Acoustics and Resonance
Acoustics and Resonance PDF Notes, Important Questions And Synopsis
Acoustics and Resonance Synopsis
Infrasonic and Ultrasonics
- The audible range of sound for human beings extends from about 20 Hz to 20000 Hz.
- This audible range varies from person to person and also with age.
- The hearing sensitivity to higher frequencies is reduced as one grows older.
- Children under the age of 5 and some animals can hear up to 30 kHz.
- The human ear is most sensitive in the range 2000 Hz to 3000 Hz.
- Sound of frequency less than 20 Hz is called infrasonic sound, whereas sound of frequency higher than 20 kHz is called ultrasonic sound.
Frequency ranges for hearing and speaking by humans and animals
Ultrasound and its Applications
- Ultrasound is high-frequency waves. It is able to travel along well-defined paths even in the presence of obstacles.
- Ultrasound is able to travel freely in solids and liquids, but its intensity reduces considerably in gases.
- It is extensively used in industries and for medical purposes.
Properties of Ultrasound
The properties of ultrasound are similar to ordinary sound, but due to high frequencies, they also have the following properties:
- The energy carried by ultrasound is very high.
- It can travel along a well-defined straight path. It does not bend much along the edges of an obstacle as it has very small wavelength.
Applications of Ultrasound
- Bats avoid obstacles in their path by producing and hearing ultrasound.
- To clean parts located in hard-to-reach places such as watches and electronic components. Objects to be cleaned are placed in a cleaning solution and ultrasonic waves are sent into the solution. Due to the high frequency, the particles of dust, grease and dirt get detached and drop out. The objects thus get thoroughly cleaned.
- To detect cracks and flaws in metal blocks. Metallic components are generally used in construction of big structures such as buildings, bridges, machines and scientific equipment. The cracks or holes inside the metal blocks, which are invisible from outside, reduce the strength of the structure. Ultrasonic waves are allowed to pass through the metal block and detectors are used to detect the transmitted waves. If there is even a small defect, then the ultrasound gets reflected, indicating the presence of the flaw or defect.
- It is used to drill holes or make cuts of desired shape in materials like glass.
- Ultrasonic waves are made to reflect from various parts of the heart and form the image of the heart. This technique is called Echocardiography.
- Ultrasonography is used to obtain the images of patient’s organs such as the liver, kidneys etc. It helps to detect stones in these organs.
- Ultrasound can be used to remove cataract and break stones in the kidneys into fine grains.
- SONAR (SOund NAvigation and Ranging) uses ultrasound to detect and find the distance of objects under water.
When a train approaches a platform at high speed while blowing the horn, for a person standing on the platform, the frequency of the horn would appear to be different from the real frequency.
The pitch of the sound appears to increase when the train approaches an observer, and appears to be lower than its true pitch when the train passes by and moves away from the observer.
Similarly, while traveling in a vehicle towards a factory blowing the siren, changes in the frequency of the sound produced can be observed.
This phenomenon of apparent change in the frequency of sound whenever there is a relative motion between the source of sound and the observer, is called Doppler Effect.
When the source is moving towards a stationary observer, the frequency of the sound heard increases.
The following table gives the expressions for each of the other cases where either the source or the observer is moving towards/away from the other.
V – velocity of sound in air.
Natural Vibrations and Forced Oscillations
- A body clamped at one point when disturbed slightly from its position of equilibrium (or rest), starts vibrating. The vibrations so produced are called the free or natural vibrations of the body.
- The period (or frequency) of vibration depends on the shape and size (or structure) of the body.
- The time period of a freely vibrating body is called it’s free (or natural period) and the frequency of the freely vibrating body is called its natural frequency.
- The amplitude of an isolated, freely vibrating body remains constant.
- The free vibrations of a body actually occur only in vacuum because the presence of medium offers some resistance due to which the amplitude of vibration does not remain constant and decreases continuously.
- Thus, we define free vibrations as, the periodic vibrations of a body of constant amplitude in the absence of any external force on it.
Examples of Free or Natural Vibrations
- If the bob of a simple pendulum is displaced slightly from its mean (or resting) position, it starts vibrating with its natural frequency which is determined by the length of the pendulum and the acceleration due to gravity at that place. A simple pendulum of length 1·0 m on earth surface, where acceleration due to gravity is 9·8 m s-2, has its natural frequency as 0·5 Hz.
- When a load suspended from a spring is stretched (or compressed) and then released, it starts vibrating with a period determined by the hardness (or force constant K) of the spring and the mass m of the load.
- When a tuning fork is struck against a hard rubber pad, it vibrates with its natural frequency.
- When we strike the keys of a piano, various strings are set in vibration at their natural frequencies.
- When the string in the instruments like sitar, guitar, violin etc., is plucked, the transverse vibrations of a definite natural frequency are produced in the string. The frequency f of vibration depends on (i) the length l, (ii) the radius r, and (iii) the tension T in the string.
Nature of Free Vibrations
- The amplitude and frequency of a freely vibrating body should remain constant. Once a body starts vibrating, it should continue with the same amplitude and same frequency forever. The figure below shows the displacement-time graph for free vibrations of a body in an ideal condition.
- In actual practice, this does not happen because the surrounding medium offers resistance (or friction) to the motion, so the vibrating body continuously loses energy due to which the amplitude of motion gradually decreases.
- When a body is made to vibrate in a medium, the amplitude of the vibrating body continuously decreases with time and ultimately the body stops vibrating. Such vibrations are called the damped vibrations.
- Thus, we define damped vibrations as the periodic vibrations of a body of decreasing amplitude in presence of resistive force.
- The amplitude of motion decreases due to the frictional (or resistive) force which the surrounding medium exerts on the body vibrating in it. The frictional force at any instant is found to be proportional to the velocity of the vibrating body and it has the tendency to resist the motion.
- As a result, the vibrating body continuously loses energy in doing work against the force of friction.
- The rate at which the energy is lost to the surroundings (or the rate of decrease of amplitude), depends on the nature (i.e., viscosity, density etc.) of the surrounding medium and also on the shape and size of the vibrating body.
- When a slim branch of a tree is pulled and then released, it makes damped vibrations.
- A tuning fork vibrating in air executes damped vibrations as its prongs stop vibrating after some time.
- A simple pendulum oscillating in air executes damped vibrations.
- A freely (or naturally) vibrating body in a medium cannot maintain its amplitude of vibration constant due to the presence of damping forces of the surrounding medium.
- However, the amplitude of vibrations can be kept constant by applying an external periodic force such that the external periodic force compensates for the loss of energy in each vibration due to the damping forces.
- The vibrations of the body under a periodic force are called the forced vibrations.
- Thus, we define forced vibrations as the vibrations of a body which take place under the influence of external periodic force acting on it.
- When an external periodic force is applied on a vibrating body, the body no longer vibrates with its own natural frequency, but it gradually acquires the frequency of the applied periodic force.
- The external applied force is called the driving force.
- If the frequency of the external force is different from the natural frequency of the body, the body oscillates with very small amplitude. But if the frequency of the external force is exactly equal to the natural frequency of the vibrating body, the body oscillates with very large amplitude.
- When the stem of a vibrating tuning fork is pressed against the top of a table, the tuning fork forces the table top to vibrate with its own frequency. Since the table top has a much larger vibrating area than the tuning fork, the forced vibrations of the table top produce a louder (or more intense) sound than that produced by the tuning fork.
- The vibrations produced in the diaphragm of a microphone sound box with frequencies corresponding to the speech of the speaker are the forced vibrations.
- When a guitar is played, the artist forces the strings of the guitar to execute forced vibrations.
- All stringed instruments are provided with a hollow sound box which contains air. In these instruments, when the strings on it are made to vibrate by plucking, vibrations are produced in air of the sound box which are forced vibrations. Since surface area of air in sound box is large, the forced vibrations of air cause a loud sound.
- When the frequency of an externally applied periodic force on a body is equal to its natural frequency, the body readily begins to vibrate with increased amplitude. This phenomenon is known as resonance. The vibrations of large amplitude are called resonant vibrations.
- The phenomenon of resonance can be demonstrated by the following experiments.
- Two tuning forks are placed on separate sound boxes. If the prong of one of the tuning forks is struck on a rubber pad, it starts vibrating. On putting the tuning fork A on its sound box, we find that the other tuning fork B also starts vibrating and a loud sound is heard. The vibrations produced in the second tuning fork B are forced vibrations and the sound is loud due to resonance.
- Four pendulums A, B, C and D are suspended from the same elastic support. A and B are of the same length. D is longer than A or B and C is shorter than A or B. They are suspended in the order shown below.
- When the pendulum A is set into vibration by displacing it to one side, normal to its length. Pendulum B also starts vibrating initially with small amplitude and in some time it acquires the same amplitude as A. When the amplitude of pendulum B becomes maximum, the amplitude of pendulum A becomes minimum since total energy is constant. After sometime, the amplitude of pendulum B decreases and that of A increases.
- The exchange of energy takes place only between the pendulums A and B because their natural frequencies are equal. The pendulums C and D also vibrate, but each of them vibrate with very small amplitude.
- Reason- The vibrations produced in pendulum A are communicated as forced vibrations to the other pendulums B, C and D through the rubber string PQ. The pendulums C and D remain in the state of forced vibrations, while the pendulum B comes in the state of resonance. This is because the natural time period of pendulum B is equal to that of A, and therefore resonance takes place. The pendulum B, therefore, vibrates with the frequency of pendulum A and it remains in phase with A.
- The arrangement for studying the resonance in an air column is shown below. It consists of a long cylindrical tube A and a cylindrical vessel B. The tube A is fixed, while the vessel B can be moved up or down and it can be clamped at a desired position. Both the tube and the vessel are connected at the lower ends by a rubber tube and they are partially filled with water.
- The vibrating source (i.e., tuning fork) is kept at the mouth of the tube A so that it works as a closed end air pipe with water surface in it forming the closed end (i.e., the reflecting surface). Thus, an air column is formed in the tube A between the water surface and its mouth. When this air column is made to vibrate, it will vibrate with its natural frequency which depends on the length of the air column. The length of the air column in tube A can be changed by moving B up or down.
- In the experiment, first the tube A is filled with water up to the top. Then the level of water in tube A is lowered while keeping a vibrating tuning fork at its mouth. It is found that a loud sound is heard at a certain level of water. On further lowering the level of water in tube A, when the length of air column becomes three times the previous one, a loud sound is heard again.
- Reason- The vibrating tuning fork when held just above the mouth of the tube A, causes the forced vibrations in the air column of the tube A. When the frequency of air column is changed by increasing the length of air column (i.e., decreasing the water level in the tube), at a certain level of water in tube A, a loud sound is heard. This happens when the frequency of the air column becomes equal to the frequency of the tuning fork, i.e., the vibrations of the air column are in resonance with those of the fork.
- Some of few examples are:
- Soldiers should break their steps while crossing a bridge. In case, the soldiers match their steps, it would prove fatal to them as the vibration of a particular large frequency is produced. The natural frequency of the suspension bridge happens to be equal to the frequency of the steps. Resonance will take place and the bridge will vibrate with large amplitude and could even crumble.
- Earthquakes cause destruction to a greater extent. It is interesting to note that sometimes in an earthquake, short and tall structures remain unaffected while medium sized structures fall down. This happens because the natural frequency of the short structures is higher and those of the taller structures are lower than the frequency of the seismic waves.
- In sitar and guitar, sound of the strings can be heard due to the presence of hollow part
The hollow portion is the sound box of these instruments; without which no sound can be heard. The natural frequency of a string corresponds to the natural frequency of the air column. The air column inside the sound box vibrates with the same frequency of string and produces sound.
G - Peg
B1 & B2 - Knife edges
P - Pulley
W - Weight hanger
M - Metre scale
S - Wire
T - Tuning fork
H - Holes for the sound box
- Sonometer is an instrument that is used for studying the vibrations of a fixed wire or string. It consists of a hollow wooden box with a wire stretched across its top. The wire is fixed at one end while the other end passes over a pulley and a load can be suspended from it. Any length of wire can be set into vibration by placing two inverted V-shaped bridges at the ends, by placing vibrating tuning fork on the sonometer.
- The wire of this sonometer can be allowed to vibrate forcibly by using a vibrating tuning fork. If the natural (fundamental) frequency of vibrating length of the string between two knife edges is same as that of the tuning fork used, then it starts vibrating with maximum amplitude, i.e., resonance occurs. If a paper rider is placed at the centre of vibrating portion of the string, then in resonating state, it is thrown off from the string.
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