8/14/2019 Waves_ Notes for a Revised)

1/16

PHYSICS NOTES

W.R.David

WAVES

WAVES:A wave is a travelling disturbance which travels from one place to another transferringenergy but no mass.Waves are either Mechanical or Electromagnetic.

Electromagnetic Waves:These are waves which dont require a medium to travel and travel a thespeed of light ie 3 108m/sMechanical Waves are waves which propagate through a material medium (solid, liquid, or gas)at a wave speed which depends on the elastic and inertial properties of that medium. All mechanicalwaves require (1) some source of disturbance, (2) a medium that can be disturbed, and (3) somephysical mechanism through which elements of the medium can influence each other.

There are two basic types of wave motion for mechanical waves: longitudinal waves and trans-verse waves.Transverse waves:A transverse wave is a moving wave that consists of oscillations occurring perpen-dicular to the direction of energy transfer. If a transverse wave is moving in the positive x-direction,its oscillations are in up and down directions that lie in the y-z plane. Examples of transversewaves:a)Waves traveling in a rope(refer fig. b)A seismic secondary wave which is called the S-wave isa transverse wave c)Light Waves are also transverse(remember-light waves are not mechanicalwaves-but electromagnetic waves)

Figure 1: A transverse pulse traveling on a stretched rope. The direction of motion of any element Pof the rope is perpendicular to the direction of propagation.

Longitudinal waves are the waves in which the particles of the medium vibrate along the di-

rection of wave motion.Examples of Longitudinal waves: a)Sound waves b)waves in spring(refer fig.2) Longitudinal

1

8/14/2019 Waves_ Notes for a Revised)

2/16

Figure 2: A longitudinal pulse along a stretched spring.

wave can be expressed as waves travelling in a long spring(slinky) the graphical representation of thelongitudinal wave is represented below:Graphical representation of waves:

Figure 3: Graphical representation of longitudinal waves

Wave terminology:

Amplitude:The amplitude of a wave is the maximum displacement of a particle from its

undisturbed position.

Phase difference():The phase difference between two particles along the wave is thefraction of the cycle one moves behind the other.Phase difference is measured in degrees orradians.2 radians correspond to one full cycle of the wave.

Wave length(:)The wave length is the distance from one particle to the next particle inphase with it.

Frequency(f):The frequency f of the wave motion is the number of wave crests passing agiven position each second.The unit of frequency is Hertz(Hz)

Time period(T):The time between one wave crest and the next arriving at the same point iscalled the time period(T).Frequency and time period are related as shown:-

f =1

T

Phase,wavelength and wavelength:Remember

2 radians corresponds to meters which C orresponds to T seconds

Thus if two particles are at a distance x apart then the phase difference between them is :-

=2x

2

8/14/2019 Waves_ Notes for a Revised)

3/16

8/14/2019 Waves_ Notes for a Revised)

4/16

Wave Speed(c):The wave speed of a progressive wave is the distance traveled by its wavecrests in one second.

Relation Between wave speed(c),wave length() and frequency(f)Consider a wave crest in a wave train the wave crest travels a distance of meters in a time of T(the

time period) seconds.Speed(c) =

distance

time

c =

T

since f = 1T

c = f

Intensity of Waves:We know that waves carry energy. The power-per-area of waves is given aname. It is called the intensity of the wave.Consider the wave source to be point source(like abuzzer) the waves travel in the form of imaginary spherical shells (imagine in 3D),centered around

the point source .Then the intensity I at the point (and at any other point on the spherical shell) issimply the power of the source divided by the area of the spherical shell(4r2):

I =P

4r2(1)

where the r is the radius of the imaginary spherical shell, but, more importantly, it is the distanceof the point at which we wish to know the intensity, from the source.The unit of wave intensity isW m2Another important result is that the intensity of the wave is proportional to thesquare of the amplitude

intensity (amplitude)2

Principle of Superposition of waves:If two or more traveling waves are moving through a medium,the resultant value of the wave displacement at any point is the algebraic sum of the values of thedisplacement of the individual waves.

total Displacement at a point = Sum of individual displacements

Figure 6: Reinforcement and cancellation of waves

Standing waves or Stationary waves:Waves carry energy.In certain conditions wave energy canbe localised,which means it can be stopped from moving.When this happens,the waves are said tobe standing waves or stationary waves.Cause of Standing Waves:Standing waves are formed when two waves moving in opposite direc-tions, each having the same amplitude and frequency. The phenomenon is the result of interference-that is, when waves are superimposed, their energies are either added together or canceled out.Standingwave pattern can be observed in stretched strings.Nodes:Every stationary wave Pattern has points with no displacement these are called as nodes.

4

8/14/2019 Waves_ Notes for a Revised)

5/16

Antinodes: The points where the displacement is maximum in a standing wave pattern arecalled as Antinodes.

Figure 7: Nodes and antinodes

Standing Waves on Strings

When a string with one end attached to a variable frequency vibrator and the other end attachedto a fixed point is made to vibrate then standing waves are produced in the string.[refer fig.8]

First harmonic or Fundamental frequency fo: The lowest possible frequency with which

Figure 8: fundamental frequency(fo) and harmonics

the string vibrates is called the fundamental frequency or first harmonic,at this frequency( fo) thepattern in the string has two nodes and one anti node .[refer.fig.8a].The antinodes positionalong the rope vibrates up and down from a maximum upward displacement from rest to a maximumdownward displacement as shown in fig.8a. The vibration of the rope in this manner creates the

appearance of a loop within the string.Harmonics:At frequencies the higher than the fundamental frequency the standing wave patternchanges these are called as harmonics.The Second harmonic:The second harmonic pattern consists of two anti-nodes. Thus, there aretwo loops within the length of the string,hence the frequency is twice the fundamental frequency(fo)i.e2fo [refer Fig. 8b].

Similarly for the other harmonics.The frequency is given by First harmonic(f0),second harmonic(2fo),third harmonic (3f0)....The first harmonic has one loop the second two loops on the string,thethird harmonic has three loops and so on...

5

8/14/2019 Waves_ Notes for a Revised)

6/16

8/14/2019 Waves_ Notes for a Revised)

7/16

The distance between adjacent nodes is equal to 2

.

The distance between a node and an adjacent antinode is 4

.

The frequency of any pattern is equal to the number of loops the fundamental

frequency.

For the nth harmonic L =n

2

WAVE PROPERTIES:

1. Interference

2. Diffraction

3. Polarisation

INTERFERANCE:The combination of separate waves in the same region of space to produce aresultant wave is called interference.Constructive Interference: When the crest and crest and trough of two waves meet they reinforceeach other an a new wave with bigger amplitude results this is called constructive interference(reffigure 6).Destructive Interference:When the crest of one wave and the trough and trough of another wavemeet they cancel each other. this is called Destructive interference(ref figure 6).Interference takes place in all kinds of waves be it longitudinal(sound) or transverse (light).INTERFERENCE OF LIGHTConditions for Interference of Light:In order to form an interference pattern, the incident lightmust satisfy the following conditions:

The light sources must be coherent. This means that the plane waves from the sources mustmaintain a constant phase relation. For example, if two waves are completely out of phase with = , this phase difference must not change with time.Note:Light emitted from an incandescent light bulb is incoherent because the light consists ofwaves of different wavelengths and they do not maintain a constant phase relationship. Thus,no interference pattern is observed.

The light must be monochromatic. This means that the light consists of just one wavelength()Youngs Double-Slit Experiment:In 1801 Thomas Young carried out an experiment in which thewave nature of light was demonstrated. A schematic diagram of the apparatus that Young used isshown in Figure 10 . Plane light waves arrive at a barrier that contains two parallel slits S1 and S2.These two slits serve as a pair of coherent light sources because waves emerging from them originatefrom the same wave front and therefore maintain a constant phase relationship. The light from S1and S2 produces on a viewing screen a visible pattern of bright and dark parallel bands called fringes(Fig.11). When the light from S1 and that from S2 both arrive at a point on the screen such thatconstructive interference occurs at that location, a bright fringe appears. When the light from thetwo slits combines destructively at any location on the screen, a dark fringe results.The experimental

setup is shown below:-

7

8/14/2019 Waves_ Notes for a Revised)

8/16

Figure 10: The youngs Double slit Experiment- Experimental setup.

Figure 11: Schematic diagram of Youngs double-slit experiment. Slits S1 and S2 behave as coherent

Figure 12: Formation of dark and bright bands.Constructive interference (a) at P(Crest and Crest),and (b) at P1 (Crest and Crest). (c) Destructive interference at P2(Crest and trough)

.

8

8/14/2019 Waves_ Notes for a Revised)

9/16

Figure 13: Photograph of interference fringes

PATH DIFFERENCE:The difference in distance travelled by the waves from the slits S1 andS2 is called path difference.For example the path difference in the diagram below [fig.14]is given by:-

Path difference = S1R S2R

Figure 14: Path Difference

Condition for a point on the Screen to be Bright:Based on path difference between the raysfrom the two slits we can find out whether a particular point on the screen is bright or dark.

For the point on the screen to be Bright

Path difference = m (m = 0, 1, 2, 3...)

For a point on the screen to be Dark

Path difference = (m +1

2) (m = 0, 1, 2, 3,...)

m is called the order of the fringes or order number as illustrated below in [Fig.15]

Fringe spacing (also called fringe width:)The distance between two adjacent Bright or darkfringes is called fringe spacing or fringe width.[see Fig.15] The fringe spacing is given by

y =D

d

The distance of the mth bright fringe from the central bright fringe is given by [see Fig.15]

ym = mDd

9

8/14/2019 Waves_ Notes for a Revised)

10/16

15]

Figure 15: Order of the fringes and Fringe spacing

The distance of the mth dark fringe from the central bright fringe is given by

ym = (m +1

2)

D

d

Note:It Should be noted that the spacing between any two fringes is the sameKey points about Interference of light.

Condition:Coherence,Monochromatic.

Constructive interference Bright fringes.

Destructive interference Dark fringes.

Youngs Double slit Experiment.

Fringe width y = Dd

The distance of the mth bright fringe from the central bright fringe is given by ym =mD

d

The distance of the mth dark fringe from the central bright fringe is given by ym = (m + 12)D

d

It Should be noted that the spacing between any two fringes is the same.

sound waves which are longitudinal also undergo interference.

10

8/14/2019 Waves_ Notes for a Revised)

11/16

Diffraction

What is Diffraction:When light or any wave passes through a slit or any sharp edge, it spreadsout in arcs from the slit this is called Diffraction. The amount of diffraction increases the closerthe slit width is to the wavelength of the wave .i.e when the slit width is almost equal to the

wavelength the diffraction is pronounced.This is shown in fig.16

Figure 16: Diffraction is pronounced when the width of the slit is almost equal to the wavelength.

Diffraction of light through single slit:When light is passed through a narrow slit light bends

and if a screen is kept in front of the slit then dark and bright fringes are formed on the screen.asshown in fig.17

Figure 17: Diffraction through single slit

Calculating the angles at which fringes occur:Consider narrow slit of width d and and diffractedby an angle then the angle of diffraction,the wavelength are related by the following equation i.ethe position of the dark fringe can be found using

For Dark Bands

d sin = m where m = 0, 1, 2, 3...

Here it should be noted that d sin is the path difference between the waves from the top edge of theslit and the bottom edge of the slit as shown in fig.19Diffraction Grating:A diffraction grating is a plate with many closely spaced lines.Key points about the grating.

The closer the slits,the more widely spaced are the diffracted beams

The longer the wavelength used the more widely spaced are the diffracted beams.

White light gives a multi-spectrum of colours

11

8/14/2019 Waves_ Notes for a Revised)

12/16

Figure 18: The diffraction angle

Figure 19: Path difference between the wave from the upper edge(A) and bottom edge (B)

The Diffraction Grating Equation:The diffraction grating equation is given by

d sin m = m

where d is the slit spacingm is the angle of diffraction for the m

th orderm is the order number of the beam.

is the wave length of the beam.

Figure 20: Grating-Showing the order of the beams

Polarsation:Polarisation is the process which restricts the oscillation of a transverse wave to oneplane at a right angle to the direction of travel.Only transverse waves can be polarised

12

8/14/2019 Waves_ Notes for a Revised)

13/16

Polarisation of a wave travelling on a rope:

Figure 21: Polarised and unpolarised wave in a rope

Figure 22: When two polarisers are crossed the wave dies out

Polarisation of light:Light is a transverse wave and can be polarised using Certaian syntheticfilms like Poloroids and certain natural crystals like tourmaline. Note:By keeping two polarisers

Figure 23: Polarisation of light-Polarisation proves that light is transvers.

perpendicular to each other the intensity of light coming out of the second polariser can be reducedto almost zero as shown in fig.23 above.

End

13

8/14/2019 Waves_ Notes for a Revised)

14/16

Check if you have understood

1. Suppose that a string is 1.2 meters long and vibrates in the first, second and third harmonicstanding wave patterns. Determine the wavelength of the waves for each of the three pat-terns.[ans First harmonic: 2.4 m, Second harmonic: 1.2 m ,Third harmonic: 0.8 m]

2. The string below is 1.5 meters long and is vibrating as the first harmonic. The string vibratesup and down with 33 complete vibrational cycles in 10 seconds. Determine the frequency,period, wavelength and speed for this wave.

3. The string below is 6.0 meters long and is vibrating as the third harmonic. The string vibratesup and down with 45 complete vibrational cycles in 10 seconds. Determine the frequency,

period, wavelength and speed for this wave.

4. The number of nodes in the standing wave shown in the diagram at the below is .and the number of antinodes in the standing wave shown in the diagram below is .

5. Suppose that light passes through two Polaroid filters whose polarization axes are parallel toeach other. What would be the result?

6. The diagram below depicts the results of Youngs Experiment. The appropriate measurementsare listed on the diagram. Use these measurements to determine the wavelength of light innanometers. (GIVEN: 1 meter = 109 nanometers)

7. A student uses a laser and a double-slit apparatus to project a two-point source light interference

pattern onto a whiteboard located 5.87 meters away. The distance measured between the centralbright band and the fourth bright band is 8.21 cm. The slits are separated by a distance of0.150 mm. What would be the measured wavelength of light?

8. Minute after minute, hour after hour, day after day, ocean waves continue to splash onto theshore. Explain why the beach is not completely submerged and why the middle of the oceanhas not yet been depleted of its water supply.

9. A transverse wave is transporting energy from east to west. The particles of the medium willmove .

10. A wave is transporting energy from left to right. The particles of the medium are moving backand forth in a leftward and rightward direction. This type of wave is known as a .

14

8/14/2019 Waves_ Notes for a Revised)

15/16

11. Consider the diagram below in order to answer questions 1 to 2.

(a) The wavelength of the wave in the diagram above is given by letter .

(b) The amplitude of the wave in the diagram above is given by letter .

(c) choices (a) A to C.,(b)B to D (c) A to G (d) C to G

12. A wave is introduced into a thin wire held tight at each end. It has an amplitude of 3.8cm, a frequency of 51.2 Hz and a distance from a crest to the neighboring trough of 12.8 cm.Determine the period of such a wave.

13. Frieda the fly flaps its wings back and forth 121 times each second. The period of the wingflapping is sec.

14. . Non-digital clocks (which are becoming more rare) have a second hand which rotates aroundin a regular and repeating fashion. The frequency of rotation of a second hand on a clock is Hz.

15. Fig.below shows the variation with time t of the displacement y of a wave W as it passes apoint P. The wave has intensity I.A second wave X of the same frequency as wave W also passespoint P. This wave has intensity 1

2I. The phase difference between the two waves is 60. On

Fig.below, sketch the variation with time t of the displacement y of wave X.

16. In a double-slit interference experiment using light of wavelength 540 nm, the separation of theslits is 0.700 mm. The fringes are viewed on a screen at a distance of 2.75 m from the doubleslit, as illustrated in Fig.below (not to scale).Calculate the separation of the fringes observedon the screen.

(a) State the effect, if any, on the appearance of the fringes observed on the screen when thefollowing changes are made, separately, to the double-slit arrangement in (b). (i) Thewidth of each slit is increased but the separation remains constant.

(b) The separation of the slits is increased what happens .

15

8/14/2019 Waves_ Notes for a Revised)

16/16

17. Explain what is meant by (i) interference,(ii) coherence.

18. Red light of wavelength 644 nm is incident normally on a diffraction grating having 550 linesper millimetre, as illustrated in Fig. below. Red light of wavelength is also incident normallyon the grating. The first order diffracted light of both wavelengths is illustrated in Fig. 4.1.(i)Calculate the number of orders of diffracted light of wavelength 644 nm that are visible on eachside of the zero order,

16