Chapter 8 Wave Motion And Sound 6o2r6h

CHAPTER 8 WAVE MOTION AND SOUND 1. 2. 3. 4. 5. 6.

Contents: SIMPLE HARMONIC MOTIONIN THE SPRING. CIRCULAR MOTION AND SIMPLE HARMONIC MOTION. CALCULATION OF TIME PERIOD CALCULATION OF VELOCITY. CLCULATION OF ENERGY. SIMPLE PENDULUM.

TECHNECHAL TERM RELATIVE DEFINITIONS 1. Vibration: A complete round trip about mean position is called “vibration”. 2. Vibratory Motion: For ward and back ward motion, about the mean position is called “vibratory motion”. 3. Displacement : Path between the two positions that an object moves from its equilibrium position denoted by “x” 4. Amplitude: The maximum displacement between the two positions that an object moves from its equilibrium position, denoted by “xo” 5. Time period ( T ): The time that it taken form an oscillator to execute one complete cycle of its motion 6. Frequency (f): The number of cycles (or oscillations) the object completes per unit time. Say, 7. Angular Frequency: The angular frequency is the characteristic of angular velocity.

ω=

T=

1 f.

2π =2πf T

8. Periodic motion : Periodic motion of a body is that motion which is repeated identically after a fixed interval of time. The fixed interval of time after which the motion is repeated is call period of motion. For Examples:  The revolution of earth around the sun is a periodic motion. Its period of revolution is one year.  The rotation of earth about its polar axis is a periodic motion. Its period of rotation is one day. 9. Oscillatory motion: Oscillatory or Vibratory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point(called mean position or equilibrium position), in a definite interval of time). Thus a periodic and bounded motion of a body about a fixed point is called an oscillatory or vibratory motion. 10. Oscillation: The oscillatory motion can be expressed in of sine and cosine functions or their combinations. It is due to this reason that the oscillatory motion is called a harmonic motion. Examples 1. The motion of the pendulum of a wall clock is oscillatory motion. 2. The motion of the bob of a simple pendulum. 3. The motion of a loaded spring. 4. The motion of liquid contained in U-tube. 11. Simple harmonic motion: Simple Harmonic Motion is characterized by the acceleration a being oppositely proportional to the displacement, a-x

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DESCRIPTIVE PART 1.  Calculation of Simple Harmonic Motion: Consider a spring–mass system, in which a mass is attached to a spring. If the spring is unscratched, there is no net force on the mass. If the mass is displaced from equilibrium, the spring will exert a restoring force, which is a force that tends to restore the mass to the equilibrium position. In the case of the spring–mass system, this force is the elastic force, which is given by Hooke's Law. F = − k x … (i) Where “F” is the restoring force, “x” is the displacement, and “k” is the spring constant. Any system that undergoes simple harmonic motion according second law of motion, F = ma … (ii) By comparing equation (i) & (ii) ma = – k x

 k a=-   x  m a = - (constant) x a-x

 

Or This shows that, Acceleration is directly proportional to the displacement, Motion is to wards the mean position Such type of vibratory motion is called “Simple harmonic motion”. 2.  Simple Harmonic motion and Circular motion: Suppose a particle is rotating from position A to P, on the circumference of a circle with uniform angular velocity “” about fixed radius “R”. When a particle is at a point “Q”, then projection located on to diameter. The angular displacement covered is “”. We know that, centripetal acceleration, is given by, ac = R 2 The motion of the projection a particle is to wards the mean position. Therefore,

 

 x   aQ = - R2  R

a = - 2x or a = -  2x Where, 2 is constant. Hence, a = - (constant) x a - x This shows that, the projection of particle has, The, acceleration is directly proportional to displacement. And, motion is towards the mean position, indicated by negative sign. It means the projection of a particle performs the “Simple Harmonic Motion”. 3.  Calculation of Time period: Let us consider a particle is moving in circular motion with uniform angular velocity “”, about fixed radius “r”. When it is at a position point, covers angular displacement “”, from position A. The projection is to be observed a Q, on to diameter. The time required to complete one rotation is called “Time Period”, denoted by “T”. S=vt 2 r = r T

T=

We know that,

2π ω

a=-k

… (i)

xo m and a = - 2xo

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k x o = ω2 x o m or

Thus equation # 1 becomes,

k = ω2 m We get, 2π T= k m

T = 2π Or

f= And

k = ω m

m k

1 T

1 m 2π k 1 k f= 2π m f=

4.  Calculation of velocity: Let us consider a particle has circular motion with uniform angular velocity “”, about fixed radius “r”. When a particle is at position P, the angular displacement is “”, its projection is to be observed at Q on to diameter. The particle has tangential velocity “vp”, at P, resolve the velocity in to its rectangular components. We get, vQ = vp Sin  … (i) 2 2 We know that, Sin  + Cos  = 1 Sin 2  = 1 - Cos2

Sinθ =

1 - Cos 2θ

The equation # 1 can be written as,

1-

x2 r2

vQ = r ω 1 -

x2 r2

Sinθ =

Or

vQ = r ω 

vQ = r ω  

vQ = ω We know that, And, By comparing, we get

r2 - x2 r2 1 r2 - x2 r

r2 - x2

x a=-k o m a = - 2 xo

-k

xo = - ω2 x o m

k = ω2 m k = ω m or The equation # 2 can be written as when, r = xo

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… (2)

k m

v=

x o2 - x 2

If the velocity is maximum (vmax = v), then x= 0 

k m

v max = x o

vmax = x o

 1

0  x o2

k m

5.  Calculation of Energy in Simple Harmonic Motion :   There is constant interchange between kinetic and potential energy in toand-fro motion. If the system does not have to work against forces such as friction, the total energy will remain constant. Let us consider a ball of mass “m” is to be attached at one end of elastic spring having spring constant “k”, the other end is fixed with rigid . The spring is placed on to smooth horizontal surface. When the spring is to be stretched, from its mean position, then it performs simple harmonic motion, about its mean position. The work is to be done on to the ball when force “F”, produces displacement “x”, at position A. 1. Potential energy: Work = Fd The force on to the ball at mean position is zero (F=0) and “F” at Q, according to the hooks law.

Fi + Ff 2 0+F Fave = 2 F Fave = 2 Fx Work = 2  k x x Work = 2 Fave =

Or Therefore,

Work =

k x2 2

The work is to be stored in the form of potential energy, into spring.

P.E P =

1 k x2 2

Hence, When potential energy is maximum ( P.E.max) then, x = xo

P.E max =

Therefore, 2. Kinetic energy:

K.E =

1 k x o2 2

1 m v2 2

We know that, This energy possesses by a ball due, its motion towards mean position The kinetic energy at position “Q”, is given by,

K.E P =

1 m v 2P 2 1  K.E P = m  2  1  K.E P = m  2 

K.E P =

k m

x



- x   k 2  xo - x2  m  2 o



1  k  x o2 - x 2  2 

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2



2

Therefore, When, K.E is maximum then, x =0.

1 2 1 2 kx o - kx 2 2

K.E P =

1 K.E max = k x o2 2

Therefore, 3. Total energy: A ball possesses total energy at any position during its motion on to horizontal surface, say at “Q” Total energy =K.EP + P.EP. 1  1  k x 2o - k x 2 2  2 

T.E = 

T.E =

+

1 k x2 2

1 2 k xo 2

This shows that, total energy of the system remains conserved, during its simple harmonic motion. 6.  The Simple Pendulum A heavy point mass tied to a perfectly flexible and inextensible string and suspended with rigid , such that it can oscillate without friction, the arrangement is called a “simple pendulum.” Simple Pendulum performs SHM: Consider a small bob of mass m hanging from a very light string, length L, which in turn hangs from a fixed point. If it is pulled to one side through a small angle  it will swing with a to-and-fro movement in the arc of a circle.The angular displacement from the mean position is, at an instant. The weight of the bob, can be resolved in to components, we get [mg cos], along the string which balance the tension. And [mg sin], which tries to bring the bob towards its mean position, this force is called the “restoring force.” F = - mg sin ma = - mg sin a= - g sin

 x a=-g    L  g a = -   (x)  L a = - constant (x) a  -x This shows that,  The acceleration is directly proportional to the displacement,  And the motion is towards the mean position. It means the motion of simple pendulum executes “Simple Harmonic motion.” Calculation of Time Period: The time required to complete one vibration is called time period denoted by “T”. We know that, Where,

T=

2π ω

… (1)

a = - 2 x And By comparing two equation Or

 g a=-    L

x

 g - ω2 x = -    L

x

 g ω2 =    L

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 g  L

ω =

By putting in equation# (1),

2π

T=

 g  L

We get,

 L T = 2π    g

Or This shows that, the time period of simple pendulum is independent of its mass.

f= f=

f=

1 T 1

 L 2π    g 1 2π

 g  L

What is Seconds pendulum? A pendulum whose time period is two seconds is called “seconds pendulum”. What is Compound pendulum? The length L of the ideal simple pendulum, used for calculating the period, is the distance from the pivot point to the center of mass of the bob. For a real pendulum consisting of a swinging rigid body, called a “compound pendulum”, the length is more difficult to define. A real pendulum swings with the same period as a simple pendulum with a length equal to the distance from the pivot point to a point in the pendulum called the center of oscillation. This is located under the center of mass, at a distance called the radius of gyration, that depends on the mass distribution along the pendulum. However, for the usual sort of pendulum in which most of the mass is concentrated in the bob, the center of oscillation is close to the center of mass

WAVE MOTION Contents: 1 FUNDANENTAL FREQUENCY AND FUNDAMENTAL HARMONICS. 2 Ripple Tank 3 Determination of Fundamental Frequency from Laws of vibrating String: 4 SONOMETER.

TECHNECHAL TERM RELATIVE DEFINITIONS 1. Wave motion: Wave motion is a disturbance that moves from place to place in some medium, carrying energy with it.

2. Wave Length: The wavelength of a wave is simply the length of one complete wave cycle. 3. Crest: The crest of a wave is the point on the medium which exhibits the maximum amount of positive or upwards displacement from the mean position. 4. Trough: The trough of a wave is the point on the medium which exhibits the maximum amount of negative or downwards displacement from the rest position. 5. Amplitude: The height of the wave is called its amplitude. Some areas consider the middle of the wave to its peak as the amplitude, while others consider peak-to-peak as the amplitude.

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6. Frequency: Frequency is refers to how many waves are made per time interval. This is usually described as cycles per second. So, one cycle per second is one Hertz, as in:1 s = 1 Hertz. 7. Phase: Phase of a vibrating particle at any instant is a physical quantity which completely expresses the position and direction of motion of the particle at that instant with respect to its mean position. In Oscillatory motion, the phase of a vibrating particle is the argument of sine or cosine function involved to represent the generalized equation of motion of the vibrating particle. 8. Initial phase: It is the phase of a vibrating particle corresponding to time t=0. When t = 0, its unit is radians. 9. Phase difference: Between two vibrating particles tells the lack of harmony in the vibrating states of the two particles at a given instant. It is measured as the difference in phase angles of the two vibrating particles at any instant. 10. Zero phase difference (i) When the two vibrating particles cross their mean positions at the same time, moving in the same direction. The phase difference between them is zero. 11. 180o-Phase difference (ii) When the two vibrating particles cross their mean position at the same time, moving in the opposite direction and the particle A is ahead of particle B by half vibration. The phase difference between them is  rad or 180o 12. 90o –Phase difference The phase difference between them is /2 rad or 90o. 13. Standing waves: When two identical progressive waves (having same amplitude and frequency) travel l with same velocity along same straight along line but in opposite direction, the resultant wave formed by their superposition is called a “standing wave” or stationary wave Types of Waves: There are three types of waves 1. Mechanical waves:  Such wave requires a material medium to travel (air, water, ropes). These waves are divided into three different types  Traveling waves: The particles of a medium are disturbed from the mean position; they perform simple harmonic motion and cover distance as time increases. This type of disturbance is called “wave”. For example, a stone is to be dropped in water, waves produced are known as “Traveling waves”.  Longitudinal wave: A wave that causes the particles of the surrounding medium to vibrate in the same direction as that in which the wave is moving.  Transverse wave: A wave that causes the particles of the surrounding medium to vibrate in a direction at right angles to the direction of the wave motion. 2. Electromagnetic waves : Such wave does not require a medium to travel (light, radio). 3. Matter waves: Such wave is produced by electrons and particles Surface waves: Such wave is both transverse waves and longitudinal waves mixed in one medium.

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1. Rarefaction: A region of space with a lower-than-normal density. 2. Condensation: A region of space with a higher-than-normal density. 3. Relation between wavelength, speed and frequency: The, distance covered by the wave during one vibration =  Therefore, the distance covered by the wave during  vibration = f  Hence, Distance covered in one Second = f  v=f

f=

v 

or The wavelength () is inversely proportional to the frequency (f), under constant speed (v)

f 

f= Or

Speed of wave =

v 

1 

v= f

distance covered by one wave time taken to complete one wave .

DESCRIPTIVE PART 1. Wave Motion: Wave Motion is mechanism by which energy is conveyed from one place to another in mechanically propagated waves without the transference of matter. At any point along the path of transmission a periodic displacement, or oscillation, occurs about a neutral position. The oscillation may be of air molecules, as in the case of sound traveling through the atmosphere; of water molecules, as in waves occurring on the surface of the ocean; or of portions of a rope or a wire spring. In each of these cases the particles of matter oscillate about their own equilibrium position and only the energy moves continuously in one direction. Such waves are called mechanical because the energy is transmitted through a material medium, without a mass movement of the medium itself. The only form of wave motion that requires no material medium for transmission is the electromagnetic wave; in this case the displacement is of electric and magnetic fields of force in space. 1.  Calculation of Fundamental Frequency and Fundamental Harmonics:  Let us consider a stretched rubber cord, which is very thin, perfectly flexible and uniform in diameter of length “L” fixed at both ends. When traveling standing wave is produced in it, then velocity “v” is produced, which remain constant. Standing waves on a stretched string are such waves, must have nodes at the transmutation points. The middle point of the string has the maximum displacement, called anti- node. The distance between

λ consecutive nodes or antipodes is 2 . A node is always produced at displacement is zero and anti node at which displacement is maximum. If stretched rubber cord is in normal mode of vibration,

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λ 2 String of length L fixed at both ends:

L=

 = 2L

v λ v λ= f1

f1 =

2L= So

f1 =

v f1

v 2 L , known as Fundamental frequency.

Hence, The stretched rubber cord consists of two loops or two-segment. Let

L=

λ λ + 2 2

λ 2 v f L= 2 2 2 v L= 2 2 f2 v f2 = 2 2L L= 2

Or f2 = 2 f1 is the frequency of First Overtone, and Second Harmonic Similarly, f 3 = 3f1 is the frequency of Second Overtone and Third Harmonic fn = n f1 is the frequency of (n-1) overtone and nth Harmonic 2. Ripple Tank: It is an instrument to generate the waves and study the waves. It consists a transparent glass tank with a motor thin elastic spring. A source of light placed above the tank. A tank is filled with water and elastic spring dipped in water. When motor operated the circular waves are moving in the water tank. The light produced the shadow of water waves on the screen placed below the water tank. On the screen crest appears s a light line and trough as dark line. 3.  Determination of Fundamental Frequency from Laws of vibrating String:  Let us consider a stretched string, which is fixed between two nails. Suppose the string is plucked at its center, the fundamental mode of vibration is set in the string with nodes at its two ends. The frequency of such vibration will be fundamental frequency, obeys the following laws, LAW # 1 (Law of Length): For the given martial of string, the frequency is inversely proportional to its length,

f 

1 L

LAW # 2 (Law of mass per unit length):

f 

1 μ

LAW # 3 (Law of Tension): The length of given string, the frequency is directly proportional as square root, of the tension in the string.

f 

T

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f Hence,

f=K

1 L

  

1   L 

T  μ T  μ

1 Where K= 2 , constant of proportionality. f=

1   2L 

T  μ

Thus, 2. Sonometer: It is an instrument used for verification of laws of standing waves. It consists of wooden box on which a steel wire stretched one end of which is fixed to a peg and other end es over of pulley. The open end carries a hanger on which different weights can be hanged. Two sharp ended bridges are placed below the wire. They are used to change the length of vibrating wire. A horizontal scale gradated in millimeter is fixed below the wire on the box by which the length of loop is measured

SOUND Contents: WHAT IS SOUND SPEED OF SOUND. MUSICAL SOUND AND NOISE WITH CHARACTERSTICS OF MUSICAL SOUND. BEATS. DOPPLES EFFECT. EQUATIONS. DIMENSIONS. SHORT QUESTIONS AND ANSWERS.

1. 2. 3. 4. 5. 6. 7. 8.

TECHNECHAL TERM RELATIVE DEFINITIONS 1. Sound or Acoustics: Sound is a form of energy that we receive with the help of our ears and feel a specific type of sensation in our brain. It originates from a mechanically vibrated body; the vibration of the body causes disturbances in its surrounding medium. The disturbance when reaches our ears, the ear drum is also set into similar vibration and we have a sensation of sound with our auditory organs. 2. Reflection of sound: Sound waves reflect from a surface obeying the laws of reflection of light waves. 3. Echo: In a large open space or in a valley, when a loud sound is produced, the sound repeats several times which are perceived clearly after the initial sound. This repetition of a sound, distinguishable from the original sound is called Echo. At higher temperature velocity of sound increases, so this distance for the reflector also increases. 4. Tone and Note: Sound of one frequency is Tone. Note is a sound mixed of different frequencies. 5. Acoustics: It is the interdisciplinary science that deals with the study of sound, ultrasound and infrasound (all mechanical waves in gases, liquids, and solids). A scientist who works in the field of acoustics is an acoustician. The application of acoustics in technology is called acoustical engineering. There is often much overlap and interaction between the interests of acousticians and acoustical engineers. 6. Creating and detecting sounds: Creating and detecting sounds are similar effects, but opposite. They demonstrate the duality of nature. 7.

Creating sound:

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Whenever an object in air vibrates, it causes longitudinal or compression waves in the air. These waves move away from the object as sound. Detecting sound: When a sound wave strikes an object, it can cause the object to vibrate. This leads to the method to detect sound, which requires changing that vibration into some other type of signal—usually electrical. The main way you detect or sense sounds is through ears. The sound waves vibrate your ear drum, which goes to the inner ear and is changed to nerve signals you can sense. We can also feel sounds. Mach number: The ratio of the speed of the aircraft to the speed of sound in the gas determines the magnitude of many of the compressibility effects. Because of the importance of this speed ratio, aerodynamicists have designated it with a special parameter called the Mach number

DESCRIPTIVE PART

i)

1. Origin of sound:  Sound is a form of energy that we receive with the help of our ears and feel a specific type of sensation in our brain. It required following things for the propagation Sounding body (tuning fork) ii) Material medium (air) iii) Receiver (ear) Frequency Ranges: The normal human ear cannot hear sound of all frequencies. The ear can give easily response to sound of frequency greater than 20 hertz and less than 20,000hertz Ultra-sonic waves, having frequency range greater than 20,000 hertz. Audible waves, having frequency range between 20 hertz and 20,000 hertz, are sensitive to human ear and can be heard. Infrasonic waves, having frequency range below 20 hertz. They cannot be heard. Progressive wave: A wave which transfers energy by moving away from its source is called “progressive wave”. Periodic wave: A continuous regular and rhythmic disturbances in a medium result from periodic vibration of a source cases “periodic wave” in that medium. 2.  Calculation of Speed of sound: A sound wave is a pressure disturbance which travels through a medium by means of particle-to-particle interaction. As one particle becomes disturbed, it exerts a force on the next adjacent particle, thus disturbing that particle from rest and transporting the energy through the medium. Sound waves in air, which produce a sensation in our ears, are progressive longitudinal waves. In these waves sound propagated from one place to another in the form of compressions and rarefaction. Great scientist Newton first studied the factors, on which the velocity of sound in a medium depends. According to Newton speed of sound depends on to

Elastic properties and inertial properties of a material (solids, liquids & gases.). The velocity of sound in the

v= medium is given by,

elastic property inertial property

E d For propagation of sound in solids stress E= strain force /area E= Change in length

v=

E is known as young’s modulus

or For propagation of sound of sound in fluids, volume elasticity is used in a place of “E”

“B” is known as Bulk modulus Defects of Newton’s formula:

v=

B d

B=

force /area Change in volume

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According to Newton, when sound propagated through air, compression sand rarefaction is produced very slowly so that the heat produced during compression (or absorbed during rarefaction)

P d , In this expression speed of sound is determined in air

v=

According to Newton, a sound travel through in air under isothermal conditions and Boyle’s law is applicable. Lap lace’s Correction: According to Lap lace, Newton’s idea of assuming compression and rarefaction to be isothermal was not correct. Hence in the Newton’s formula for velocity of sound we must use adiabatic elasticity.

γP d - - - - - - - ► (1) Specific heat capacity at constant pressure γ= Specific heat capacity at constant volume

v= According to Lap lace,

    c v

γ= 

= 1.4 According to the ideal gas equation, pV = nRT

P=

v = By putting “p” in equation #1, we get

nRT V nRT γ V d

γnRT m m V d (density) = V V Because, m (given mass) n 1 n= = M (molecular mass) m M v =

We know that,

v =γ

n RT m

v =

γ RT M

v =

γ R M

Therefore,

The factor

T

γ R M is constant

v 

T

This shows that, speed of sound is directly proportional to square root of absolute temperature, if The absolute temperature of medium is made four times; the speed of sound will be doubled. 3. Musical sound and Noise: The different feelings and sensations created in our brain on reception of sound may be divided in two classes: The different feelings and sensations created in our brain on reception of sound may be divided in two classes Noise: Noise is characterized as being a periodic or having a non-repetitive pattern. There are many different types of noise, depending primarily on the random distribution of frequencies. If a sound creates a pleasing sensation it is a musical sound. If a sound is annoying and causes displeasure, it is called noise. Noise is the sound emitted by a source vibrating in irregular, periodic and discontinuous manners. Examples: Sounds of shouting, thunder claps, gunfire, etc. Musical: A sound is considered musical if it is generated from a source that vibrates continuously and periodically. This sound comprise a series of sounds with smooth, regular, patterns producing pleasant effect Examples of Musical sound: sounds of musical instruments, sitar, violin, etc.

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Characteristics of musical sound are: (a) Intensity, (b) Loudness(c) Pitch, (d) Quality or Timbre (a) Intensity: The amount of energy which is transported past a given area of the medium per unit of time is known as the “intensity of the sound wave”. If greater the amplitude of vibrations of the particles of the medium, then greater the rate at which energy is transported through it and the more intense that the sound wave is. Intensity is the energy/time/area; and since the energy/time ratio is equivalent to the quantity power, intensity is simply the power/area. Typical units for expressing the intensity of a sound wave are Watts/meter2. Special units have been designed to measure sound intensity (the bel

B = log

I Io where I is the intensity of interest in W/m2, and I = o

and the decibel). The intensity in bel is found by 10-12 W/m2 (which is assumed to be the threshold of human hearing).

(b) Loudness: The characteristic of musical sound in which “the magnitude of auditory sensation produced of sound.” Intensity is that characteristic which gives an idea about loudness of sound. The difference between two sound intensities is called “intensity level. Its unit is “bel”. Usually a smaller unit decibel is used, that is one tenth of "bel”. The intensity of sound is 40 to 80 decibels. (c) Pitch: The characteristics of musical sound which “A shrill sound can be distinguished from grave sound.” Pitch is the property of sound which determines whether the sound is high or low. Pitch is determined by the number of vibrations per second made by the sounding body. Comparatively slow vibrations produce a low sound, Pitch is a term used to describe the frequency of sound waves. The normal range of human hearing is from about 20 Hz to 20 000 Hz. d) Quality or Timbre: The characteristic of musical sound which “Anyone can distinguish between two notes of same frequency and intensity”. The quality of sound depends on the number and relative intensity of harmonics produced by the sounding body. The quality of sound depends on waveform of sound The property of sound by which we are able to distinguish this difference is called quality. Quality depends upon the form of the vibrations. 4. Principle of Superposition: When two waves in a same medium travel, the net displacement of the medium caused by the resultant wave at any point is equal to the algebraic sum of the displacements of the entire wave. The is called principle of superposition. Y= Y1+Y2 + - - - - If two waves which cross each other have opposite phase, their resultant displacement will be, Y= Y1 -Y2 If Y1 =Y2 then result displacement Y=0. Principle of superposition leads to many phenomenon with wave 1: Two waves have same frequency and traveling in the same direction (interference) 2: Two waves of slightly different frequencies and traveling in the same direction (beats). 3: Two waves of equal frequency traveling in opposite direction (stationary wave) 5. Interference: Superposition of two waves having the same frequency and traveling in the same direction results in a phenomenon called “interference”. Whenever path difference is an integral multiple of wavelength, of two waves are added up. This effect called “constructive interference”. Path difference = n  , where, n= 0,1,2,3, - - And points where the displacements of two waves cancel each other‘s effect, the path difference is an odd integral multiple of half the wavelength. This effect is called “destructive interference”.

λ Path difference = (2 n +1) 2 where, n= 0,1,2,3, - - -

5.  Beats: A beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies. It means, “The periodic alterations of sound between maximum and minimum loudness are called beats.”

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Let us consider two tuning forks “A” and “B” of frequencies 32 Hz and 30 Hz respectively. They are placed at equal distances from the ear and start to vibrate at same time. When, t = 0 Both tuning forks are rights of mean position and prodded compression .The loud sound is to be heard, because After this time the prongs of both tuning fork are in phase. After, t = ¼ sec. After this time, the prong of tuning fork “A” completes 8 vib. to the right of mean position, produces compression and tuning fork “B” completes 7½ vib. to the left of mean position, produces rarefaction. Hence intensity decreases and “minimum” sound is to be heard because both are in opposite phase. After, t = ½ sec . After this time tuning fork “A” competes16 vib. to the right and produces compression and tuning fork “B” completes 15 vib. also to the right of mean position. Hence intensity increases and “maximum” sound is to be heard because both are in phase. After, t =3/4 sec. After this time tuning fork “A” completes 24vib. to the right and again produces compression and tuning fork “B” completes 22 ½ vib. Produces rarefaction to the left of mean position. The prongs are in opposite directions. Hence intensity decreases and “minimum” sound is to be heard both are in opposite phase. After, t = 1sec. After this time Tuning fork “A” completes 32 vib. and tuning fork “B” completes 30 vib. The prongs of both tuning forks are right of mean position produces compression. Hence intensity increases and “maximum” sound is to be heard because both are in phase. From above explanation it is observed that two beats are heard in one second. The difference between the frequencies is equal to the number of beats. far - fib = Number of beats /second 7.  Doppler’s effect: When the source of sound and observer are in relative motion with respect to each other, pitch of the sound heard by the observer appears to be different from the actual frequency of the source. This apparent change of pitch is called “Doppler’s effect” Explanation: Let us consider a source of sound in the state of rest is emitting “f” waves per second, with velocity, “v”. The sound waves approach the observer, at rest. The distance covered in one second, v = s .The rest observer will receive same real frequency “f”.

λ=

Distance occupied by υ waves f number of waves v λ= f

Case 1(a) When observer moves towards stationary sounding source: Let us consider an observer is moving with velocity “vo”, towards stationary sounding body. In the state of rest an

vo t vt observer receives λ waves and after moving he receives λ additional waves. Total number of waves f= time vt vt + o λ f´ = λ t  v + vo  t f´ = λt  v + vo  f´ = λ

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v f  v + vo  f´ = v f  v + vo  f f´ = v λ=

We know that,

Therefore,

This shows that apparent frequency will be greater than real frequency. (b)When observer is moving away from stationary source: Suppose an observer is moving away from the stationary source. The number of waves decreases by v ote/ as compared to the original waves.

f '' =

 v - vo  f v

This shows that apparent frequency will be smaller than real frequency. Case 2 (a) When source of sound is moving towards the stationary observer: Let us consider a source of sound is moving with a velocity “vs.”. The effect is observed that wavelength decreases. Hence the wave length receives by stationary observer is given by,

λ' =

We know that, Therefore, Or

Distance in which υ waves are contained number of waves v - vs λ' = f v λ' = f´ v v - vs = f´ f vf f´= v - vs

This shows that apparent frequency will be greater than real frequency. (b) When source of sound is moving away from stationary observer: Suppose a source of sound is moving away from stationary observer, the wavelength increases by (v s +v).

f '' =

vf v + vs

Therefore, This shows that apparent frequency will smaller than real frequency. Case 3 (a) When an observer and source of sound are moving closer to one another: Suppose an observer is moving with velocity vo closer to the source of sound moving with velocity vs. along the same line.

f´ =

 v + vo  f  v - vs 

The apparent frequencies heard by an observer will increases. (b)When observer and source of sound are moving away from one another: Suppose an observer and source of sound, after approaching closer to one another; move away from each other, along the line ing in opposite direction.

f '' =

 v - vo  f  v + vs 

The apparent frequencies heard by an observer will decreases. Applications of Doppler’s effect:

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1) Radar waves are reflected from an airplane, so that speed of traffic is to be measured. 2) The ultrasonic instruments are designed. 3) Motion of sub marine is to be detected. 4) The speed and direction of motion of the earth satellite can be detected. 8. Shock waves and production: When speed of source of sound ( Vs ) comparable to the speed of sound waves in the medium (V)then the source keeps pace with the outgoing waves in direction of medium. When the speed of source is exactly equal to the wave speed (Vs= V ) the wave can never move ahead of the source . They continue to pile up on top of each other. This causes a large amount of wave energy to be concentrated in the area just ahead of the wave source. This concentration of the waves is called “shock wave”. When speed of source exceeds the wave speed (Vs> V) then the source gets ahead of the wave. The plane wave front now takes the shape of cone with the source at its apex. This cone is known at “Match cone”.

Equations

1. F = k x

5.

v max = x o

k m

 L T = 2π    g 9. E v= d 13. 17.

 v ± vo  f f" =  v ± vs 

 k a=-   x  m 2. 3. a = - 2xo m 1 k T = 2π f= k 2π m 6. 7.

11.

RT M

v= 14.

15.

vQ = ω

r2 - x2

1 k x o2 2 8. T.E= K.E = P.E = Work =

B d

v= 10. fn = n f1

4,

v= 12.

P d

Energy Area. Time 16. f a - f b = Number of beats /second

Intensity =

v f  v± vs 

f" = 18,

Dimensions PHYSICAL QUANTITY

Hooks constant

SYMBOL

DIMENSION

 M L T 

UNIT

-2

F k= x

 L

 M T   M

-2

Acceleration

 k a=-   x  m

Time period

T

 T

f

 T 

 M T -2 =

 L

 L T -2

=

N/m=kgsec-2

m sec-2 sec

-1

Frequency

hertz

SHORT QUESTION Q. # 1: What is the relation between frequency and time period? The relation between them is, f =1/T Q. # 2: Give the characteristics of S.H.M. 1. The motion should be vibratory, 2. Acceleration should be directly proportional to the displacement 3. Motion should be towards the mean position. Q. # 3: Why explosions taking place in the sun are not heard on earth? Because, sound travels through a medium in the form of longitudinal waves hence, between sun and earth some part is vacuum. So that sound cannot travel through such, free space. The explosions taking place in the sun are not heard on the earth.

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Q. # 4: Define time period and frequency. Time period: The time required to complete one vibration is called “time period”. Frequency: The number of vibrations completed in one second is called “frequency”. Q. # 5: When two-tuning fork are sounded together, two beats are produced in one second. If the frequency of one fork is 256 Hz then find the frequency of the other fork? Answer: fA - fB= Number of beats completed in one second 256 – fB= 2 256 - 2 = fB fB = 254 Q. # 6: What is ideal simple pendulum? A heavy point mass tied with a perfectly flexible and inextensible string and suspended with rigid , such that it can oscillate without friction, the arrangement is called an “ideal simple pendulum.” Q. # 7: Would you keep the amplitude of a simple pendulum small or large? Explain. The amplitude of simple pendulum must be kept very small. Thus, the value of acceleration due to gravity is same while oscillating, at every position. Q. # 8: Define frequency and wavelength. Frequency: The number of vibration completed in one second is called “frequency”. Wave length: The path covered a traveling wave is called “wavelength”. Q. # 9: Distinguish between longitudinal and transverse waves. Transverse Waves: 1. In these waves’ particles of the medium executes simple harmonic motion at right angles to the direction of propagation of the waves. 2. The highest point above the mean line is called crest and the lowest point trough. 3. These waves do not required any medium for propagation 4. These waves can be set up only in rigid media (solids) Longitudinal waves: 1. In these waves’ particles of the medium execute simple harmonic motion parallel to the direction of propagation of the waves. 2. The region where the crowding of particles is maximum is called compression and the region where the distance between the particles is minimum is called rarefaction. 3. These waves required material medium for the propagation 4. These waves can be set up in solids as well as fluids. Q. # 10: The number of beats is 3 per second when two forks are producing sound. If the frequency of one fork is 251 Hz, find the frequency of other fork. Difference between frequencies = Number of beats per second 251- f = 3 per second 251- 2 = f f = 249

Q. #11: Define Hooks law. Destroying force is directly proportional to the extension, is called Hooks law. Fx Q. # 12: Define intensity of sound. The energy propagated through the unit area perpendicular in unit tine is called “Intensity of sound”.

Intercity =

energy area  time

Q. # 13: Define amplitude and time period.

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Amplitude: The maximum displacement covered by a particle to the right and left of mean position, while oscillating periodically. It is denoted by xo Time period: The time required to complete on oscillation is called time period, denoted by T. Q. # 14: Define loudness The magnitude of auditory sensation produced by sound is called “loudness”. Q. # 15: Distinguish between noise and musical sound. Noise: A sound that is not pleasant to our ears. Such sound produced not regular periodic motion of vibrating bodies. Hence sound is called “non-musical sound or noise”. Q. # 16: Define beats. The periodic alternation between maximum and minimum sound heard in one second is called “beat”. Difference between frequencies = number of beats heard per second. Q. # 17: Define simple harmonic motion. 1. The, motion of the projection of a particle is vibratory, 2. The, acceleration is directly proportional to displacement. 3. And, motion is towards the mean position, indicated by negative sign. It means the projection of a particle performs the “Simple Harmonic Motion”. Q. # 18: Define standing waves. A wave travel in a medium along same line but opposite directions, the two ends being at rest and the central portion vibrating with maximum amplitude. A wave has been set up on a cord that keeps it oscillating in this fashion. Such wave is called “Standing wave”. Q. #19: What is Sonometer? The laws of transverse vibrations in strings, can be easily verified by an instrument is called “sonometer”. Q. # 20: Differentiate between the : Amplitude and Displacement. Amplitude: The maximum displacement covered by a particle to the right and left of mean position, while oscillating

x >x

periodically is called “Amplitude”. It is denoted by xo, where o Displacement: The minimum distance covered by a particle to the right and left of equilibrium position, while oscillating periodically is called “displacement” , denoted by “x”. Where, x < x o . Q. # 21: Give the important uses of ultrasonic. Due to higher energy ultrasonic are used as, Non destructive testing of methods, Cleaning of metal parts, Degassing of liquids, Acceleration of Oxidation processes, Bactericidal properties and Medical and surgical applications Q. # 22: Why are the soldiers directed to break their steps while crossing a bridge? The frequency of regular footsteps of columns of soldiers, marching on a bride, coincides with its natural frequency; a vibration of dangerously large amplitude may build up. Therefore while crossing the bridge the soldiers are ordered to break their steps in order to avoid resonance. Q. # 23: What is ultrasonic? Sound of frequency higher than about 20,000 hertz are not audible to human ear, called “ultrasonic”. Q. #24: Show that the time period of simple pendulum is independent of mass of bob?

T = 2π WE know that

L g

.This shows that time period is independent of mass of bob.

Q. # 25: Have the Doppler’s affected any utility? Explain. Answer: 1) Radar waves are reflected from an airplane, so that speed of traffic is to be measured. 2) The ultrasonic instruments are designed. 3) Motion of sub marine is to be detected. 4) The speed and direction of motion of the earth satellite can be detected.

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Q. # 26: What is the length of second’s pendulum? A pendulum whose length is one meter, frequency 0.5 hertz and time period 2secod. Such pendulum is called “Seconds pendulum.” Q. # 27: Explain the : crest and trough. The highest point on a wave is called “crest”. And the portion below the mean position is called “trough”. Q. # 28: What is the frequency of second’s pendulum? The frequency of second’s pendulum is always 0.5hertz. Q. # 29: Why does sound travel faster in solids than in gas? Because, sound travel through a material medium, as longitudinal waves. Hence, close the molecules, faster the speed of sound. So that sound travels faster in solids than the gas. Q. #30: Why does the speed of sound wave in a gas change with temperature? Because, v 

T , This shows that temperature varies , the speed of sound in air changed.

Q. #31: It has been claimed that a certain singer could shelter a glass goblet by singing a particular note in it. Could this be true? Yes, this is true .A glass goblet has a natural frequency of vibration. If a singer sins a note whose frequency is equal to the natural frequency of goblet resonance is produced. As resonance is accompanied by sudden increase of amplitude of vibration, the goblet, may overtone and break. Q. #32: Will a pendulum that keeps correct time at Karachi, be accurate at Muree or Mount Everest? The time period of pendulum depends on the value of”g” at a place. The value of “g” decreases as height increases from the sea level. Thus, the pendulum that keeps correct time at Karachi will not give correct time at Muree or at Mount Everest. Q. # 33: Give some examples of motion that are simple harmonic. Guitar, Swing, wall clock pendulum and there are many other examples of simple harmonic motion. Q. # 34: What is diffraction? Diffraction of sound is “bending” of the pressure wave around objects, obstacles and walls. Diffraction is greatest with low frequency sound or where the wavelength is large compared to the object it strikes. As illustrated above, diffraction of sound results in a less pronounced acoustic shadow zone. Q. # 35: What is Noise? Noise is unwanted sound which may be hazardous to health, interfere with speech and verbal communications or is otherwise disturbing, irritating or annoying. Q. # 36: What is sound? Sound is defined as any pressure variation in air, water or other fluid medium which may be detected by the human ear Q. # 37: What are the characteristics of sound? The two most important characteristics which must be known in order to evaluate the sound or noise are it's amplitude and frequency. The amplitude or height of the sound wave from peak to valley determines the loudness or intensity. The wave length determines the frequency, pitch or tone of the sound. Q. # 38: What are wavelengths? Sound wavelengths are the linear measurement of one full cycle of displacement where the motion of air molecules is first compressed and then rare field or expanded. The wavelength is determined by the ratio of the speed of sound to the frequency. Q. # 39: What is difference between ultrasonic and supersonics?

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The sound frequency higher than 20,000 Hz is called ultrasonic. These vibration process high energy and cannot be detected by human. Supersonics is the motion object at a speed greeter than the speed of sound. Q. # 40: Why does sound travel faster in solids than in gases?

v=

E ρ . Since the modulus of elasticity (E) is much greater in materials with density

The speed of sound waves is and closely packed molecules in metals than that of gases. Hence the speed of sound waves is greater through solids than in gases. v 

E

Q. # 41: Differentiate between frequency and pitch of sound? Frequency of sound Pitch of sound 1. The number of sound waves per second. 1. The stimulus of frequency to which ear responds the sensation 2. It is physical quantity. 2. It is subjective quantity. 3. Its unit is hertz. 3. Pitch depends on frequency of sound. Q. # 42: Differentiate between intensity and loudness of sound? Intensity: 1. It is physical quantity. . 2. It is energy ing through unit area in unit time. 3. Its unit is watt/m2. 4. It is proportional to the square of amplitude of sound wave 5. The human ear is most sensitive to sound near 3000 Hz. its sensitivity drops to zero near 20 Hz and 20 kHz. 6. The ear can hear a range f intensities from 10-12W/m2(threshold of hearing) to10-2W/m2(threshold of pain The loudest safe sound is 102W/m2 Loudness: 1. It is subjective quantity. 2. The magnitude of auditory sensation produced by sound on the hearing organ (ear) 3. It is measuring by specifying the intensity of sound. It is measured in decibel and related to the logarithm of intensity. Ear is the logarithmic detector 4. It is proportional to the amplitude of the sound waves. 5. An increase of loudness level of 10dB means that intensity has increased by factor of 10. Hence an increase of 20dB means increase in intensity by factor of 100. 6. The range of sound level on dB scale for audible threshold is from 0dB to 120d B. The sound level of a jet engine is 140dB. Q. # 43: Differentiate between interference and beats? Interference: 1. It is produced by the superposition of two sound waves of identical frequency and wavelength. 2. Three are region of maximum and minimum sound. 3. There are regions of low and high intensities of sound. 4. It is the superposition of sound waves, which gives rise to interference in space, in case of standing waves. Beats: 1. It is produced by the superposition of two similar sound waves of identical amplitude and different frequencies. 2. There are times of maximum and minimum sound. 3. Complete silence is produced only when two sound waves have the same amplitude. 4. Beats is superposition of sound waves that gives rise to interference in time. The amplitude of the resultant wave is not constant but varies with time. Q. # 44: Differentiate between Periodic motion and Oscillatory motion? The difference between periodic motion and oscillatory motion is this that all oscillatory motions are periodic motions because each oscillatory motion is completed in a definite interval of time. But all periodic motions may not be oscillatory. For example, the revolution of earth around the sun.

AUDIOBLE FREQUENCY OF DIFFERENCE SPECIES Species human dog

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Approximate Range (Hz) 64-23,000 67-45,000

cat cow horse sheep rabbit rat mouse gerbil guinea pig hedgehog raccoon ferret opossum chinchilla bat beluga whale elephant porpoise goldfish catfish tuna bullfrog tree frog canary parakeet cockatiel owl chicken

45-64,000 23-35,000 55-33,500 100-30,000 360-42,000 200-76,000 1,000-91,000 100-60,000 54-50,000 250-45,000 100-40,000 16-44,000 500-64,000 90-22,800 2,000-110,000 1,000-123,000 16-12,000 75-150,000 20-3,000 50-4,000 50-1,100 100-3,000 50-4,000 250-8,000 200-8,500 250-8,000 200-12,000 125-2,000

MACH NUMBER Aerodynamics is the study of moving air and how objects interact with it. In this field, the speed of an object is often measured relative to the speed of sound. This ratio is known as the Mach number. The speed of sound is roughly 295 m/s (660 mph) at the altitude at which commercial jet aircraft normally fly. The British Airways and Air supersonic airplane, Concorde cruises at 600 m/s (1340 mph). Simple division shows that this speed is roughly twice the speed of sound or Mach 2.0. A Boeing 777, in comparison, cruises at 248 m/s (555 mph) or Mach 0.8, which is still pretty fast.

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