AP Physics 2- 14.3 Boundary Behavior of Waves and Polarization- Study Notes- New Syllabus
AP Physics 2- 14.3 Boundary Behavior of Waves and Polarization – Study Notes
AP Physics 2- 14.3 Boundary Behavior of Waves and Polarization – Study Notes – per latest Syllabus.
Key Concepts:
- Interaction Between a Wave and a Boundary
- Polarization of a Wave
Interaction Between a Wave and a Boundary
When a wave pulse or a periodic wave reaches a boundary, part of the wave energy may be reflected, transmitted, or both. The nature of the interaction depends on the properties of the boundary and the medium beyond it.
Key Concepts:
Fixed Boundary:
- The wave pulse is inverted upon reflection.
- The reflected wave has the same speed, frequency, and wavelength as the incident wave, but opposite amplitude direction.
Free Boundary:
- The wave pulse reflects without inversion.
- The reflected wave has the same speed, frequency, and wavelength as the incident wave.
Boundary Between Two Media:
Part of the wave is transmitted into the second medium, and part is reflected. The transmitted wave speed depends on the medium’s properties:
$ \mathrm{v = \sqrt{\dfrac{F_T}{\mu}}} $ where \(\mathrm{F_T}\) is string tension, and \(\mathrm{\mu}\) is mass per unit length.
- If the second medium is denser (larger \(\mu\)), the transmitted wave slows down and its wavelength decreases.
- If the second medium is less dense (smaller \(\mu\)), the transmitted wave speeds up and its wavelength increases.
Energy Considerations:
- Total energy is shared between reflected and transmitted waves.
- Amplitude decreases if part of the energy is transmitted.
Example:
A wave pulse travels along a string and hits a fixed boundary. Describe the characteristics of the reflected pulse.
▶️ Answer/Explanation
The pulse will be inverted (upward becomes downward), reflected back with the same speed, wavelength, and frequency. Amplitude remains the same in magnitude but opposite in direction.
Example :
A wave on a light string (\(\mathrm{\mu_1 = 0.01 \, kg/m}\)) encounters a boundary with a heavier string (\(\mathrm{\mu_2 = 0.04 \, kg/m}\)). If the tension is \(\mathrm{F_T = 100 \, N}\), find the speed of the wave in both media and predict what happens.
▶️ Answer/Explanation
Step 1: Speed in the first medium: $ \mathrm{v_1 = \sqrt{\dfrac{F_T}{\mu_1}} = \sqrt{\dfrac{100}{0.01}} = 100 \, m/s} $
Step 2: Speed in the second medium: $ \mathrm{v_2 = \sqrt{\dfrac{F_T}{\mu_2}} = \sqrt{\dfrac{100}{0.04}} = 50 \, m/s} $
Step 3: Since the second medium is denser, the transmitted wave slows down and has shorter wavelength, while the reflected wave is inverted.
Polarization of a Wave
Polarization is the restriction of the vibrations of a transverse wave to a single direction. Only transverse waves (such as electromagnetic waves) can be polarized, since their oscillations are perpendicular to the direction of wave propagation.
Unpolarized Wave: A wave in which the oscillations occur randomly in all perpendicular directions to the propagation.
Linearly Polarized Wave: A wave in which vibrations are restricted to one plane perpendicular to the propagation direction.
Methods of Polarization:
- Polarizing Filter: Allows oscillations in only one direction to pass.
- Reflection: Light reflecting off certain surfaces (like water or glass) becomes partially polarized.
- Scattering: Molecules scatter light, producing partial polarization (e.g., blue sky polarization).
Malus’s Law:
If polarized light of intensity \(\mathrm{I_0}\) passes through a polarizer at angle \(\mathrm{\theta}\) relative to the polarization direction, the transmitted intensity is:
$ \mathrm{I = I_0 \cos^2 \theta} $
Applications:
- Polarized sunglasses reduce glare by blocking horizontally polarized light.
- Stress analysis in materials using polarized light.
- 3D movie glasses use perpendicular polarizations for each eye.
Example :
Unpolarized light of intensity \(\mathrm{I_0}\) passes through a single polarizer. What is the transmitted intensity?
▶️ Answer/Explanation
Unpolarized light has equal vibration components in all perpendicular directions. A polarizer passes only half of the intensity.
\(\mathrm{I = \dfrac{I_0}{2}}\)
Final Answer: Transmitted intensity is half the incident intensity.
Example :
Polarized light of intensity \(\mathrm{20 \, W/m^2}\) passes through a polarizer oriented at \(\mathrm{60^\circ}\) relative to its polarization axis. Find the transmitted intensity.
▶️ Answer/Explanation
Step 1: Apply Malus’s Law: \(\mathrm{I = I_0 \cos^2 \theta}\)
Step 2: \(\mathrm{I = (20)(\cos^2 60^\circ) = 20 (0.25)}\)
Step 3: \(\mathrm{I = 5 \, W/m^2}\)
Final Answer: Transmitted intensity = \(\mathrm{5 \, W/m^2}\).