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Stage 3 – Page 7 : Standing Waves on Strings & Air Columns (JEE Advanced)
1️⃣ Standing Waves – Basic Concept
A standing wave is formed due to the superposition of two identical waves travelling in opposite directions with same frequency and amplitude.
✔ Energy does not propagate along the medium
✔ Fixed points of zero displacement appear
2️⃣ Nodes & Antinodes
- Node: Point of zero displacement
- Antinode: Point of maximum displacement
Distance between consecutive nodes = λ/2
Distance between node and adjacent antinode = λ/4
3️⃣ Standing Waves on a Stretched String
For a string fixed at both ends:
- Both ends are nodes
- Only certain wavelengths are allowed
Allowed Wavelengths
λn = 2L / n, n = 1, 2, 3, …
Corresponding Frequencies
fn = n × (v / 2L)
👉 Fundamental frequency (n = 1) is called the first harmonic
4️⃣ Speed of Wave on a String
v = √(T / μ)
- T = Tension in the string
- μ = Mass per unit length
✔ Increasing tension increases frequency
✔ Increasing mass density decreases frequency
5️⃣ Standing Waves in Air Columns
Sound waves form standing waves inside pipes due to reflection at boundaries.
6️⃣ Open Pipe (Both Ends Open)
- Both ends act as antinodes
- All harmonics are present
λn = 2L / n
fn = n × (v / 2L)
7️⃣ Closed Pipe (One End Closed)
- Closed end → node
- Open end → antinode
- Only odd harmonics are present
λn = 4L / n, n = 1, 3, 5…
fn = n × (v / 4L)
8️⃣ End Correction (Advanced Concept)
Due to air bulging at open ends, effective length increases:
Leffective = L + 0.6r
⚠ Ignoring end correction leads to wrong answers in JEE Advanced
9️⃣ Comparison Table
| System | Harmonics | Fundamental Frequency |
|---|---|---|
| String (fixed ends) | All | v / 2L |
| Open Pipe | All | v / 2L |
| Closed Pipe | Odd only | v / 4L |
🔟 JEE Advanced Traps
- Confusing node/antinode positions
- Forgetting odd harmonics rule in closed pipe
- Ignoring end correction
- Wrong relation between n and frequency
🎯 Final Takeaway
✔ Standing waves depend on boundary conditions
✔ Harmonics differ for strings, open pipes & closed pipes
✔ Extremely high-weight topic in JEE Advanced
Stage 3 – Page 7 Completed Successfully ✅
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