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Stage 2 – Page 3 : Standing Waves on a String (Harmonics & Modes)
1️⃣ What is a Standing Wave?
A standing wave is formed due to the superposition of two identical waves travelling in opposite directions.
Key Feature:
Energy does not propagate — oscillations remain localized.
2️⃣ Nodes and Antinodes
| Point | Property |
|---|---|
| Node | Zero displacement |
| Antinode | Maximum displacement |
Distance between two consecutive nodes (or antinodes):
λ / 2
3️⃣ String Fixed at Both Ends
This is the most common JEE configuration.
Boundary Condition:
Both ends are nodes.
Allowed Wavelengths
λₙ = 2L / n
Allowed Frequencies
fₙ = n × (v / 2L)
where n = 1, 2, 3, ...
4️⃣ Fundamental Mode (First Harmonic)
- n = 1
- Lowest frequency
- Longest wavelength
f₁ = v / 2L
5️⃣ Higher Harmonics
| Harmonic | Frequency | Nodes |
|---|---|---|
| 1st | f | 2 |
| 2nd | 2f | 3 |
| 3rd | 3f | 4 |
6️⃣ Numerical Example
Question:
A string of length 1 m is fixed at both ends. Wave speed is 100 m/s.
Find fundamental frequency.
Solution:
f₁ = v / 2L = 100 / (2 × 1)
f₁ = 50 Hz
7️⃣ Important Relations for JEE
- fₙ ∝ n
- λₙ ∝ 1/n
- Energy ∝ amplitude²
- Frequency independent of amplitude
8️⃣ Common JEE Traps
- Confusing harmonic number with number of loops
- Forgetting both ends are nodes
- Using v = fλ incorrectly
9️⃣ Fast Problem-Solving Tips
- Draw the wave pattern
- Count loops → harmonic number
- Use ratios instead of absolute values
🔟 Next Page Preview
- String fixed at one end, free at other
- Odd harmonics only
Stage 2 – Page 3 Completed ✅
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