoretically, such vibrations are possible only in vacuum. However, practically, such vibrations are not possible because of the presence of a medium.

Natural time period: The time period of a body executing natural vibrations is called natural time period.

Natural frequency: The number of vibrations executed per second by a freely vibrating body is called natural frequency.

Examples of free vibrations:

1. a freely suspended pendulum vibrating about its mean position
2. a metal blade clamped at one end being gently disturbed
3. a tuning fork on being struck on a rubber pad

When a simple pendulum is set in motion, it is observed that its amplitude continuously decreases, till it comes to rest. Similarly, the amplitude of any freely vibrating body decreases with the passage of time. However, it is noticed that the frequency or time period of the vibrating body remains the same, and thus the motion is periodic in nature.The damping (decrease in amplitude) occurs due to frictional force, which the surrounding medium exerts on the vibrating body.

Definition: The periodic vibrations of continuously decreasing amplitude in the presence of the resistive force are called damped vibrations.

The frictional force is found proportional to the velocity of the body and the nature of the surrounding material, such as its density, viscosity, etc. Due to frictional force, the vibrating system loses energy, with the result that its amplitude gradually decreases, till it comes to rest. The energy lost by the vibrating body gradually changes into heat energy and is dissipated in the surrounding medium.

Examples of damped vibrations:

1. vibrations of tuning fork in air
2. vibration of stringed instruments in air
3. vibration of simple pendulum in air.

Definition: The phenomenon of damping does not allow a freely vibrating body to maintain its amplitude of vibration. If the amplitude of vibration has to be maintained, then an external periodic force has to be applied. The vibrations, so produced by the external periodic force, are called forced vibrations.   

Characteristics of forced vibrations:

1. The body on which the external periodic force is applied does not vibrate with its natural frequency, but with the frequency of an external periodic force.   

2. The amplitude of vibrations remain constant with time, but its magnitude depends upon the frequency of the driving force.   

3. If the frequency of the external force is much different from the natural frequency of the body, the amplitude of oscillations is very small.   

4. If the frequency of the external force is exactly equal to, or is an integral multiple of, the natural frequency of the vibrating body, the amplitude of oscillations is very large.

Examples of forced vibrations:

1. When the handle of a vibrating tuning fork is pressed against a table top, a loud sound is heard. It is because the vibrating tuning fork forces the table top to vibrate with its own natural frequency. As the table top is very large, the forced vibrations produce larger sound.   

2. When the wire of a sitar or a guitar is plucked, its board and wind box make forced vibrations.   

3. All stringed instruments are provided with a wind box. When the vibrations produced by strings are impressed on the enclosed air, they produce forced vibrations and a loud sound is heard.   

4. When the needle of a gramophone player moves in the grooves of a record, it produces forced vibrations.

Definition: It is defined as a phenomenon when the frequency of an applied external force is equal to, or is an integral multiple of, the natural frequency of the body on which the force is applied, such that it readily takes up the vibrations and begins to vibrate with an increased amplitude.   

Conditions for the phenomenon of resonance:

1. The natural frequency of the given body must be equal to, or is an integral multiple of, the frequency of the external force.   
2. The vibrating body must have sufficient force so that it can set the other body into vibrations.

  Examples of resonance:

1. It is a common experience that the frame of a motor cycle starts vibrating violently when driven at some particular speed. The reason is that at this time the natural frequency of the frame matches with the frequency of the piston of the engine, causing resonance.

2. Soldiers are often asked to break their steps while crossing a bridge. This precaution is taken to prevent any sudden collapse of the bridge, which may occur by the matching of the frequency of the impressed force due to footsteps of the soldiers with the natural frequency of bridge.

3. When the radio station broadcasts at some particular wavelength or frequency, these waves are received by the antenna in which they produce forced vibrations. When the person at the receiving end tunes his radio, he is in a way changing the natural frequency of the radio set. When the natural frequency of the radio set corresponds to the frequency of the broadcasting station, resonance takes place. Thus, the radio signals are amplified. These signals are further processed in the radio, and hence the sound is amplified.

Whenever a string is plucked, it starts vibrating with a specific frequency. The frequency of the note depends upon the following factors:

• Law of length: The frequency of a note produced by a stretched string is inversely proportional to its length.
• Law of tension: The frequency of a note produced by a stretched string is directly proportional to the square root of the tension in the string.
• Law of mass: The frequency of a note produced in a stretched string is inversely proportional to the square root of the mass per unit length of the string.

Terms to remember:

1. Node: Node is a point on the vibrating string with maximum tension and minimum displacement.

2. Antinode: Antinode is the point on the vibrating string with maximum displacement and zero tension.

3. Principal note or first harmonic or fundamental note: If a wire is made to vibrate in $2$ nodes and $3$ antinodes, then the note produced is called pricipal note or fundamental note or first harmonic.
   First harmonic

4. Second harmonic or first octave: If a wire is made to vibrate in $3$ nodes and $2$ antinodes, then the note produced is called principal note or second harmonic or first octave. Second harmonic

Two strings of guitar are of length $80\text{ cm}$ and $60\text{ cm}$, respectively, are made from the same material, and are of the same thickness and under the same tension. If the frequency of the longer wire is $300\text{ Hz}$, find the frequency of the smaller wire.

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We have $f \propto \frac {1}{l},$ so $\frac {f_1}{f_2}= \frac {l_2}{l_1}.$ Substituting the values,
$\begin{aligned} \dfrac {300}{f_2} &= \dfrac {60}{80}\\ f_2 &= \dfrac {300×80}{60}\\ &= 400\text{ (Hz)}.\ _\square \end{aligned}$

Sound waves which produce pleasant sensation in our ears and are acceptable are called musical sounds. Sound waves which produce unpleasant sensation in our ears and are unacceptable are called noise. However, it is difficult to differentiate between musical sound and noise because a particular sound may be pleasant and acceptable to someone, but it may be unpleasant and unacceptable to others. Generally, sound waves which are produced by regular periodic vibrations are musical in character, but those produced by irregular non-periodic vibrations of a body are non-musical in character or noise.

Characteristics of Musical Sound

1. Pitch
   $$
   This characteristic of musical sound enable us to differentiate between two sounds of equal loudness, coming from different sources and having different frequencies:
   $$
   $\begin{aligned} \text {Pitch} &\propto \text {Frequency}\\\\ \text {Pitch} &\propto \dfrac{1}{\text {Wavelength}}. \end{aligned}$ $$

2. Loudness or intensity
   $$
   Intensity of sound is the time rate at which the sound energy flows through a unit area. Loudness is the sensation produced in the brain by the combined effect of intensity and response of the ear:
   $$
   $\begin{aligned} \text {Intensity} &\propto {\text {Amplitude}}^2\\\\ \text {Intensity} &\propto \dfrac {1}{{\text {Distance}}^2}\\\\\\ \text {Intensity} &\propto \text {Density of the medium}. \end{aligned}$ $$
   Unit for measuring loudness of sound: The loudness of sound is measured in decibels (dB). One decibel is defined as the change in loudness of sound when the intensity increases by $26\%$. $$

3. Quality or timbre
   $$
   The notes of different musical instruments are distinguished by this characteristic. It is because different waveforms are produced by different musical instruments. Sound produced by musical instruments differ in wavelength, loudness and waveform. The waveforms differ because a musical instrument produces a number of subsidiary notes other than the main note and this difference depends upon the shape, size and material of the instruments. This is the same reason why few people are really singers because the quality of sound produced by them is far superior and has a more pleasant sensation on our ears.
   Waveforms of different musical instruments