NEB Class 12 Physics Wave in pipes and strings Notes in PDF Complete Handwritten. Physics Notes 2081: All Chapters | New Curriculum | Class 12 Physics Notes download.
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NEB Class 12 Physics Wave in pipes and strings Note Handwritten in PDF
Stationary Waves in a Closed Organ Pipe
A closed pipe has one end closed (node) and one end open (antinode).
Condition for modes: L = (2n−1)λ/4 where n = 1, 2, 3…
- 1st harmonic (Fundamental): L = λ/4 → f₁ = v/4L
- 2nd mode (1st overtone): L = 3λ/4 → f = 3v/4L = 3f₁
- 3rd mode (2nd overtone): L = 5λ/4 → f = 5v/4L = 5f₁
Only odd harmonics are produced. Missing: 2f₁, 4f₁, 6f₁…
Stationary Waves in an Open Organ Pipe
An open pipe has antinodes at both ends.
Condition for modes: L = nλ/2 where n = 1, 2, 3…
- 1st harmonic (Fundamental): L = λ/2 → f₁ = v/2L
- 2nd harmonic (1st overtone): L = λ → f = 2v/2L = 2f₁
- 3rd harmonic (2nd overtone): L = 3λ/2 → f = 3f₁
All harmonics are present (both even and odd).
Comparing open and closed pipes of same length L:
- f₁(open) = v/2L
- f₁(closed) = v/4L
- f₁(open) = 2 × f₁(closed)
Open pipe produces a richer, fuller tone due to all harmonics present.
End Correction
The antinode at an open end forms slightly outside the pipe:
- End correction per open end: e ≈ 0.6r (r = pipe radius)
- Open pipe effective length: L_eff = L + 2e
- Closed pipe effective length: L_eff = L + e
Including end correction:
- Open pipe: f = v / 2(L + 2e)
- Closed pipe: f = v / 4(L + e)
Velocity of Transverse Waves in a String
For a stretched string under tension T with linear mass density μ (mass per unit length):
v = √(T/μ)
Frequency of Vibrating String — Harmonics
For a string of length L fixed at both ends:
- Both ends must be nodes
- fₙ = n/(2L) × √(T/μ) for the nth harmonic
- Fundamental (1st harmonic): f₁ = 1/(2L) × √(T/μ)
- 2nd harmonic: f₂ = 2f₁
- nth harmonic: fₙ = nf₁
All harmonics are present (like open pipe).
Laws of Vibration of a Fixed String
Three laws derived from fₙ = n/(2L) × √(T/μ):
1. Law of Length: f ∝ 1/L (at constant T and μ) → Shorter string → higher frequency → higher pitch → This is why pressing a guitar fret shortens the string and raises pitch
2. Law of Tension: f ∝ √T (at constant L and μ) → Tighter string → higher frequency → Tuning pegs on guitar change T to tune the pitch
3. Law of Mass (Linear density): f ∝ 1/√μ (at constant L and T) → Thicker string → higher μ → lower frequency → lower pitch → Bass strings are thicker than treble strings
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Frequently Asked Questions
What is the difference between open and closed organ pipe?
Closed pipe: one end is node, open end is antinode — only odd harmonics (f₁, 3f₁, 5f₁). Fundamental: f₁ = v/4L. Open pipe: both ends are antinodes — all harmonics present (f₁, 2f₁, 3f₁). Fundamental: f₁ = v/2L. For same length, open pipe fundamental is twice that of closed pipe.
What are harmonics and overtones in Class 12 Physics?
Harmonics are integer multiples of fundamental frequency f₁. 1st harmonic = f₁ (fundamental), 2nd harmonic = 2f₁, 3rd = 3f₁. Overtones are harmonics above the fundamental — 1st overtone = 2nd harmonic = 2f₁ for open pipe. In closed pipe, 1st overtone = 3rd harmonic = 3f₁ (skips even harmonics).
What is end correction in organ pipes Class 12?
The antinode at an open pipe end forms slightly outside the pipe. End correction per open end: e ≈ 0.6r where r is pipe radius. Effective length of open pipe: L + 2e. Effective length of closed pipe: L + e. Including end correction gives accurate frequency calculations and is tested in NEB numericals.
What is the formula for frequency of vibrating string?
Frequency of nth harmonic of stretched string: fₙ = n/2L × √(T/μ). Where L is length, T is tension, and μ is mass per unit length. Fundamental frequency (n=1): f₁ = 1/2L × √(T/μ). All harmonics are present in a string fixed at both ends — both odd and even harmonics exist.
What are the laws of vibration of string Class 12?
Three laws: Law of length — f ∝ 1/L at constant T and μ (shorter string gives higher pitch). Law of tension — f ∝ √T (tighter string gives higher pitch). Law of mass — f ∝ 1/√μ (thicker string gives lower pitch). These three laws explain how all stringed instruments are tuned and played.
What is the velocity of transverse wave in string?
Velocity of transverse wave in stretched string: v = √(T/μ). Where T is tension in Newtons and μ is linear mass density in kg/m. Higher tension gives faster wave. Higher mass per unit length gives slower wave. This is the fundamental formula for all string vibration problems in NEB Physics.
Why does a closed pipe produce only odd harmonics?
In a closed pipe the closed end must always be a node and the open end an antinode. This means only odd numbers of quarter wavelengths fit: L = λ/4, 3λ/4, 5λ/4. This restricts frequencies to f₁, 3f₁, 5f₁ — only odd harmonics. Even harmonics would require an antinode at the closed end which is impossible.
Which topics from waves in pipes are most important for Exam?
Most important: comparison of open and closed pipes with harmonics (4 marks), frequency formula for vibrating string (2 marks), three laws of string vibration (2 marks), end correction numericals (2 marks). Questions comparing harmonics of open and closed pipes appear in nearly every NEB board paper.