Chapter 2. Wave Dynamics - Propagation of Waves
Recommended Article : 【Physics】 Physics Index
4. Interference
1. Reflection of Waves
⑴ Fixed End Reflection
① When a wave is reflected from a denser medium to a rarer medium
② Phase of the wave changes by π
⑵ Free End Reflection
① When a wave is reflected from a rarer medium to a denser medium
② Phase of the wave remains unchanged
2. Refraction of Waves
⑴ Refraction of Surface Waves
① Thought Experiment: If we consider a wave as a line segment with width, where one end is faster and the other end is slower, the slower end bends towards the slower side.
② Conclusion: Waves refract towards the slower side.
① Light has thickness, and considering reinforcement interference, the above thought experiment is valid.
② Conclusion: Regardless of elastic waves or particle waves, waves refract towards the slower side.
⑶ Example 1: Waves Approaching a Coastline
① Waves progressing towards concave portions of the coastline move faster due to the greater depth of water compared to the protruding parts.
② Consequently, as waves approach the coastline, their crests become more parallel to the shore.
⑷ Example 2: Refraction of Day and Night Sounds
① The speed of sound increases with higher air temperatures.
② During the day, sound waves bend upwards, while during the night, they bend downwards.
⑸ Example 3: Speed of Sound in Seawater
Figure 1. Deep-sea Sound Velocity Profile
① Case 1: Depth ＜ 1,000 m: Sound refracts downward.
② Case 2: Depth ＞ 1,000 m: Sound refracts upward.
③ Sound Channel: Exists around 1,000 m.
○ Sound propagates directionally in the direction parallel to the sea surface at around 1,000 m, creating a sound channel.
○ Utilized by whales and submarines.
⑹ Example 4: Light Refraction by Water - Objects appear shallower than their actual depth in water.
⑺ Example 5: Light Refraction by Lenses
⑻ Example 6: Light Refraction by the Atmosphere
① Mirage
○ The speed of light decreases with higher air temperatures.
○ Cold air above and hot air below: Light from the sky is bent downward when passing through hot air layers.
○ Cold air below and hot air above: Light traveling upward is bent downward, creating the illusion of floating objects.
② Phenomenon of the Sun being visible below the horizon during sunrise and sunset.
③ Phenomenon of the Sun appearing elliptical: The Sun near the horizon appears elliptical.
④ Mirage
⑤ Twinkling stars
3. Principle of Superposition
⑴ Principle of Superposition
① Principle 1: F(x + y) = F(x) + F(y)
② Principle 2: F(ax) = aF(x)
③ Examples of F: ax, d/dx, ∫ dx
⑵ Superposition of Wave Phenomena
① Wave Equation
② Superposition of Waves: If φ1 and φ2 are solutions to the wave equation, then φ = aφ1 + bφ2 is also a solution.
4. Interference
⑴ Overview
① Definition: Phenomenon reflecting the principle of superposition.
② Type 1: Constructive Interference: When two waves with different phases overlap in a way that their amplitudes add up.
③ Type 2: Destructive Interference: When two waves with different phases overlap in a way that their amplitudes cancel out.
⑵ Mathematical Representation
① Euler’s Representation
○ A: Amplitude
○ eiφ: Phase
② Superposition of Waves
③ Interference of Waves
○ Interference generally refers to a situation where A1 ≃ A2 = A.
○ Wave function Ψ
○ Amplitude: 2A cos((φ1 - φ2) ÷ 2)
○ Phase term: exp(i(φ1 + φ2) ÷ 2)
○ Interference term: cos((φ1 - φ2) ÷ 2)
○ 0 ≤ ψ ≤ 2A
○ Constructive Interference: Δx = nλ
○ Destructive Interference: Δx = (n + ½) λ
5. Standing Waves
⑴ Overview
① Definition: Two continuous waves with different velocities along a straight line appear stationary.
② Nodes: Points of no vibration in a standing wave.
③ Antinodes: Points of maximum amplitude in a standing wave.
⑵ Mathematical Representation
⑶ Case 1: Closed-End Standing Waves
① Nodes form at both tied ends.
② Depending on the mode n, the wavelength of standing waves formed in the string of length ℓ is 2ℓ / n.
③ Pythagorean Scale: Pythagoras used irrational numbers to create the scale.
○ Principle 1: Perfect Fifth Relation: Sol is 2/3 the length of Do.
○ Principle 2: Octave: A higher Do is 1/2 the length of the lower Do.
○ These principles can be used to create the entire scale.
○ These irrational numbers are sometimes approximated as fractions.
⑷ Case 2: Open-End Standing Waves
① Antinodes form at the ends of an open pipe.
② Depending on the mode n, the wavelength of standing waves formed in a pipe of length ℓ is 2ℓ / n.
⑸ Case 3: Closed-End Standing Waves
① In a closed pipe with one end closed and one end open, the closed end forms a node and the open end forms an antinode.
② Depending on the mode n, the wavelength of standing waves formed in a pipe of length ℓ is 4ℓ / (2n - 1).
Figure 2. Open-End and Closed-End Standing Waves
6. Doppler Effect
⑴ Formulation
① vs: Speed of the source (e.g., sound source)
② vd: Speed of the detector
③ v0: Speed of the wave (e.g., sound wave)
④ f0: Frequency of the wave
⑤ f: Observed frequency
⑥ (Note) Use intuition to determine the signs.
⑵ Moving Source: Measured wavelength changes, measured speed is constant.
⑶ Moving Detector: Measured wavelength remains constant, measured speed changes.
Input : 2019.05.03 20:05