As waves propagate, they frequently encounter boundaries, changes in medium, or openings/slits. These encounters lead to phenomena such as reflection, refraction, diffraction, and interference. Understanding these phenomena is critical for explaining anything from echoes in acoustics to the colors in soap bubbles.

1. Law of Reflection. When a wavefront strikes a boundary at an angle, it is reflected such that the angle of incidence θi equals the angle of reflection θr, measured relative to the normal at the surface. In one dimension (e.g., a wave on a string hitting a fixed end), the boundary condition may cause an inversion of phase; at a free end, reflection occurs without phase inversion.
2. Phase Changes upon Reflection.
3. Fixed (Rigid) Boundary: If the end of the medium is rigidly fixed (e.g., a string attached to a rigid support), the reflected wave is inverted (phase change of π radians).
4. Free Boundary: If the medium’s end is free to move (e.g., a string attached to a ring that can slide without friction on a rod), the reflected wave is not inverted (no phase change).
5. Applications. Echoes in Acoustics: Sound waves reflect off surfaces like walls or canyon faces, producing echoes if the round-trip travel time exceeds approximately 0.1 s.

1. Definition. Refraction is the bending of a wave as it passes from one medium into another in which its speed is different.
2. Common examples: Light going from air into glass (optics). Sound passing from warm to cold air (acoustics).
3. Index of Refraction (Optics Context). Although IB’s focus may be broader than just optics, it is instructive to introduce the concept of the index of refraction n, defined by n=cv, where c is the speed of light in vacuum and v is the speed of light in the medium.
4. Snell’s Law (General Form). When a wave passes from medium 1 (speed v1) to medium 2 (speed v2), the angles of incidence θ1 and refraction θ2 satisfy: sin θ1/v1 = sin θ2/v2
5. Critical Angle and Total Internal Reflection (Optics). There exists a critical angle θc beyond which no refraction into medium 2 occurs; instead, the wave undergoes total internal reflection.
6. Mathematically: sin θc = v2/v1(for v1>v2).
7. Applications. Refraction explains why swimming pools appear shallower than they really are.

1. Definition. Diffraction is the spreading of a wave when it encounters an obstacle whose size is comparable to its wavelength or passes through an aperture comparable to its wavelength.
2. In general, the smaller the aperture (relative to λ), the more pronounced the diffraction.
3. Single-Slit Diffraction. Consider a monochromatic wave of wavelength λ passing through a slit of width a. On a distant screen, a central maximum appears, flanked by a series of minima and secondary maxima.
4. The minima occur at angles θ satisfying: a sin θ = m λ.
5. Diffraction Grating (Brief Introduction). A diffraction grating has many equally spaced parallel slits (or grooves). The condition for constructive interference (principal maxima) is: d sin θ = n λ.
6. Physical Insight. Diffraction can be understood via Huygens’ principle: each point in a wavefront acts as a secondary source of spherical wavelets.

1. Principle of Superposition. When two or more waves overlap in space, the resultant displacement at any point is the algebraic sum of the displacements due to each wave (superposition principle).
2. Two-Source Interference (Young’s Double-Slit).
3. Setup: Two narrow slits separated by distance d are illuminated by coherent light of wavelength λ.
4. Path Difference: Δ = d sin θ.
5. Constructive Interference (Bright Fringes): Occurs when d sin θ = m λ.
6. Destructive Interference (Dark Fringes): Occurs when d sin θ = (m + 1/2) λ.
7. Fringe Spacing on Screen. If the screen is a distance D from the slits (with D ≫ d), then the distance y_m from the central maximum to the mth bright fringe is: y_m ≈ mλD/d.

1. Diffraction Gratings in Spectroscopy. Used to disperse light into its component wavelengths with high resolution. Spectrometers employ this principle to analyze atomic and molecular spectra.
2. Interferometry (e.g., Michelson Interferometer). By splitting a beam of light into two paths and recombining them, one can measure extremely small length differences or changes in refractive index.
3. Acoustic Applications. Noise-Cancelling Headphones reduce perceived sound by producing sound waves 180° out of phase with ambient noise.
4. Radio Wave Propagation. Interference and diffraction affect radio signal reception, especially at night when certain wavelengths can travel longer distances.