Waves and Sound

Wave

A wave is a vibratory disturbance in a medium which carries energy from one point to another point without any actual movement of the medium. There are three types of waves

1. Mechanical Waves Those waves which require a material medium for their propagation, are called mechanical waves, e.g., sound waves, water waves etc.
2. Electromagnetic Waves Those waves which do not require a material medium for their propagation, are called electromagnetic waves, e.g., light waves, radio waves etc.
3. Matter Waves These waves are commonly used in modern technology but they are unfamiliar to us. Thses waves are associated with electrons, protons and other fundamental particles.

Nature of Waves

(i) Transverse waves A wave in which the particles of the medium vibrate at right angles to the direction of propagation of wave, is called a transverse wave.

These waves travel in the form of crests and troughs.

(ii) Longitudinal waves A wave in which the particles of the medium vibrate in the same direction in which wave is propagating, is called a longitudinal wave.

These waves travel in the form of compressions and rarefactions.

Some Important Terms of Wave Motion

1. (i) Wavelength The distance between two nearest points in a wave which are in the same phase of vibration is called the wavelength (λ).
2. (ii) Time Period Time taken to complete one vibration is called time period (T).
3. (iii) Frequency The number of vibrations completed in one second is called frequency of the wave.
   
   Its SI unit is hertz.
4. (iv) Velocity of Wave or Wave Velocity The distance travelled by a wave in one second is called velocity of the wave (u).
   Relation among velocity, frequency and wavelength of a wave is given by v = fλ.
5. (v) Particle Velocity The velocity of the particles executing SHM is called particle velocity.
   

Sound Waves

Sound waves of all the mechanical waves that occur in nature, the most important in our everyday lives are longitudinal waves in a medium, usually air, called sound waves.

Sound waves are of three types

(i) Infrasonic Waves The sound waves of frequency lies between 0 to 20 Hz are called infrasonic waves.

(ii) Audiable Waves The sound waves of frequency lies between 20 Hz to 20000 Hz are called audiable waves.

(iii) Ultrasonic Waves The sound waves of frequency greater than 20000 Hz are called ultrasonic waves.

Sound waves are mechanical longitudinal waves and require medium for their propagation. Sound waves can travel through

[sound waves cannot propagate through vacuum.

If V_s, V_i and V_g are speed of sound waves in solid, liquid and gases, then

V_s > V_i > V_g

Sound waves (longitudinal waves) can reflect, refract, interfere and diffract but cannot be polarised as only transverse waves can polarised.]

Velocity of Longitudinal (Sound) Waves

Velocity of longitudinal (sound) wave in any medium is given by

where, E is coefficient of elasticity of the medium and ρ is density of the medium.

Newton’s Formula

According to Newton, the propagation of longitudinal waves in a gas is an isoth. ermal process. Therefore, velocity of longitudinal (sound) waves in gas should be

where, E_T is the isothermal coefficient of volume elasticity and it is equal to the pressure of the gas.

Laplace’s Correction

According to Laplace, the propagation of longitudinal wave is an adiabatic process. Therefore, velocity of longitudinal (sound) wave in gas should be

where, E_S, is the adiabatic coefficient of volume elasticity and it is equal to γ p.

Factors Affecting Velocity of Longitudinal (Sound) Wave

(i) Effect of Pressure The formula for velocity of sound in a gas.

Therefore, (p/ρ) remains constant at constant temperature.

Hence, there is no effect of pressure on velocity of longitudinal wave.

(ii) Effect of Temperature Velocity of longitudinal wave in a gas

Velocity of sound in a gas is directly proportional to the square root of its absolute temperature.

If v₀ and v_t are velocities of sound in air at O°C and t°C, then

(iii) Effect of Density The velocity of sound in gaseous medium

⇒

The velocity of sound in a gas is inversely proportional to the square root of density of the gas.

(iv) Effect of Humidity The velocity of sound increases with increase in humidity in air.

Shock Waves

If speed of a body in air is greater than the speed of sound, then it .s called supersonic speed. Such a body leaves behind it a conical “egion of disturbance which spreads continuously. Such a disturbance is Called a shock wave.

Speed of Transverse Motion

On a stretched string v = √(T/m)

where, T = tension in the string and m = mass per unit length of the string.

Speed of transverse wave in a solid v = √(η/ρ)

where, η is modulus of rigidity and ρ is density of solid.

Plane Progressive Simple Harmonic Wave

Equation of a plane progressive simple harmonic wave

where,

• y = displacement,
• a = amplitude of vibration
• λ = wavelength of wave, of particle,
• T = time period of wave,
• x = distance of particle from the origin and
• u = velocity of wave.

Important Relation Related to Equation of Progressive Wave

Relation between phase difference and path difference and time difference

Echo

The repetition of sound caused by the reflection of sound waves is called an echo.

Sound persists on ear for 0.1 s.

The minimum distance from a sound reflecting surface to hear an echo is 16.5 m.

If first echo be heard after It second, second echo after ~ second, then third echo will be heard after (t₁ + t2)s.

Superposition of Waves

Two or more progressive waves can travel simultaneously in the medium without effecting the motion of one another. Therefore, resultant displacement of each particle of the medium at any instant is equal to vector sum of the displacements produced by two waves separately. This principle is called principle of superposition.

Interference

When two waves of same frequency travel in a medium simultaneously in the same direction, then due to their superposition, the resultant intensity at any point of the medium is different from the sum intensities of the two waves. At some points the intensity of the resultant wave is very large while at some other points it is very small Or zero. This phenomenon is called interference of waves.

Constructive Interference

Phase difference between two waves = 0, 2π, 4π
Maximum amplitude = (a + b)
Intensity ∝ (Amplitude)² ∝ (a + b)²

In general, amplitude = √a² + b² +2abcos φ

Destructive Interference

Phase difference between two waves = π, 3π, 5π

Minimum amplitude = (a ~ b) = Difference of component amplitudes.

Intensity ∝ (Amplitude)²

∝ (a – b)²

A vibrating tuning fork, when rotated near ear, produced loud sound and silence due to constructive and destructive interference.

Beats

When two sound waves of nearly equal frequencies are produced simultaneously, then intensity of the resultant sound produced by their superposition increases and decreases alternately with time. This rise and fall intensity of sound is called beats.

The number of maxima or minima heard in one second is called beats frequency.

[The difference of frequencies should not be more than 10. Sound persists on human ear drums for 0.1 second. Hence, beats will not be heard if the frequency difference exceeds 10]

Number of beats heard per second = n₁ – n₂ = difference of frequencies of two waves.

Maximum amplitude = (a₁ + a₂)

Maximum intensity = (Maximum amplitude)² = (a₁ + a₂)²

For loudness, time intervals are

Stationary or Standing Waves

When two similar waves propagate in a bounded medium in opposite directions, then due to their superposition a new type of wave is obtained, which appears stationary in the medium. This wave is called stationary or standing waves.

Equation of a stationary wave

Nodes and antinodes are obtained alternatively in a stationary waves.

• At nodes, the displacement of the particles remains minimum, strain is maximum,pressure and density variations are maximum.
• At antinodes, the displacement of the particles remains maximum, strain is minimum, pressure and density variations are minimum.
• The distance between two consecutive nodes or two consecutive antinodes = λ/2.
• The distance between a node and adjoining antinode = λ/4.
• All the particles between two nodes vibrate in same phase. particles on two sides of a node vibrate in opposite phase.
• n consecutive nodes are separated n by ((n -l)λ/2) .

Vibrations in a Stretched String

Velocity of a transverse wave in a stretched string.

Where,T is tension in the string and m is mass per unit length of the string.

Fundamental frequency or frequency of first harmonic

Frequency of first overtone or second harmonic

n₂ = 2(v/2l) = 2n₁

Frequency of second overtone or third harmonic

n₃ = 3(v/2l) = 3n₁

n₁ : n₂ : n₃ : … = 1 : 2 : 3 : …

Organ Pipes

Organ pipes are those cylindrical pipes which are used for produe musical (longitudial) sounds. Organ pipes are of two types

1. Open Organ Pipe Cylindrical pipes open at both ends.
2. Closed Organ Pipe Cylindrical pipes open at one end closed at other end.

Fundamental Note

It is the sound of lowest frequency produced in fundamental note., vibration of a system.

Overtones Tones having frequencies greater than the runoamen note are called overtones.

Harmonics When the frequencies of overtone are integral multiples of the fundamental, then they are known as harmonics. Thus note of lowest frequency n is called fundamental note or first harmonics. The note of frequency 2n is called second harmonic or first overtone.

Vibrations in Open Organ Pipe

fundamental frequency or frequency of first harmonic

n₁ = (2v/l)

frequency of first overtone or second harmonic

n₂ = (2v/2l) = 2n₁

Frequency of second overtone or third harmonic

n₃ = (3v/2l) = 3n₁

n₁ : n₂ : n₃ : …. = 1 : 2 : 3 …

Therefore,even and odd harmonics are produced by an open organ pipe.

Vibrations in Closed Organ Pipe

Fundamental frequency or frequency of first harmonic

n₁ = (v/4l)

Frequency of first harmonic or third harmonic

n₃ = 5(v/4l) = 5n₁

n₁ : n₂ : n₃ : … 1 : 3 : 5 : …

Frequency of second harmonic or fifth harmonic

n₃ = (3v/2l) = 3n₁

Therefore only even harmonics are produced by a closed organ pipe.

End Correction

Antinode is not obtained at exact open end but slightly above it. The distance between open and antinode is called end correction.

It is denoted by e.

• Effective length of an open organ pipe = (l + 2e)
• Effective length of a closed organ pipe = (1 + e)
• If r is the radius of organ pipe, then e = 0.6 r

Factors Affecting Frequency of Pipe

1. Length of air column, n ∝ (1/l)
2. Radius of air column, n ∝ (1/r)
3. Temperature of air column, n ∝ √T
4. Pressure of air inside air column, n ∝ √p
5. Density of air, n ∝ (1/√ρ)
6. Velocity of sound in air column, n ∝ v

Resonance Tube

Resonance tube is a closed organ pipe in which length of air coluDf can be changed by changing height of liquid column in it.

Melde’s Experiment 

In longitudinal mode, vibrations of the prongs of tuning fork are the length of the string.

In transverse mode, vibrations of tuning fork are at 90° to the length of string.

In both modes of vibrations, Melde’s law

p²T = constant, is obeyed.

Characteristics of Musical Sound

Musical sound has three characteristics

(i) Intensity or Loudness Intensity of sound is energy transmitted per second per unit area by sound waves. Its SI unit is watt/metre². Intensity is measured in decibel (dB).

(ii) Pitch or Frequency Pitch of sound directly depends on frequency.

A shrill and sharp sound has higher pitch and a grave and dull sound has lower pitch.

(iii) Quality or Timbre Quality is the characteristic of sound that differentiates between two sounds of same intensity and same frequency .

Quality depends on harmonics and their relative order and intensity.

Doppler’s Effect

The phenomena of apparent change in frequency of source due to a relative motion between the source and observer is called Doppler’s effect.

(i) When Source is Moving and Observer is at Rest When source is moving with ‘velocity towards an observer at rest, then apparent frequency

(ii) When Source is at Rest and Observer is Moving When observer is moving with velocity VO’ towards a source at rest, then apparent frequency.

(iii) When Source and Observer Both are Moving

(a) When both are moving in same direction along the direction of propagation of sound, then

Transverse Doppler’s Effect

(i) The Doppler’s effect in sound does not take place in the transverse direction.

(ii) As shown in figure, the position of a source is S and of observer is O. The component of velocity of source towards the observer is V cos θ. For this situation, the approach frequency is

f ‘ which will now be a function of θ so, it will no more constant.

Similarly, if the source is moving away from the observer as shown above, with velocity component V_s cos θ then,

(iii) If θ = 90°, the V_s cos θ = 0 and there is no shift in the frequency.

Thus, at point P, Doppler’s effect does not occur.

Effect of Wind

If wind is also blowing with a velocity w in the direction of sound, then its velocity is added to the velocity of sound. Hence, in this condition the apparent frequency is givenby

Applications of Doppler’s Effect

The measurement of Doppler shift has been used

1. by police to check overspeeding of vehicles.
2. at airports to guide the aircraft.
3. to study heart beats and blood flow in different parts of the body.
4. by astrophysicist to measure the velocities of plants and stars.

Oscillations

Periodic Motion

A motion which repeats itself identically after a fixed interval of time is called periodic motion. e.g., orbital motion of the earth around the sun, motion of arms of a clock, motion of a simple pendulum etc.

Oscillatory Motion

A periodic motion taking place to and fro or back and forth about a fixed point, is called oscillatory motion, e.g., motion of a simple pendulum, motion of a loaded spring etc.

Note Every oscillatory motion is periodic motion but every periodic motion is not oscillatory motion.

Harmonic Oscillation

The oscillation which can be expressed in terms of single harmonic function, i.e., sine or cosine function, is called harmonic oscillation.

Simple Harmonic Motion

A harmonic oscillation of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean Position is called Simple Harmonic Motion (SHM).

A simple harmonic oscillation can be expressed as

y = a sin ωt

or y = a cos ωt

Where a = amplitude of oscillation.

Non-harmonic Oscillation

A non-harmonic oscillation is a combination of two or more than two harmonic oscillations.

It can be expressed as y = a sin ωt + b sin 2ωt

Some Terms Related to SHM

(i) Time Period Time taken by the body to complete one oscillation is known as time period. It is denoted by T.

(ii) Frequency The number of oscillations completed by the body in one second is called frequency. It is denoted by v.

Its SI unit is ‘hertz’ or ‘second^-1‘.

Frequency = 1 / Time period

(iii) Angular Frequency The product of frequency with factor 2π is called angular frequency. It is denoted by ω.

Angular frequency (ω) = 2πv

Its SI unit is ‘hertz’ or ‘second^-1‘.

(iv) Displacement A physical quantity which changes uniformly with time in a periodic motion. is called displacement. It is denoted by y.

(v) Amplitude The maximum displacement in any direction from mean position is called amplitude. It is denoted by a.

(vi) Phase A physical quantity which express the position and direction of motion of an oscillating particle, is called phase. It is denoted by φ.

Simple harmonic motion is defined as the projection of a uniform circular motion on any diameter of a circle of reference.

Some Important Formulae of SHM

(i) Displacement in SHM at any instant is given by

y = a sin ωt

or y = a cos ωt

where a = amplitude and

ω = angular frequency.

(ii) Velocity of a particle executing SHM at any instant is given by

v = ω √(a² – y²)

At mean position y = 0 and v is maximum

v_max = aω

At extreme position y = a and v is zero.

(iii) Acceleration of a particle executing SHM at any instant is given by

A or α = – ω² y

Negative sign indicates that the direction of acceleration is opposite to the direction in which displacement increases, i.e., towards mean position.

At mean position y = 0 and acceleration is also zero.

At extreme position y = a and acceleration is maximum

A^max = – aω²

(iv) Time period in SHM is given by

T = 2π √Displacement / Acceleration

Graphical Representation

(i) Displacement – Time Graph

(ii) Velocity – Time Graph

(iii) Acceleration – Time Graph

Note The acceleration is maximum at a place where the velocity is minimum and vice – versa.

For a particle executing SlIM. the phase difference between

(i) Instantaneous displacement and instantaneous velocity

= (π / 2) rad

(ii) Instantaneous velocity and instantaneous acceleration

= (π / 2) rad

(iii) Instantaneous acceleration and instantaneous displacement

= π rad

The graph between velocity and displacement for a particle executing SHM is elliptical.

Force in SHM

We know that, the acceleration of body in SlIM is α = -ω² x

Applying the equation of motion F = ma,

We have, F = – mω² x = -kx

Where, ω = √k / m and k = mω² is a constant and sometimes it is called the elastic constant.

In SHM, the force is proportional and opposite to the displacement.

Energy in SHM

The kinetic energy of the particle is K = 1 / 2 mω² (A² – x²)

From this expression we can see that, the kinetic energy is maximum at the centre (x = 0) and zero at the extremes of oscillation (x ± A).

The potential energy of the particle is U = 1 / 2 mω² x²

From this expression we can see that, the potential energy has a minimum value at the centre (x = 0) and increases as the particle approaches either extreme of the oscillation (x ± A).

Total energy can be obtained by adding potential and kinetic energies. Therefore,

E = K + U

= = 1 / 2 mω² (A² – x²) + 1 / 2 mω² x²

= 1 / 2 mω² A²

where A = amplitude

m = mass of particle executing SHM.

ω = angular frequency and

v = frequency

Changes of kinetic and potential energies during oscillations.

The frequency of kinetic energy or potential energy of a particle executing SHM is double than that of the frequency in SHM.

The frequency of total energy of particles executing SHM is zero as total energy in SHM remains constant at all positions.

When a particle of mass m executes SHM with a constant angular frequency (I), then time period of oscillation

T = 2π √Inertia factor / Spring factor

In general, inertia factor = m, (mass of the particle)

Spring factor = k (force constant)

How the different physical quantities (e.g., displacement, velocity, acceleration, kinetic energy etc) vary with time or displacement are listed ahead in tabular form.

Simple Pendulum

A simple pendulum consists of a heavy point mass suspended from a rigid support by means of an elastic inextensible string.

The time period of the simple pendulum is given by :

T = 2π √l / g

where l = effective length of the pendulum and g = acceleration due to gravity.

If the effective length l of simple pendulum is very large and comparable with the radius of earth (R), then its time period is given by

T = 2π √Rl / (l + R)g

For a simple pendulum of infinite length (l >> R)

T = 2π √R / g = 84.6 min

For a simple pendulum of length equal to radius of earth,

T = 2π √R / g = 60 min

If the bob of the simple pendulum is suspended by a metallic wire of length l, having coefficient of linear expansion α, then due to increase in temperature by dθ, then

Effective length l’ = l (1 + α dθ)

Percentage increase in time period

(T’ / T – 1) * 100 = 50 α dθ

When a bob of simple pendulum of density ρ oscillates in a fluid of density ρ_o (ρ_o < p), then time period get increased.

Increased time period T’ = T √ρ / ρ – ρ_o

When simple pendulum is in a horizontally accelerated vehicle, then its time period is given by

T = 2π √1 / √(a² + g²)

where a = horizontal acceleration of the vehicle.

When simple pendulum is in a vehicle sliding down an inclined plane, then its time period is given by

T = 2π √l / g cos θ

Where θ = inclination of plane.

Second’s Pendulum

A simple pendulum having time period of 2 second is called second’s Pendulum.

The effective length of a second’s pendulum is 99.992 em of approximately 1 metre on earth.

Conical Pendulum

If a simple pendulum is fixed at one end and the bob is rotating in a horizontal circle, then it is called a conical pendulum.

In equilibrium T sin θ = mrΩ²

Its time period T = 2π √mr / T sin θ

Compound Pendulum

Any rigid body mounted, so that it is capable of swinging in a vertical plane about some axis passing through it is called a physical or compound pendulum.

Its time period is given by

T = 2π √l / mg l

where, I = moment of inertia of the body about an axis passing through the centre of suspension,

m = mass of the body and

l = distance of centre of gravity from the centre of suspension.

Torsional Pendulum

Time period of torsional pendulum is given by

T = 2π √I / C

where, I = moment of inertia of the body about the axis of rotation and

C = restoring couple per unit twist.

Physical Pendulum

When a rigid body of any shape is capable of oscillating about an axis (mayor may not be passing through it). it constitutes a physical pendulum.

T = 2π √I / mgd

• The simple pendulum whose time period is same as that of a physical pendulum is termed as an equivalent simple pendulum.

T = 2π √I / mgd = 2 π √l / g

• The length of an equivalent simple pendulum is given by l = I / md

Spring Pendulum

A point mass suspended from a massless (or light) spring constitutes a spring pendulum. If the mass is once pulled downwards so as to stretch the spring and then released. the system oscillated up and down about its mean position simple harmonically. Time period and frequency of oscillations are given by

T = 2π √m / k or v = 1 / 2π √k / m

If the spring is not light but has a definite mass m_s, then it can be easily shown that period of oscillation will be

T = 2π √(m + m_s / 3) / k

Oscillations of Liquid in a U – tube

If a liquid is filled up to height h in both limbs of a U-tube and now liquid is depressed upto a small distance y in one limb and then released, then liquid column in U-tube start executing SlIM.

The time period of oscillation is given by

T = 2π √h / g

Oscillations of a floating cylinder in liquid is given by

T = 2π √l / g

where I = length of the cylinder submerged in liquid in equilibrium.

Vibrations of a Loaded Spring

When a spring is compressed or stretched through a small distance y from mean position, a restoring force acts on it.

Restoring force (F) = – ky

where k = force constant of spring.

If a mass m is suspended from a spring then in equilibrium,

mg = kl

This is also called Hooke’s law.

Time period of a loaded spring is given by

T = 2π √m / k

When two springs of force constants k₁ and k₂ are connected in parallel to mass m as shown in figure, then

(i) Effective force constant of the spring combination

k = k₁ + k₂

(ii) Time period T = 2π √m / (k₁ + k₂)

When two springs of force constant k₁ and k₂are connected in series to mass m as shown in figure, then

(i) Effective force constant of the spring combination

1 / k = 1 / k₁ + 1 / k₂

(ii) Time period T = 2π √m(k₁ + k₂) / k₁k₂

Free Oscillations

When a body which can oscillate about its mean position is displaced from mean position and then released, it oscillates about its mean position. These oscillations are called free oscillations and the frequency of oscillations is called natural frequency.

Damped Oscillations

Oscillations with a decreasing amplitude with time are called damped oscillations.

The displacement of the damped oscillator at an instant t is given by

x = x_oe^{– bt / 2m} cos (ω’ t + φ)

where x_oe^{– bt / 2m} is the amplitude of oscillator which decreases continuously with time t and ω’.

The mechanical energy E of the damped oscillator at an instant t is given by

E = 1 / 2 kx²_oe^{– bt / 2m}

Un-damped Oscillations

Oscillations with a constant amplitude with time are called un-damped oscillations.

Forced Oscillations

Oscillations of any object with a frequency different from its natural frequency under a periodic external force are called forced oscillations.

Resonant Oscillations

When an external force is applied on a body whose frequency is an integer multiple of the natural frequency of the body, then its amplitude of oscillation increases and these oscillations are called resonant oscillations.

Lissajous’ Figures

If two SHMs are acting in mutually perpendicular directions, then due to then: superpositions the resultant motion, in general, is a curvelloop. The shape of the curve depends on the frequency ratio of two SHMs and initial phase difference between them. Such figures are called Lissajous’ figures.

1. Let two SHMs be of same frequency (e.g., x = a₁ sinωt and y = a₂ sin (omega;t + φ), then the general equation of resultant motion is found to be

x² / a²₁ + y² / b²₂ – 2xy / a₁a_{2/sub> cos φ = sin² φ}

The equation represents an ellipse. However, if φ = O° or π or nπ, then the resultant curve is a straight inclined line.

2. Let two SHMs be having frequencies in the ratio 1 : 2, then, in general, the Lissajous figure is a figure of eight (8).