1. 4.4 Wave Behavior
More interesting things that waves do!

2. Objectives & Understandings
Sketch & interpret incident, reflected and transmitted waves at boundaries between media
Solve problems involving reflection at a plane interface, Snell’s law, critical angle, and total internal reflection
Determine refractive index experimentally

3. Nature of Science: Wave or Particle?
Late 17th century – two rival theories of the nature of light proposed by Newton & Huygens.
Newton –
light is a particle; supported by the facts that it apparently travels in a straight line & can travel through a vacuum (waves need a medium)
individual particles decide whether to reflect or transmit (not a strong argument – wave theory became predominant)
Huygens –
light is a wave; supported by Grimaldi’s work in showing that light diffracts around small objects and through narrow openings;
argued that when a wave meets a boundary the total incident energy is shared by the reflected and transmitted waves
In the 21st century light is treated as both a wave & a particle in order to explain the full range of its properties.
[Sheldon clip]

4. Introduction
Now that we have looked at how to describe waves, we are in a position to look at wave properties/behavior
We have already heard and used some of these ideas, and now we’ll try to understand them better.

5. Reflection & Refraction of Waves
Ray diagrams ignore wavefronts (so we can’t see what happens to the wavelength of the waves), but they could be added at any time.
Three laws:
The reflected and refracted rays are in the same plane as the incident ray and the normal
The angle of incidence equals the angle of reflection (measured from normal)
Snell’s Law: For waves of a particular frequency and for a chosen pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant called the (relative) refractive index. sin 𝜃 1 sin 𝜃 2 = 𝑛 2 𝑛 1 =1𝑛2

6. Reflection & Refraction of Waves
1. The reflected and refracted rays are in the same plane as the incident ray and the normal

7. Reflection & Refraction of Waves
2. The angle of incidence equals the angle of reflection (measured from normal)

8. Reflection & Refraction of Waves
3. Snell’s Law: For waves of a particular frequency and for a chosen pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant called the (relative) refractive index. sin 𝜃 1 sin 𝜃 2 = 𝑛 2 𝑛 1 =1𝑛2

9. Refraction
When a wave moves from one medium to another, the speed of the wave changes and the wavelength of the wave changes, but the frequency stays the same.
Optical density – measured in terms of refractive index; higher refractive index = higher optical density
SFA– high to low optical density -> speeds up (fast) -> away
FST – low to high optical density-> slows down (slow) -> toward

10. What happens to light at an interface between two media
This is a complex process, but generally, when charges are accelerated, for example when they are vibrated, they can emit energy as an EM wave. In moving through a vacuum, this wave moves at a speed of 3.00 x 108 m s-1.
When the wave reaches an atom, energy is absorbed and causes electrons within the atom to vibrate. All particles have frequencies at which they tend to vibrate most efficiently – called the natural frequency.
When the frequency of the EM wave does not match the natural frequency of vibration of the electron, then the energy will be re-emitted as an EM wave.
This EM wave has the same frequency as the original wave, and will travel at the usual speed in the vacuum between atoms.

11. What happens to light at an interface between two media
This process repeats as the new wave comes into contact with other atoms of unmatched natural frequency. With the wave travelling at 3.00 x 108 m s-1 in space but being delayed by the absorption-re-emission process, the overall speed of the wave will be reduced.
In general, the more atoms per unit volume in the material, the slower the radiation will travel.

12. What happens to light at an interface between two media
When the frequency of the light does match that of the atom’s electrons, the re-emission process occurs in all directions and the atom gains energy, increasing the internal energy of the material.

13. Refractive Index & Snell’s Law
The absolute refractive index (n) of a medium is defined in terms of the speed of electromagnetic waves as 𝑛= 𝑐 𝑣 = 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑤𝑎𝑣𝑒𝑠 𝑖𝑛 𝑎 𝑣𝑎𝑐𝑢𝑢𝑚 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑤𝑎𝑣𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑚𝑒𝑑𝑖𝑢𝑚
The refractive index depends on the frequency of the EM radiation and, since the speed of the light in a vacuum is as fast as it can go, the absolute refractive index is always greater than 1
For all practical purposes the absolute refractive index of air is 1 so it is not necessary to perform refractive index experiments in a vacuum.

14. Some useful refractive indices

15. Example
Calculate the angle of refraction when the angle of incidence at a glass surface is 55° (refractive index of the glass = 1.48). F: θr G: θi = 55°, n2 (glass) = 1.48, n1 (air) = 1.00 E: sin 𝜃 1 sin 𝜃 2 = 𝑛 2 𝑛 1 W: sin 55° sin 𝜃 2 = A: θr=33.6° ≈ 34°

16. Reversibility of light
Light will take the same path with the same angle of incidence regardless of the side of the block.
Mathematically
sin 𝜃 1 sin 𝜃 2 =1𝑛2∴ sin 𝜃 2 sin 𝜃 1 =2𝑛1 and 1𝑛2= 1 2𝑛1

17. Example
The absolute refractive index of water is 1.3 and that of glass is 1.5.
Calculate the relative refractive index from glass to water.
1𝑛2= sin 𝜃 1 sin 𝜃 2 = 𝑛 2 𝑛 1 = =0.87
Explain what this implies regarding the refraction of light rays.
If the relative refractive index is less than one, it means that light travels faster in water than glass, and therefore bends away from normal
Draw a wavefront diagram to show how light travels through a plane interface from glass to water.

18. Lab: Determining Refractive Index
Measure the angles of refraction and use Snell’s law to determine the refractive index of an unknown material and attempt to identify it.
Independent variable
Dependent variable
Controlled variables
At least 5 values for IV and at least 5 trials. You may work together in groups of 3-4 to design a method and collect data. Run it by me before you start collecting data.

19. Critical Angle
When a wave moves from a higher optically dense medium to a lower optically dense medium, the speed and wavelength of the wave increase (frequency is constant) and the direction of the wave moves away from the normal – the angle of refraction being greater than the angle of incidence, so as the angle of incidence increases, the angle of refraction will approach 90°
The angle of incidence when the angle of refraction is 90°

20. Total Internal Reflection
TIR - When the angle of incidence is larger than the critical angle, the light wave does not move into the new medium at all, but is reflected back into the original medium
For all angles smaller than the critical angle there will always be a reflected ray; although this will only carry small portion of the incident energy.
Laser demo w/ triangle

21. Critical angle
When changing media, white light can disperse into the colors of the rainbow because each of the colors, of which white light is comprised, has a different frequency. The refractive index for each of the colors is different
Calculating the critical angle: 𝐬𝐢𝐧 𝜽 𝒄 = 𝟏 𝒏 𝟏
sin 𝜃 1 sin 𝜃 2 =1𝑛2 (medium 1 must be more optically dense than medium 2)
When 𝜃 1 = 𝜃 𝑐 , then 𝜃 2 =90° so sin 𝜃 2 =1
sin 𝜃 𝑐 =1𝑛2
1𝑛2= 𝑛 2 𝑛 1 ; when the less dense medium is a vacuum or air 𝑛 2 =1 so sin 𝜃 𝑐 = 1 𝑛 1

22. Example
Calculate the (average) critical angle for a material of (average) absolute refractive index 1.2. F: 𝜃 𝑐 G: n=1.2 E: sin 𝜃 𝑐 = 1 𝑛 1 W: 𝜃 𝑐 = sin −1 1 𝑛 1 = sin − A: 𝜃 𝑐 =56.4°≈56°

23. Another interesting phenomenon
Water waves move faster in deeper water.
Frequency does not change when refracted.
C= wave speed

24. Diffraction
The bending of a wave that passes around an obstacle or through a narrow slit causing the lights to spread.

25. Objectives
Qualitatively describe the diffraction pattern formed when plane waves are incident normally on a single slit, and double-slit interference patterns
Understand the terms reflection, refraction, critical angle, total internal reflection, diffraction, interference & path difference
Understand Snell’s Law, diffraction through a single and double slit & around objects

26. Diffraction Remember the
double slit demonstration
Veritasium video
First detailed description/observation by Italian priest Francesco Grimaldi – work published in 1665, two years after his death
When waves pass through a narrow gap or slit (called an aperture), or when their path is partly blocked by an object, the waves spread out into what we would expect to be the shadow region. (human hair width)

27. Diffraction

28. Diffraction Further observation of diffraction has shown
The frequency, wavelength, and speed of the waves each remains the same after diffraction
The direction of propagation and the pattern of the waves change
The effect of diffraction is most obvious when the aperture width is approximately equal to the wavelength of the waves
The amplitude of the diffracted wave is less than that of the incident wave because the energy is distributed over a larger area.

29. Example: Radio waves
AM waves (λ between 600 and 200 m) easily diffract around buildings (size ≈ 100 m)
FM waves (λ between 5.5 and m) and cellular (λ between .375 and m) easily blocked in tunnels and buildings.
Longer wavelengths diffract more easily
Check your understanding: Why is diffraction much more evident for sound waves than light waves in our everyday environment

30. Diffraction
Explaining diffraction by a single slit is complex – you only need to know a qualitative description like the Huygens-Fresnel principle
Plane waves travelling towards the slit behave as if they were sources of secondary wavelets (Huygen’s principle). These sources each spread out as circular waves. The tangents to these waves will now become the new wavefront

31. Diffraction
The central image is bright and wide, beyond it are further narrower bright images separated by darkness.

32. Diffraction of Sound Waves
How we hear hallway noise through the doorway or window

33. Double-Slit Diffraction
An instance of the superposition principle
When two or more waves meet they combine to produce a new wave – this is called interference (happens to all waves, not just light)
Constructive interference– resultant wave has a larger amplitude than any of the individual waves
Destructive interference – resultant wave has a smaller amplitude than any of the individual waves
Achieved using two similar sources of all types of waves
Usually only observable if the two waves have a constant-phase relationship (not necessarily in phase) – same frequency – coherent

34. Double-Slit Interference
Where the two diffracted beams cross, interference occurs as seen above.
A pattern of equally spaced bright and dark fringes is obtained on a screen positioned in the region where the diffracted beams overlap
When a crest meets a crest or a trough meets a trough constructive interference occurs

35. Double-Slit Interference
1801 Young’s Double-Slit Experiment (English physicist)
The coherent beam is achieved by placing a single slit close to the source of light – this means that the wavefronts spreading from the single slit each reach the double slit with the same phase relationship and so the secondary waves coming from the double slit retain the constant phase relationship.
Demonstrates the wave nature of light

36. Path Difference & the Double-Slit Equation
𝑠= λ𝐷 𝑑 gives the separation of successive bright fringes (or bands of loud sound)
In general, for two coherent beams starting in phase, if the path difference is a whole number of wavelengths, we get constructive interference and if it’s an odd number of half wavelengths we get destructive interference.

37. Path Difference & the Double-Slit Equation
In summary:
𝑠= λ𝐷 𝑑
Constructive interference 𝑠=𝑛λ; where n=0,1,2…
Destructive interference 𝑠= 𝑛 λ; where n=0,2,4…
n is known as the order of the fringe, n=0 being the zeroth order, n=1 is the first order, etc.
Check your understanding: What happens to the distance between fringes if the separation between two slits (d) is increased?

38. Example
In a double-slit experiment using coherent light of wavelength λ, the central bright fringe is observed on a screen at point O, as shown above. At point P, the path difference between light arriving at P from the two slits is 7λ.
Explain the nature of the fringe at P.
Path difference is an integral number of wavelengths = bright fringe
State and explain the number of dark fringes between O and P.
Destructive interference = odd number of half-wavelengths means dark fringes at λ 2 , 3λ 2 , 5λ 2 , 7λ 2 , 9λ 2 , 11λ 2 , 13λ 2 (7 total)

39. Example
Two coherent point sources S1 and S2 oscillate in a ripple tank and send out a series of coherent wavefronts as shown in the diagram. State and explain the intensity of the waves at points P and Q.
Assuming each wavefront is a crest and the waves have the same amplitude, then two crests meet at point P and superpose constructively giving a wave with twice the amplitude of either wave. 𝐼∝ 𝐴 2 , so the intensity at P is 4 times that of either wave alone.
At Q, a blue crest meets a red trough so they interfere destructively resulting in zero intensity.

40. Summary
Complete the diffraction assignment

41. Diffraction Simulation
Use the google doc to guide you
Submit to Canvas by the due date!

42. Double-Slit Experiment (HL)
Determine the wavelength of the laser by making measurements of its interference pattern.
IV:
DV:
Control: