14.3 Boundary Behavior of Waves and Polarization

When a wave hits the boundary between two media, part of it reflects and part transmits. The reflected wave inverts if the wave slows down in the new medium, frequency stays the same across the boundary, and only transverse waves can be polarized. These ideas help you explain wave diagrams, boundary changes, and polarization effects on the AP Physics 2 exam.

Why This Matters for the AP Physics 2 Exam

Boundary behavior shows up in both multiple-choice questions and free-response work where you analyze wave diagrams and explain what happens to a pulse at a junction. You will often need to reason from a picture: predict whether a reflected pulse flips, track how wavelength changes while frequency stays fixed, and connect those changes to wave speed in each medium. The translation between representations question rewards being able to move between a verbal description, a sketch, and a relationship like $\lambda = v/f$. Polarization questions test whether you understand that transverse waves can be restricted to one plane and that this can reduce intensity.

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Key Takeaways

• A wave hitting a boundary splits into a reflected wave and a transmitted wave; how much goes each way depends on the two media.
• The reflected wave inverts when the wave moves into a medium where it travels slower, and stays upright when it speeds up.
• Frequency does not change across a boundary. Wave speed and wavelength can change, and they stay linked by $\lambda = v/f$.
• Only transverse waves can be polarized. Longitudinal waves like sound cannot, because there is no oscillation perpendicular to the travel direction.
• Polarization can lower a wave's intensity, where intensity is the average power per unit area over one period.

Wave-Boundary Interaction

Transmission and Reflection

When a wave reaches a boundary between two different media, it can be transmitted into the new medium, reflected back, or both. How the energy splits depends on the properties of the two media. A classic setup is a wave traveling along a string that connects to a second string of different mass per length.

• A wave moving from a low-mass string to a high-mass string produces both a reflected wave and a transmitted wave.
• The reflected wave is inverted (a 180 degree phase shift) when the wave moves into a medium where its speed decreases.
• The reflected wave is not inverted when the wave moves into a medium where its speed increases.
• Frequency stays the same as the wave crosses the boundary. Only the wave speed and wavelength can change in the new medium.

Because frequency is fixed and $\lambda = v/f$, a change in wave speed forces a matching change in wavelength. If the wave slows down, the wavelength gets shorter; if it speeds up, the wavelength gets longer.

Wave Polarization

Polarization applies only to transverse waves, where the oscillation is perpendicular to the direction the wave travels.

• A transverse wave can oscillate in any direction perpendicular to its motion. Polarization restricts that oscillation to a single plane.
• Transverse waves may become polarized when they reflect off a surface, refract through a medium, or pass through specific openings or filters.
• Longitudinal waves, such as sound, cannot be polarized. Their oscillation is parallel to the travel direction, so there is no perpendicular component to restrict.

Light is the most common example of a transverse wave that can be polarized. When unpolarized light passes through a polarizing filter, only the part of the wave oscillating in the allowed plane gets through.

Polarization and Intensity

Polarizing a wave can reduce how much energy it carries, measured as intensity.

• Intensity is the average power transferred per unit area over one full period of the wave.
• Written as a relationship: $I = \frac{P_{avg}}{A}$ where $I$ is intensity, $P_{avg}$ is the average power, and $A$ is the area.
• Filtering out part of the oscillation removes part of the wave's power, so the transmitted intensity drops.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Identify the direction of travel first. Ask whether the wave is moving into a faster or slower medium, then decide if the reflected pulse inverts.
• Keep frequency constant across any boundary. If a question changes the wave speed, update the wavelength using $\lambda = v/f$, not the frequency.
• For a string, a heavier (denser) string slows the wave, which means a reflected pulse from a light-to-heavy junction inverts.

Free Response

• When a sketch is given, describe the reflected and transmitted pulses separately, including whether the reflected one is flipped.
• Justify inversion by naming the cause: the wave speed decreased in the new medium.
• For polarization, state clearly that the wave is transverse, that polarization limits oscillation to one plane, and that this can lower intensity.

Common Trap

• Do not say frequency changes at a boundary. It does not. Wavelength and speed are the quantities that change.

Practice Problem 1: Wave Reflection at Boundaries

Solution

The wave moves from a medium where it travels faster to one where it travels slower (8 m/s to 4 m/s), so the reflected pulse is inverted. Part of the wave reflects and part transmits into the heavier string. Frequency does not change at the boundary; only the wave speed and wavelength in the new medium change.

Practice Problem 2: Polarization and Intensity

Solution

When unpolarized light passes through a polarizing filter, the intensity is cut in half: $I_1 = \frac{I_0}{2} = \frac{100 \text{ W/m²}}{2} = 50 \text{ W/m²}$

When this polarized light passes through a second polarizer at angle $\theta$ to the first, Malus's Law applies as an application of the intensity idea: $I_2 = I_1\cos^2\theta$

With $\theta = 30°$: $I_2 = 50 \text{ W/m²} \times \cos^2(30°) = 50 \text{ W/m²} \times (0.866)^2 = 50 \text{ W/m²} \times 0.75 = 37.5 \text{ W/m²}$

The final intensity after both polarizers is 37.5 W/m².

Common Misconceptions

• "Frequency changes when a wave enters a new medium." It does not. Frequency is set by the source and stays constant across the boundary. Speed and wavelength are what change.
• "The reflected wave always flips." It only inverts when the wave moves into a medium where it slows down. Moving into a faster medium leaves the reflected pulse upright.
• "Sound can be polarized." No. Sound is longitudinal, so its oscillation is along the travel direction and there is no perpendicular component to restrict.
• "Polarization just blocks light, so it has nothing to do with energy." Polarization reduces intensity, which is power per unit area, so it directly reduces the energy delivered per second.
• "A wave either reflects or transmits at a boundary." Both usually happen at the same time; the energy splits between a reflected wave and a transmitted wave.

Related AP Physics 2 Guides

• 14.2 Periodic Waves
• 14.6 Wave Interference and Standing Waves
• 14.7 Diffraction
• 14.1 Properties of Wave Pulses and Waves
• 14.4 Electromagnetic Waves
• 14.5 The Doppler Effect