Boundary Behavior of Waves and Polarization
Revisit polarization and boundary behavior for transverse light before comparing optical patterns.
Before this connection: Transverse and longitudinal waves; Refraction and wave phase.
AP Physics 2 Topic 14.3 · College Board CED printed p. 114, 115 / PDF p. 121, 122 · See this topic in the course outline
At a boundary, track the reflected disturbance, transmitted disturbance, and energy. Their directions and phases carry different information.
Before you start
- Wave speed
- Frequency
- Power and area
A stationary junction divides a wave
At a junction of two strings, some incident energy can be reflected and some transmitted. With equal tension, a heavier string has lower wave speed: an upward incident pulse reflects as a downward pulse (a phase reversal of π) when entering that slower string. From the heavier, slower string into a lighter, faster string, the reflected pulse remains upward. A fixed end is the inverted limiting case; a loose end is the noninverted limiting case. The transmitted pulse keeps its displacement orientation in this simple string-junction model.
The reflected wave travels back in its original medium at the original speed. The transmitted wave travels at the new medium's speed. A stationary boundary cannot change how often the source makes cycles, so incident, reflected, and transmitted frequencies match; their wavelengths can differ. The reflected-speed phase rule stated here assumes this string/ordinary optical boundary setting. In a general wave system the impedance, not speed alone, controls reflection, so specify the medium rather than treating the shorthand as universal.
Select a transverse direction
A transverse wave has more than one possible orientation perpendicular to its travel direction. Polarization selects a direction or plane of oscillation. A vertical slot can pass vertical rope motion while blocking horizontal rope motion; a light polarizer selects one direction of electric-field oscillation. Reflection and refraction can also preferentially select orientations. A longitudinal fluid sound wave has only the parallel oscillation direction, so it cannot be polarized in this sense.
Draw light traveling along z: allowed electric-field directions lie in the x–y plane. A vertical filter transmits the y component. A second perpendicular filter blocks that component ideally. These axes refer to field orientation, not to the direction light moves.
Intensity is power per area
Intensity is the time-averaged energy-transfer rate per area perpendicular to propagation: , in W/m². Averaging over a period removes the instant-to-instant oscillation. If 0.60 W crosses 0.020 m², I is 30 W/m².
A polarizer generally reduces intensity because it removes some orientations/components. An ideal first polarizer passes half the intensity of unpolarized light. For already polarized light and an analyzer angle θ, explains why aligned axes pass most and crossed axes block it. This algebraic supporting model assumes ideal filters; real filters also absorb. A smaller transmitted intensity means less average power per area, not a lower light frequency.
Crossing onto a heavier string
Two strings have equal tension, but the second has four times the mass per length. A 12 Hz wave on the first string travels at 24 m/s. Describe reflection and transmission.
, so v₂ = 12 m/s.
Why this step? Equal tension lets the mass-per-length ratio determine speed.
The reflected displacement is inverted because the transmitted wave enters the slower string. Reflected speed is still 24 m/s in string 1.
Why this step? Separate phase from direction and identify which medium each wave occupies.
f remains 12 Hz; λ₁ = 2.0 m and λ₂ = 1.0 m.
Why this step? A stationary junction preserves source timing; λ = v/f changes with v.
Answer and check
Inverted reflection; transmitted wave has 12 m/s, 12 Hz, and 1.0 m wavelength.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
Could crossed polarizers block a longitudinal sound wave in air by selecting oscillation planes?
Worked answer and reason
No. Its disturbance lies parallel to travel, so it does not have transverse orientation choices to select.
Polarization requires transverse degrees of freedom.
Question 2
An ideal analyzer rotates from aligned to perpendicular to polarized light. What changes: intensity, frequency, or propagation direction?
Worked answer and reason
Transmitted intensity falls to zero ideally; frequency remains the same, and the transmitted light still travels along the beam.
The filter removes a field component and its average power per area.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- An inverted reflected wave has negative frequency.
- Inversion is a π phase reversal of signed displacement; frequency remains the positive repetition rate.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 5 · mixed
- Problem 6 · mixed
- Problem 21 · mixed
Scientific calculator workspace
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Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
For the teacher
- Use two strings at the same tension to establish the speed/phase rule. Do not generalize that rule to arbitrary impedance systems without specifying the model.
Source objectives and scope
- 14.3.A — Describe the interaction between a wave and a boundary.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- AP Physics 2 Course and Exam Description, official Unit 14 — Official topic/LO/EK scope and boundaries; local pinned source pages 117–134, equations visually checked on 118,120,130,132,134.
- OpenStax University Physics 1, Energy and Power of a Wave — Verification of fixed-amplitude mechanical-wave frequency scaling and distinction from electromagnetic intensity.
- OpenStax University Physics 1, Interference of Waves — Linear superposition and pointwise addition model.
- OpenStax University Physics 3, Single-Slit Diffraction — Verification of aperture minima and small-angle conditions.
- OpenStax University Physics 3, Interference in Thin Films — Verification of reflection phase and normal-incidence thin-film paths.
Glossary and deeper reading · 9 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Tension
Tension is a pulling force transmitted through a stretched rope, string, or similar connector. It acts along the connector away from the object it pulls, and an ideal massless rope has the same tension throughout an uninterrupted segment. For example, a rope supporting an accelerating load can have a tension different from the load’s weight.
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Wave
A wave is a propagating disturbance that can transfer energy without transporting the medium as a whole. Mechanical waves require a material medium, while electromagnetic waves can propagate through vacuum. For example, a pulse travels along a rope while individual rope segments move locally around their equilibrium positions.
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Frequency
Frequency is the number of complete repeating cycles per unit time. It is measured in hertz, meaning cycles per second, and is the reciprocal of the period. For example, a vibrating string with a higher frequency produces more cycles each second.
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Wavelength
Wavelength is the spatial distance between neighboring points at the same phase of a periodic wave. It connects propagation speed and frequency through speed equal to frequency times wavelength. For example, neighboring crests of a sinusoidal water wave are separated by one wavelength.
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Transverse wave
In a transverse wave, the disturbance is perpendicular to the direction of propagation. The disturbance direction and the energy-transfer direction must therefore be distinguished. For example, a rope pulse can travel horizontally while rope segments move up and down.
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Wave polarization
Wave polarization specifies the orientation of a transverse wave’s oscillation. For light, an ideal linear polarizer selects an electric-field component along its transmission direction. For example, crossed ideal linear polarizers block light transmitted through the first polarizer.
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Wave intensity
Wave intensity is the time-averaged energy-transfer rate per area perpendicular to propagation. It is measured in watts per square meter and differs from total power because area matters. For example, spreading the same sound power over a larger area reduces the intensity there.
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Oscillation period
An oscillation period is the time required for one complete repeating cycle. It is the reciprocal of frequency and is measured in seconds. For example, measuring the time for several pendulum swings and dividing by the number of cycles estimates the period.
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Vector component
A vector component describes the part of a vector along a chosen axis. Its signed value changes when the axes change, even though the physical vector remains the same. For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.