Fundamental Wave Properties

Why This Matters

Waves are everywhere in AP Physics 2—from the electromagnetic radiation that carries energy through space to the quantum mechanical behavior of electrons in atoms. Understanding fundamental wave properties isn't just about memorizing definitions; it's about grasping the universal language that describes how energy and information travel through the universe. You'll see these concepts resurface in geometric optics, diffraction, interference patterns, and even the Bohr model's standing wave interpretation of electron orbits.

The exam tests whether you can connect wave properties to real phenomena. Can you explain why diffraction is more pronounced when slit width matches wavelength? Can you use the wave equation to predict how frequency and wavelength trade off when wave speed is fixed? Don't just memorize that $v = f\lambda$—know why this relationship matters and how it constrains wave behavior across every topic you'll encounter.

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The Core Wave Equation and Its Variables

Every wave property connects back to one fundamental relationship: $v = f\lambda$. Master these three quantities and their interdependence, and you'll have the foundation for everything else.

Wave Speed

• The rate at which wave disturbances propagate through a medium—calculated as $v = f\lambda$, where $v$ is in meters per second
• Medium-dependent property: sound travels ~343 m/s in air but ~1480 m/s in water; light travels at $c = 3 \times 10^8$ m/s in vacuum
• Fixed by medium properties, meaning frequency and wavelength must adjust to each other when a wave enters a new medium

Wavelength

• Distance between consecutive points of identical phase (crest to crest, compression to compression)—symbolized by $\lambda$ (Greek letter lambda)
• Inversely related to frequency when wave speed is constant: doubling wavelength halves frequency
• Critical for diffraction: waves spread most when $\lambda$ is comparable to obstacle or opening size

Frequency

• Number of complete cycles passing a point per second—measured in hertz (Hz), where 1 Hz = 1 cycle/second
• Inversely related to period: $f = \frac{1}{T}$, so higher frequency means shorter period
• Determined by the source, not the medium—frequency stays constant when a wave crosses boundaries

Period

• Time for one complete wave cycle—measured in seconds and related to frequency by $T = \frac{1}{f}$
• Determines timing of oscillations: a 2 Hz wave has a period of 0.5 s between crests
• Essential for standing wave problems where you need to track how quickly nodes and antinodes form

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Energy and Intensity Relationships

Waves carry energy, and the amount depends critically on amplitude. This section connects wave properties to the energy they transport.

Amplitude

• Maximum displacement from equilibrium position—determines how much energy the wave carries
• Energy scales with amplitude squared: doubling amplitude quadruples energy, making this a nonlinear relationship
• Affects perceived intensity: larger amplitude means louder sound or brighter light

Wave Energy and Intensity

• Intensity is power per unit area, measured in $\text{W/m}^2$—tells you how much energy flows through a given cross-section
• Proportional to amplitude squared: $I \propto A^2$, so small amplitude changes produce large intensity changes
• Decreases with distance from a point source as energy spreads over larger areas (inverse-square law for 3D spreading)

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Wave Interactions: Superposition and Interference

When waves meet, they combine according to the superposition principle. This creates interference patterns that appear throughout AP Physics 2, from double-slit experiments to standing waves.

Superposition Principle

• Resultant displacement equals the algebraic sum of individual displacements—waves pass through each other without permanent change
• Foundation for all interference phenomena: constructive and destructive interference both emerge from this principle
• Applies to all wave types: sound, light, water waves, and even quantum mechanical probability waves

Interference (Constructive and Destructive)

• Constructive interference: waves in phase (path difference = $m\lambda$) combine to produce larger amplitude
• Destructive interference: waves out of phase (path difference = $(m + \frac{1}{2})\lambda$) cancel to reduce or eliminate amplitude
• Produces observable patterns: bright/dark fringes in light, loud/quiet zones in sound—key to double-slit and single-slit problems

Standing Waves

• Formed when identical waves travel in opposite directions—creates stationary pattern of nodes and antinodes
• Nodes have zero displacement; antinodes have maximum displacement—spacing between adjacent nodes is $\frac{\lambda}{2}$
• Connects to Bohr model: electron orbits must fit an integer number of de Broglie wavelengths, creating standing wave patterns

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Wave Behavior at Boundaries

What happens when waves encounter obstacles, openings, or new media? These behaviors—reflection, refraction, and diffraction—explain everything from echoes to rainbows to the patterns in single-slit experiments.

Wave Reflection

• Wave bounces back when encountering a boundary—angle of incidence equals angle of reflection
• Partial or total depending on boundary properties: hard boundaries invert the wave, soft boundaries don't
• Foundation for geometric optics: mirrors, echoes, and sonar all rely on predictable reflection

Wave Refraction

• Bending of wave direction when speed changes at a boundary—occurs because different parts of the wavefront slow down at different times
• Snell's law governs the angles: $n_1 \sin\theta_1 = n_2 \sin\theta_2$, connecting wave behavior to medium properties
• Frequency stays constant while wavelength changes—critical concept for understanding how waves cross boundaries

Diffraction

• Spreading of waves around obstacles or through openings—most pronounced when opening size $a$ is comparable to wavelength $\lambda$
• Single-slit minima occur at $a \sin\theta = m\lambda$ for $m = \pm 1, \pm 2, ...$ —know this equation cold
• Central maximum is twice as wide as other maxima—the intensity pattern follows a sinc-squared function

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Advanced Wave Phenomena

These topics connect wave properties to more sophisticated applications, including relative motion effects and the quantum nature of light.

Doppler Effect

• Apparent frequency shift due to relative motion between source and observer—approaching sources have higher observed frequency
• Blue shift (higher frequency) when approaching; red shift (lower frequency) when receding
• Applications span physics: radar guns, astronomical redshift measurements, and medical ultrasound all use Doppler shifts

Polarization

• Orientation of transverse wave oscillations—only transverse waves (like light) can be polarized
• Achieved through filters, reflection, or scattering—Malus's law ($I = I_0 \cos^2\theta$) governs intensity through polarizers
• Proves light is transverse: longitudinal waves cannot be polarized, so polarization experiments confirm light's wave nature

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The Wave Equation in Action

This fundamental relationship deserves its own focus because it's the key to solving most wave problems.

Wave Equation ($v = f\lambda$)

• Connects the three core wave quantities—if you know any two, you can find the third
• Speed is medium-dependent, frequency is source-dependent, wavelength adjusts to satisfy both constraints
• Use for boundary problems: when a wave enters a new medium, speed changes, frequency stays constant, so wavelength must change proportionally

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Quick Reference Table

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Self-Check Questions

1. If a wave's frequency doubles while traveling in the same medium, what happens to its wavelength? What happens to its speed?

2. Which two wave phenomena both produce patterns of maxima and minima, and what's the key difference in how they form?

3. A wave passes from air into water. Which property stays constant (frequency, wavelength, or speed), and why?

4. Compare and contrast how standing waves in a vibrating string relate to the Bohr model's explanation of quantized electron energy levels.

5. An FRQ describes a single-slit diffraction experiment where the slit width is halved. How does the angular width of the central maximum change, and which equation supports your answer?