AP Physics 2 Unit 6 Study Notes: Waves, Sound, and Physical Optics

Wave

A traveling disturbance that transfers energy and momentum from place to place without transporting matter overall; the medium oscillates about equilibrium while the pattern moves.

Mechanical wave

A wave that requires a material medium (e.g., sound in air, waves on a string, water waves).

Electromagnetic wave

A wave that does not require a medium to travel (e.g., light) and can propagate through vacuum.

Transverse wave

A wave in which the medium oscillates perpendicular to the direction of wave travel (e.g., string waves, light).

Longitudinal wave

A wave in which the medium oscillates parallel to the direction of wave travel (e.g., sound in air).

Amplitude (A)

Maximum displacement from equilibrium (or maximum pressure variation for sound); relates to energy transfer and intensity.

Period (T)

Time for one complete cycle of oscillation.

Frequency (f)

Number of cycles per second (Hz); related to period by f = 1/T.

Wavelength (λ)

Spatial period of a wave: distance between repeating points in space (e.g., crest-to-crest or compression-to-compression).

Wave speed (v)

Speed at which a point of constant phase (like a crest) moves through space.

Wave relationship (v = fλ)

Key link between time and space behavior of a periodic wave: wave speed equals frequency times wavelength.

Medium-determined wave speed

For ordinary linear waves, the wave speed is set by the medium and wave type (e.g., tension and mass/length for a string; medium and temperature for sound), not by amplitude.

Sinusoidal traveling-wave model

A common model for a wave traveling in +x: y(x,t) = A sin(kx − ωt + φ), showing sinusoidal variation in space and time.

Phase

The relative “position in the cycle” of a wave; phase relationships determine constructive vs destructive interference.

Wave number (k)

Spatial frequency of a sinusoidal wave, related to wavelength by k = 2π/λ.

Angular frequency (ω)

Rate of phase change in time, related to frequency by $\theta = 2\pi f$.

Principle of superposition

When waves overlap in a linear medium, the resulting displacement equals the algebraic sum of the individual displacements.

Interference

The pattern produced when waves overlap and add via superposition; can increase or decrease the resultant amplitude depending on phase.

Constructive interference

Waves arrive in phase (crest with crest), producing a larger resultant amplitude.

Destructive interference

Waves arrive out of phase by half a cycle (crest with trough), producing a smaller resultant amplitude (possibly near zero); energy is redistributed, not destroyed.

Path difference (ΔL)

Difference in distances traveled by two waves to the same point; constructive if $\Delta L = m \lambda$ and destructive if $\Delta L = (m + \frac{1}{2})\lambda$, where $m = 0,1,2,…$

Coherent sources (coherence)

Sources that maintain a constant phase relationship, enabling stable, lasting interference patterns (e.g., two slits fed by one source).

Standing wave

A fixed pattern formed by two waves of the same frequency and amplitude traveling in opposite directions, producing nodes and antinodes.

Node

Point in a standing wave that always has zero displacement (always at equilibrium).

Antinode

Point in a standing wave with maximum displacement (largest oscillation amplitude).

Boundary conditions

Constraints at ends/boundaries that determine allowed standing-wave patterns (e.g., fixed string end is a displacement node; open pipe end is a displacement antinode; closed pipe end is a displacement node).

Displacement node–pressure antinode (air columns)

In sound standing waves in air columns, a displacement node corresponds to a pressure antinode, and a displacement antinode corresponds to a pressure node.

Fundamental (first harmonic)

Lowest-frequency standing-wave mode that satisfies the boundary conditions (n = 1).

Harmonics / overtones

Higher allowed standing-wave modes above the fundamental; correspond to higher frequencies that still satisfy boundary conditions.

String/open-open pipe resonant frequencies

For length L with nodes at both ends (string fixed-fixed or open-open pipe displacement pattern): f_n = n v/(2L), with n = 1,2,3,…

Closed-open pipe resonant frequencies

For a pipe closed at one end and open at the other: $f_n = \frac{(2n - 1)v}{4L}$, $n = 1,2,3,…$ (only odd harmonics appear).

Resonance

Large-amplitude response when a driving frequency matches a system’s natural frequency, allowing energy to build efficiently.

Damping

Energy loss that limits resonance amplitude; less damping gives a sharper/higher resonance peak, more damping gives a broader/lower peak.

Sound wave (in air)

A longitudinal mechanical wave of pressure and density variations traveling through air.

Compressions and rarefactions

Alternating regions of high pressure/density (compressions) and low pressure/density (rarefactions) in a longitudinal sound wave.

Intensity (I)

Power transferred per area perpendicular to wave travel: I = P/A.

Inverse-square law (point source)

For a point source radiating uniformly: $I = \frac{P}{4\pi r^2}$, so intensity decreases as $\frac{1}{r^2}$ with distance.

Decibel level (β)

Logarithmic sound intensity level: $\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$; +10 dB corresponds to 10× intensity, and +3 dB is about 2× intensity.

Reference intensity (I₀)

Standard reference for sound level: $I_0 = 1.0 \times 10^{-12} \text{W/m}^2$.

Pitch vs loudness

Pitch is primarily related to frequency (higher f → higher pitch); loudness is related to intensity/amplitude (perception is not linear).

Doppler effect

Observed frequency shift due to relative motion of source and observer: $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$; motion toward increases $f'$, motion away decreases $f'$ (choose signs to match this meaning).

Beats

Periodic variation in loudness when two close frequencies overlap; beat frequency is $f_{\text{beat}} = |f_1 - f_2|$.

Young’s double-slit interference

Two coherent slits separated by d produce bright fringes when $d \sin(\theta) = m \lambda$ and dark fringes when $d \sin(\theta) = (m + \frac{1}{2})\lambda$.

Small-angle approximation (fringes)

For small $\theta$, $\sin \theta \text{≈ tan} \theta \approx \frac{y}{L}$, relating screen position $y$ to angle $\theta$ when the screen is distance $L$ away.

Fringe position formula

Using the small-angle approximation, the mth bright fringe is at $y_m = \frac{m \lambda L}{d}$ in a double-slit setup.

Single-slit diffraction minima

For slit width a, dark fringes occur at a sinθ = mλ with m = 1,2,3,… (m does not start at 0).

Diffraction grating

Many equally spaced slits producing very sharp principal maxima; maxima satisfy $d \sin(\theta) = m \lambda$ (same angle condition as double slit, but sharper peaks).

Diffraction-limited resolution

Fundamental limit on resolving two close sources due to diffraction; improved by using smaller wavelength (λ) and larger aperture.

Polarization

Orientation of oscillations in a transverse wave; light can be polarized (transverse behavior), while sound in air cannot (longitudinal).

Malus’s law

For polarized light through a polarizer at angle $\theta$: $I = I_0 \cos^2 \theta$; unpolarized light through one ideal polarizer transmits $I = \frac{1}{2}I_0$.