Reflection occurs when a wave hits a surface and bounces back, changing direction. The core rule is simple: the angle of incidence equals the angle of reflection. Both angles are measured from the normal, which is an imaginary line perpendicular to the surface at the point where the wave hits. Mirrors reflect light this way, and echoes are just sound waves reflecting off walls or cliffs.

Refraction is what happens when a wave passes from one medium into another (like from air into water). Because the wave changes speed in the new medium, it also changes direction. This is why a straw in a glass of water looks bent at the surface: light traveling from water into air speeds up and bends.

The relationship between the angles is described by Snell's Law:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

$n_1$ and $n_2$ are the refractive indices of the two media (a measure of how much each medium slows light down compared to a vacuum)
$\theta_1$ is the angle of incidence, $\theta_2$ is the angle of refraction
A higher refractive index means the wave travels slower in that medium, and the wave bends toward the normal when entering it

Diffraction and Interference

Diffraction is the bending or spreading of waves around obstacles and through openings. It's most noticeable when the wavelength is close to the size of the obstacle or gap. This is why you can hear someone talking around a corner (sound waves have long wavelengths, roughly matching doorway sizes) but you can't see them (visible light wavelengths are tiny compared to a doorway).

Interference happens when two or more waves overlap in the same space. The waves combine according to the superposition principle: at any point, the total displacement is the sum of the individual wave displacements. You can observe this in a ripple tank where two sets of water waves cross, or in Young's double-slit experiment, where light passing through two narrow slits creates a pattern of bright and dark bands on a screen.

$Reflection and Refraction, Snell's law - Wikipedia$

Interference Patterns

Constructive and Destructive Interference

When two waves meet, they combine in one of two ways:

Constructive interference happens when crests line up with crests and troughs line up with troughs (the waves are in phase). The amplitudes add together, producing a wave with greater amplitude and more energy.
Destructive interference happens when crests line up with troughs (the waves are out of phase). The amplitudes cancel, reducing the combined wave's amplitude. If the two waves have equal amplitude, they can cancel completely, producing zero displacement at that point.

In experiments like the double-slit setup, these two types of interference alternate across space, creating an interference pattern. Bright bands appear where constructive interference occurs, and dark bands appear where destructive interference occurs.

Noise-canceling headphones use destructive interference. They detect incoming sound waves and produce a wave that's out of phase with the noise, canceling it out.

$Reflection and Refraction, Refraction - Snell's Law | TikZ example$

Standing Waves and Nodes

A standing wave forms when two waves with the same frequency and amplitude travel in opposite directions through the same medium. Instead of appearing to move, the wave pattern stays in place.

Nodes are fixed points where the medium doesn't move at all (zero amplitude). They occur where the two traveling waves always cancel.
Antinodes are points of maximum vibration, located halfway between nodes. These are where constructive interference is greatest.

Standing waves show up constantly in musical instruments. A guitar string vibrates as a standing wave, with nodes at each end where the string is fixed. The simplest pattern, with just one antinode in the middle, produces the fundamental frequency (the lowest pitch the string can make). Higher harmonics have additional nodes and antinodes along the string, producing higher pitches. The standing wave pattern that forms depends on the length of the string (or air column) and the wave's frequency.

Wave Phenomena

Resonance in Mechanical and Acoustic Systems

Every object or system that can vibrate has a natural frequency, the frequency at which it vibrates most easily. Resonance occurs when an external force drives the system at or near this natural frequency, causing the amplitude of vibration to build up dramatically.

A playground swing is a good example: pushing at just the right rhythm (matching the swing's natural frequency) makes it go higher and higher. A more dramatic case is a wine glass shattering when exposed to a loud sound at its resonant frequency. The sound waves drive the glass to vibrate with increasing amplitude until the glass breaks. This is also why soldiers break step when crossing a bridge: rhythmic marching could match the bridge's natural frequency and cause dangerous oscillations.

For instruments, the resonant frequency depends on physical properties:

Strings: length, tension, and mass per unit length determine the frequencies
Wind instruments: the size and shape of the air column set the resonant frequencies

Doppler Effect and Its Applications

The Doppler effect is the change in observed frequency that occurs when a wave source and an observer are moving relative to each other.

When the source approaches, each successive wave crest is emitted from a closer position, compressing the waves together. The observer detects a higher frequency (higher pitch for sound).
When the source moves away, the wave crests spread apart, and the observer detects a lower frequency (lower pitch). Think of how an ambulance siren sounds higher as it approaches and drops in pitch as it passes.

The Doppler effect formula for sound is:

$f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)$

$f'$ = observed frequency, $f$ = emitted frequency
$v$ = speed of sound in the medium
$v_o$ = speed of the observer, $v_s$ = speed of the source
Use the top signs (+ in numerator, − in denominator) when source and observer move toward each other, and the bottom signs when they move apart

The Doppler effect has wide-ranging applications:

Radar guns bounce radio waves off moving vehicles; the frequency shift reveals the vehicle's speed
Medical ultrasound uses Doppler shifts to measure blood flow velocity through vessels
Astronomy relies on Doppler shifts in light from stars and galaxies to measure how fast objects are moving toward or away from Earth, which is key to detecting exoplanets and measuring the expansion of the universe

2,589 studying →