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Stage 1 – Page 9 : Standing Waves on a String (JEE / Intermediate)
1️⃣ Standing Waves on a Stretched String
When a stretched string is fixed at one or both ends, the incident wave reflects back and forms standing waves.
Important: At a fixed end, displacement is always zero → Node
2️⃣ String Fixed at Both Ends
Both ends are fixed, hence nodes at both ends.
- Allowed vibrations are called normal modes
- Only specific wavelengths are possible
Allowed Wavelengths
L = n(λ / 2) (n = 1,2,3...)
⇒
λₙ = 2L / n
3️⃣ Frequencies of Vibration
Wave speed on string:
v = √(T / μ)
where T = tension in string μ = mass per unit length
Frequency:
fₙ = n(v / 2L)
Fundamental Frequency (n = 1):
f₁ = v / 2L
4️⃣ Harmonics & Overtones
| Mode | Name | Frequency |
|---|---|---|
| n = 1 | Fundamental | f₁ |
| n = 2 | 1st Overtone (2nd Harmonic) | 2f₁ |
| n = 3 | 2nd Overtone (3rd Harmonic) | 3f₁ |
5️⃣ String Fixed at One End & Free at Other
One end is fixed → Node Other end is free → Antinode
Key Rule: Odd harmonics only are present
6️⃣ Allowed Wavelengths (Fixed–Free)
L = (2n − 1)(λ / 4)
⇒
λₙ = 4L / (2n − 1)
(n = 1,2,3...)
7️⃣ Frequencies (Fixed–Free)
fₙ = (2n − 1)(v / 4L)
Fundamental Frequency:
f₁ = v / 4L
8️⃣ Comparison Table (Very Important)
| Property | Fixed–Fixed | Fixed–Free |
|---|---|---|
| Ends | Node–Node | Node–Antinode |
| Harmonics | All | Odd only |
| Fundamental f | v / 2L | v / 4L |
9️⃣ JEE Advanced Tips
- Always identify node/antinode at ends
- Check if odd harmonics only
- Use tension change → affects frequency via √T
🔟 One-Line Memory
Both ends fixed → all harmonics
One end free → odd harmonics
Stage 1 – Page 9 Completed ✅
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