16.3 Content
16.3.1 Reflection of Waves
When a wave reaches a boundary, some or all of it reflects back.
Fixed boundary (hard reflection): - Wave inverts upon reflection - Crest becomes trough, trough becomes crest - Examples: String fixed to a wall, sound reflecting from a wall
Free boundary (soft reflection): - Wave reflects without inversion - Crest remains crest, trough remains trough - Examples: String with a loose end, open pipe end
16.3.2 Interactive: Wave Reflection
A pulse approaching a boundary:
16.3.3 Refraction of Waves
Refraction occurs when a wave changes speed as it enters a new medium. The frequency stays constant, but wavelength changes.
Key points: - Wave bends toward the normal when slowing down - Wave bends away from the normal when speeding up - Frequency is preserved across the boundary
16.3.4 Superposition Principle
When two or more waves overlap, the principle of superposition states:
The resultant displacement at any point is the sum of the individual displacements.
\[y_{total} = y_1 + y_2 + y_3 + ...\]
This leads to interference patterns.
16.3.5 Interactive: Constructive Interference
Two waves in phase combine to produce a larger amplitude:
Constructive interference: Waves in phase (\(\Delta\phi = 0\)) → amplitudes add → larger resultant.
16.3.6 Interactive: Destructive Interference
Two waves out of phase (180°) combine to cancel:
Destructive interference: Waves 180° out of phase (\(\Delta\phi = \pi\)) → amplitudes subtract → smaller or zero resultant.
16.3.7 Types of Interference
| Condition | Phase Difference | Path Difference | Result |
|---|---|---|---|
| Constructive | \(0, 2\pi, 4\pi, ...\) | \(0, \lambda, 2\lambda, ...\) | Maximum amplitude |
| Destructive | \(\pi, 3\pi, 5\pi, ...\) | \(\lambda/2, 3\lambda/2, ...\) | Minimum amplitude |
16.3.8 Standing Waves
Standing waves form when two identical waves travel in opposite directions and interfere.
Key features: - Nodes: Points of zero displacement (destructive interference) - Antinodes: Points of maximum displacement (constructive interference) - The pattern appears stationary, but energy oscillates between nodes
The distance between adjacent nodes = \(\lambda/2\)
The distance between adjacent antinodes = \(\lambda/2\)
16.3.9 Interactive: Standing Wave Pattern
16.3.10 Standing Waves on Strings
For a string fixed at both ends, only certain wavelengths produce standing waves:
| Harmonic | Loops | Wavelength | Frequency |
|---|---|---|---|
| 1st (fundamental) | 1 | \(\lambda_1 = 2L\) | \(f_1\) |
| 2nd | 2 | \(\lambda_2 = L\) | \(f_2 = 2f_1\) |
| 3rd | 3 | \(\lambda_3 = 2L/3\) | \(f_3 = 3f_1\) |
| nth | n | \(\lambda_n = 2L/n\) | \(f_n = nf_1\) |