phase difference between two waves formula of different frequencies physics class 11

derive phase difference between two waves formula of different frequencies physics class 11 derivation ?

Wave Motion

Wave motion involves the transport of energy without any transport of matter. The key word in wave motion is ‘disturbance’. In case of mechanical waves, the disturbance is the physical displacement of particles of a medium. In case of electromagnetic waves, the disturbance is a change in electric and magnetic fields.

Types of Waves

There are two types of wave motions: (1) transverse and (2) longitudinal.

(1) Transverse Waves

In transverse waves the particles of the medium vibrate at right angles to the direction in which the wave propagates. Waves on strings, surface water waves and electromagnetic waves are transverse waves. In electromagnetic waves (which include light waves) the disturbance that travels is not a result of vibrations of particles but it is the oscillation of electric and magnetic fields which takes place at right angles to the direction in which the wave travels.

(2) Longitudinal Waves

In longitudinal waves the particles of the medium vibrate along the direction of wave propagation. Sound waves are longitudinal.

Characteristics of a Harmonic Wave

(1) Amplitude The amplitude of a wave is the maximum displacement of the particles of the medium from their mean position.

(2) Period The time period of a wave is the period of harmonic oscillations of particles of the medium. The frequency of a wave is the reciprocal of the time period.

(3) Wave Velocity Wave velocity is the distance travelled by the wave in one second.

(4) Wavelength The wavelength is defined as the distance (measured along the direction of propagation of wave) between two nearest particles which are in the same phase of vibration.

Displacement Equation for a Travelling Wave

When a plane wave travels in a medium along the positive x-direction, the displacement y of a particle located at x at time t is given by

where A = amplitude of the wave, ω (= 2πv) is the angular frequency (in rad s^–1), n is the frequency (in Hz) and k = 2π/λ is the angular wave number and λ is the wavelength of the wave. For a wave travelling along the negative x-direction

Phase and Phase Difference

The argument of the sine (or cosine) function which represents a wave is called the phase of the wave. For a wave travelling along the positive x-direction, the phase Φ at a space point x at time t is given by

Φ = wt – kx

It is clear that the phase changes with time t as well as space point x.

Phase Change with Time

The phase of a given particle (i.e. x fixed) changes with time. As time changes from t to (t + △t), the phase of a particle oscillation changes from Φ to (Φ + △Φ) where △Φ is given by

where T is the time period of particle oscillation. If △t = T, Φ = 2π.

At a given instant of time t, the phase of particles of the medium varies with position x of the particles. The phase difference at an instant t between two particles separated by x and (x + △x) is given by

The minus sign indicates that, for a wave travelling along the positive x-direction, the particles located at higher values of x lag behind in phase. If △x = λ, |△Φ| = 2π. Hence wavelength can be defined as the distance between two particles whose phases differ by 2π.

Expressions for Wave Velocity in Different Media

1. Velocity of sound in an elastic medium is given by

where E = modulus of elasticity of the medium and
ρ = density of the medium.
(a) For gases:
E = y P
where y = Cp/Cv is the ratio of the specific heat of the gas at constant pressure and that at constant volume and P is the pressure of the gas.
Thus

(b) For solids: E = Y; the Young’s modulus of the solid. Thus

(c) For liquids: E = K, the bulk modulus of the liquid. Thus

2. The velocity of sound in a gas is independent of the pressure but is directly proportional to the square root of the absolute temperature

where t is the temperature in °C.

3. The velocity of sound increases with increase in humidity. Sound travels faster in moist air than in dry air at the same temperature.

4. Velocity of a transverse wave on a stretched string is given by

where T = tension in the string and m = mass per unit length of the string. For a string of diameter d and density ρ, we have

Superposition of Waves (The Superposition Principle)

When a wave reaches a particle of a medium, it imparts a displacement to that particle. If two or more waves arrive at a particle, the resultant displacement of the particle is equal to the vector sum of individual displacements. In the particular case when the waves travelling in the same straight line superpose, the resultant displacement is equal to the algebraic sum of the individual displacements. This is called the principle of superposition. y = y1 + y2 + … + yn The following three cases of superposition are of practical importance.

Interference

The superposition of two waves of the same frequency travelling in the same direction in a medium is called interference. Consider two waves of equal amplitude a, equal angular frequency ω and equal angular wave number k but having a phase difference Φ travelling along the positive x-direction. The displacements y1 and y2 of a particle located at x at time t are

According to the superposition principle, the resultant displacement is given by y = y1 + y2

where A = 2a cos (Φ/2) is the resultant amplitude. The frequency and wavelength of the resultant wave remain the same as those of individual waves. (a) Constructive Interference: If A is maximum (positive or maximum), the interference is constructive, then A_max = ±2a. This happens if

(b) Destructive Interference: If A = 0, the interference is destructive. This happens if

Standing (or Stationary) Waves

Standing (or stationary) waves are produced when two waves of the same frequency travelling in opposite directions in a medium superpose. In actual practice, we do not send two independent waves in a medium in opposite directions. A wave is sent in a finite medium which has its boundaries, for example, a string of a finite length or a rod or a column of gas or liquid. The wave gets reflected at the boundaries and a superposition of the incident and reflected waves occurs continuously, giving rise to standing waves. When a wave is reflected from a rigid boundary, it undergoes a reversal of amplitude (which implies a phase change of π). Consider a wave travelling in the negative x-direction towards a boundary at x = 0, where it is reflected. The particle displacements due to the incident and reflected waves are given by

From the superposition principle, the resultant displacement is

Equation (i) represents a standing wave. It does not represent a travelling wave since it does not involve the combination (wt ± kx) in the argument of the sine or cosine function. Equation (ii) tells us each particle has a simple harmonic motion and Eq. (iii) tells us that the amplitude of motion is different for different particles (i.e. for different values of x). Such simple harmonic motions of the particles of a medium are called normal modes.

Nodes There are certain points in the medium which are permanently at rest. These points are called nodes. The position of nodes is given by

Distance between two consecutive nodes = λ/2

Antinodes

There are certain points in the medium which have maximum (positive or negative) amplitude. These points are called antinodes. Their position is given by

Distance between two consecutive antinodes = λ/2

note : 1. Exactly mid-way between two nodes is an antinode and vice versa.
2. The distance between a node and the next antinode = λ/4
3. There is no transfer of energy along the medium.

Normal Modes of a String Fixed at Both Ends

Consider a uniform string of length L stretched with a tension T and fixed rigidly at its ends at x = 0 and x = L. The string can vibrate in a number of modes. Figure 10.25 shows the first three harmonics. (a) Fundamental Mode (or First Harmonic) In this mode, the string vibrates in one segment. [Fig. below (a)]

In general, for a string vibrating in the nth harmonic, the frequency of vibration is

Normal Modes in Air Columns in a Pipe

A gas column in a pipe can oscillate in a number of modes

Case 1: Closed Pipe Consider a pipe of length L open at one end and closed at the other. The closed end is a node and the open end is an antinode. Figure below shows the first three modes of a closed pipe.

note : In a closed pipe only odd harmonics are present; all the even harmonics are absent.

Case 2: Open Pipe Figure 10.27 shows the first three modes of an open pipe

(a) Fundamental Mode (or First Harmonic) [Fig. above (a)]

(b) Second Harmonic (or First Overtone) [Fig. above (b)]

(c) Third Harmonic (or Second Overtone) [Fig. above (c)]

note : In an open pipe, all harmonics (even as well as odd) are present.

End Correction

We have taken the open end of a pipe to be an antinode. This is not strictly true. In fact, the particles of air just at the open end are not perfectly free because of the restriction imposed by the pipe. The true antinode is slightly away from the open end as shown in Fig. below.

The distance e is called the end correction. The effective length of the pipe is (L + e).

Beats

The periodic rise and fall of intensity of the wave resulting from the superposition of two waves of different frequencies is called the phenomenon of beats. Consider two waves of angular frequencies w1 and w2. For simplicity, we assume that they have equal amplitude a and that the observation point is at x = 0. Then

Now, intensity is proportional to A² . Therefore, the resultant intensity is maximum when

Similarly, we can show that the frequency of minima = vb. Hence the frequency of beats is vb = v1 – v2 Thus Beat frequency = difference between the frequencies of interfering waves.

Doppler Effect in Sound

The apparent change in frequency of sound heard by an observer due to a relative motion between the observer and the source of sound is called the Doppler effect. The expressions for the apparent frequencies are as follows: 1. Source approaching a stationary observer

note : 1. If the source of sound moves, the apparent change in frequency is due to change in wavelength; the speed of sound remaining the same. 2. If the observer moves, the apparent change in frequency is due to change in the speed of sound relative to the observer; the wavelength of sound remaining the same.

9. When the source of sound goes past a stationary observer, the apparent change in the frequency of sound is given by

10. When the observer goes past a stationary source of sound, the apparent change in frequency of sound is given by

11. Effect of the Motion of the Medium The velocity of material or mechanical waves is affected by the motion of the medium. If the medium is moving with a velocity um in the direction of propagation of sound, the effective velocity of sound is increased from v to (v + um). In this case, v is replaced by (v + um) in the above expressions. On the other hand, if the medium is moving with a velocity um in a direction opposite to the direction of wave propagation, v is replaced by (v – um).

Doppler Effect in Light

If a star emitting light of frequency n goes away from the earth with a speed v, the apparent frequency n ¢ of the light reaching the earth is given by

The apparent change in wavelength Dl is called the Doppler shift. If the wavelength of light reflected from a moving object decreases, the object is moving towards the observer and vice versa. The wavelength of light reflected from a galaxy is found to increase. This is called the red shift which indicates that the galaxy is receding from us. The red shift indicates that the universe is expanding.

Intensity and Intensity level of Sound

The intensity of a wave is defined as the rate at which the energy of the wave passes through a unit area held normal to it. The intensity I of a wave of amplitude a, frequency ν, travelling with a speed v in a medium of density ρ is given by

I = 2 π² ν²a² ρv

Human ear is sensitive to a very large range of intensity of sound. So a logarithmic (rather than an arithmetic) scale is used to define the intensity of sound. The intensity level β of sound is defined as

where I₀ = 10^–12 Wm^–2 is the reference intensity. Sounds of intensity I < I₀ are barely audible. I₀ is called the threshold intensity and it varies with the frequency of sound. For pure note, I = 10^–12 Wm^–2 at frequency of 1 kHz.

note : The intensity level β of a whisper is about 15 dB. For normal conversation β is about 70 dB. If β >120 dB, the sound becomes painful as in the case of a fire cracker or a jet engine.