DP IB Physics: SL

C. Wave Behaviour

C.3 Wave Phenomena

DP IB Physics: SL C. Wave Behaviour C.3 Wave Phenomena Understandings Students should understand:
a)	That waves travelling in two and three dimensions can be described through the concepts of wavefronts and rays
b)	Wave behaviour at boundaries in terms of reflection, refraction and transmission
c)	Wave diffraction around a body and through an aperture
d)	Wavefront-ray diagrams showing refraction and diffraction
e)	Snell’s law, critical angle and total internal reflection
f)	Snell’s law as given by [math]\frac{n_1}{n_2} = \frac{\sin \theta_2}{\sin \theta_1} = \frac{v_2}{v_1}[/math] Where n is the refractive index and θ is the angle between the normal and the ray
g)	Superposition of waves and wave pulses
h)	That double-source interference requires coherent sources
i)	The condition for constructive interference as given by [math] \text{path difference} = nλ[/math]
j)	The condition for destructive interference as given by [math]\text{Path difference} = \left(n + \frac{1}{2}\right)\lambda[/math]
k)	Young’s double-slit interference as given by [math]s = \frac/{λD}{d}[/math] Where s is the separation of fringes, d is the separation of the slits, and D is the distance from the slits to the screen.

a) Waves in Two & Three Dimensions: Wavefronts and Rays
A wavefront is a line (in 2D) or a surface (in 3D) that connects points of equal phase in a wave—meaning all the points on the wavefront are at the same stage in their wave cycle (like all crests or all troughs).
– In 2D, wavefronts are lines.
– In 3D, wavefronts are surfaces.
Examples of Wavefronts:
– Circular wavefronts from a point source in water.
– Plane wavefronts from a distant source (like sunlight).
Rays are imaginary lines perpendicular to wavefronts that show the direction of energy travel.
– Rays indicate the path a wave takes.
– They always point normal (90°) to the wavefront.
Figure 1 Wavefront and rays
So, in diagrams:
Wavefronts = curved/flat lines
Rays = straight arrows perpendicular to those lines
⇒ 2D vs 3D Waves:

Property	2D Wave (e.g., ripple)	3D Wave (e.g., sound/light in space)
Wavefront	Lines (circles or straight)	Surfaces (spheres or planes)
Ray	Arrows from center of circles	Arrows radiating from source in 3D
Visualization	Water ripples	Light from a bulb or sound from a point

b) Wave Behavior at Boundaries: Reflection, Refraction, Transmission
When a wave reaches the boundary between two different media, three things can happen:
⇒ Reflection
The wave bounces back into the original medium.
The angle of incidence = angle of reflection.
Example: Echo from a wall, light reflecting from a mirror.
Figure 2 Reflection
⇒ Law of Reflection:
[math]θ_i = θ_r[/math]
⇒ Refraction
The wave changes direction as it enters a new medium due to a change in wave speed.
Example: A straw looks bent in water.
⇒ Snell’s Law:
[math]n_1 sinθ_1 = n_2 sinθ_2[/math]
Where:
– n= refractive index of the medium
– θ= angle from the normal
If wave slows down (e.g. air → water): it bends toward the normal
If wave speeds up (e.g. water → air): it bends away from the normal
Figure 3 Refraction
⇒ Transmission
Part of the wave passes through the second medium.
Transmission can involve a loss in amplitude due to energy absorption or scattering.
Example: Light passing through glass.
Figure 4 Light passing through glass
c) Wave Diffraction Around Obstacles and Through Apertures
Diffraction is the spreading of a wave when it passes through a gap or moves around an obstacle.
– Occurs best when gap size or obstacle is similar to the wavelength.
– All waves can diffract: sound, water, light, even EM waves.
⇒ Main Cases:
1. Diffraction Through a Narrow Slit (Aperture)
Wave bends into the shadow region behind the slit.
The narrower the slit, the greater the spreading.
Used in diffraction grating and single slit experiments.
2. Diffraction Around an Obstacle
Wave bends around the edges of a barrier.
Sound can be heard around a wall because it diffracts.
Figure 5 Diffraction from obstacles
d) Wavefront-Ray Diagrams: Refraction & Diffraction
⇒ Refraction (Wavefront Diagram)
Before entering a new medium:
– Wavefronts are evenly spaced
– Rays are perpendicular
After entering a medium with different speed:
– Wavelength changes
– Wavefronts bend
– Rays bend toward/away from normal
Example:
Air → Water:
– Speed ↓
– Wavelength ↓
– Bends toward normal
⇒ Diffraction (Wavefront Diagram)
Wide gap (>> λ):
– Little spreading
– Wavefronts pass almost straight through
Narrow gap (≈ λ):
– Significant spreading
– Wavefronts emerge circular
Figure 6 Diffraction and interference
e) Snell’s Law: Refraction of Light
Refraction is the bending of a wave as it passes from one medium to another due to a change in speed. This happens with light, sound, and water waves.
⇒ Snell’s Law
Snell’s Law mathematically describes how light bends:
[math]\frac{n_1}{n_2} = \frac{\sin \theta_2}{\sin \theta_1} = \frac{v_2}{v_1}[/math]
Where:
– [math]n_1 \text{and} n_2[/math] : refractive indices of the two media
– [math]θ_1[/math] : angle of incidence (from normal)
– [math]θ_2[/math] : angle of refraction
– [math]v_1, v_2[/math] : speeds of light in each medium
Figure 7 Snell’s Law
⇒ Important Concepts:
If [math]n_2 > n_1[/math] (e.g. air → glass): light slows down and bends toward the normal.
If [math]n_2 < n_1[/math] (e.g. glass → air): light speeds up and bends away from the normal.
Refractive index (n) is a measure of how much a material slows down light:
[math]n = \frac{c}{v}[/math]
Where [math]c = 3.00 × 10^8 m/s[/math] is the speed of light in vacuum.
f) Critical Angle & Total Internal Reflection (TIR)
When light travels from a more optically dense medium to a less dense one (e.g. glass → air), two things happen:
1. At small angles, light refracts (bends out).
2. At a certain angle (the critical angle), light refracts at 90°.
3. Beyond that angle, no refraction occurs—light is totally reflected back inside.
This is called Total Internal Reflection (TIR).
⇒ Critical Angle Formula:
[math]\sin \theta_c = \frac{n_2}{n_1}[/math]
– Only applies when [math]n_1 > n_2[/math]
Conditions for TIR:
1. Wave moves from denser to rarer medium.
2. Angle of incidence > critical angle.
Figure 8 Total internal reflection
Real-Life Examples:
– Optical fibers
– Prisms in binoculars
– Sparkling of diamonds
g) Superposition of Waves & Pulses
Superposition is when two or more waves meet at the same place and combine.
The resulting wave is the sum of the displacements of individual waves at each point.
⇒ Types of Superposition:
1. Constructive Interference
– Waves add up
– Occurs when crests meet crests or troughs meet troughs
– Produces a larger amplitude
2. Destructive Interference
– Waves cancel out
– Occurs when crest meets trough
– Results in reduced or zero amplitude
⇒ Superposition of Pulses
Even single wave pulses can interfere:
If two pulses meet on a string:
– They temporarily combine
– Then pass through each other, continuing unaffected
Important: Superposition is temporary; the waves do not permanently change.
Figure 9 Superposition of Wave
h) Double-Source Interference & Coherent Sources
Interference occurs when two waves overlap and interact—producing regions of constructive and destructive interference.
⇒ Double-Source Interference Pattern
If two wave sources emit waves (e.g., two loudspeakers, or two slits for light):
– Bright/strong regions form where waves meet in phase → constructive
– Dark/quiet regions form where waves meet out of phase → destructive
This forms an interference pattern of alternating maxima and minima.
Figure 10 Two sources interference (Coherent sources)
⇒ Requirement: Coherent Sources
To produce a clear, stable interference pattern, the two sources must be:
1. Coherent – same frequency and constant phase difference
2. Monochromatic – same wavelength (for light)
Without coherence, the pattern is unstable or washed out.
Examples:
– Double-slit experiment with lasers (Young’s experiment)
– Water ripple tanks with synchronized dippers
– Sound from two speakers playing same tone
i) Conditions for Constructive and Destructive Interference
When two waves overlap in space, they interfere. The resulting wave depends on the phase relationship between the two.
⇒ Constructive Interference
Occurs when two waves arrive in phase—their peaks and troughs align. This results in a larger amplitude (brighter light, louder sound, etc.).
Condition:
[math]\text{Path difference} = nλ[/math]
Where n=0,1,2,3,…
– λ = wavelength
– n = order of fringe (central = 0, next bright = 1, etc.)
This means that the waves have traveled distances differing by a whole number of wavelengths.
Figure 11 Constructive and destructive interference
j) Destructive Interference
Occurs when two waves arrive out of phase—a peak aligns with a trough. This causes cancellation, reducing or eliminating amplitude (dark spots, quiet sound).
Condition:
[math]\text{Path difference} = \left(n + \frac{1}{2}\right)\lambda[/math]
Where n=0,1,2,…
This means the waves differ in path by half a wavelength, causing them to cancel.
⇒ Path Difference:
It’s the difference in distance that each wave has traveled from its source to the point where they meet. Interference depends entirely on this difference.
k) Young’s Double-Slit Experiment
Young’s double-slit experiment demonstrates wave interference using light. It proved that light behaves like a wave.
Thomas Young shone monochromatic light (one color) on two closely spaced narrow slits, which acted like two coherent sources of waves.
The light diffracted at each slit and then interfered on a screen behind.
An interference pattern:
– Bright fringes (constructive interference)
– Dark fringes (destructive interference)
– Formed alternating bands on the screen
⇒ Fringe Separation Formula:
[math]s = \frac{\lambda D}{d}[/math]
Where:
– s = separation between adjacent bright (or dark) fringes
– λ = wavelength of the light
– D = distance from slits to the screen
– d = separation between the two slits
Larger wavelength → wider fringe spacing
Larger screen distance D → wider fringe spacing
Larger slit separation d → narrower fringe spacing
Figure 12 Young double slit experiment
⇒ Key Conditions for Visible Interference:
1. The two slits must act as coherent sources (same wavelength, constant phase difference).
2. Light must be monochromatic for clear bands.
3. Slits must be narrow and close together to allow diffraction and overlap.