MHT-CET Physics · Sound

Organ Pipes, Resonance and Beats

A pipe open at both ends vibrates at every multiple of v/2L, a pipe closed at one end only at the odd multiples of v/4L; a resonance tube picks out the lengths λ/4, 3λ/4, 5λ/4…, and two close frequencies heard together beat at their difference.

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Why this matters

22 PYQs, 6 of them HARD. Fourteen compare pipes — fundamental and overtones of open and closed pipes, overtones matched between two pipes, end corrections, the harmonic a pipe picks out. Eight are resonance tubes, strings and beats: water poured into a tube, a string resonating with a pipe, sets of tuning forks, and a wire's tension changed without changing the beats. Two cards.

Concept 1 of 2: Open and Closed Pipes

An open pipe has antinodes at both ends and fits any whole number of half wavelengths: fₙ = nv/2L, all harmonics, the fundamental twice a same-length closed pipe's. A closed pipe has a node at the closed end and fits odd quarter wavelengths: f = (2n − 1)v/4L, odd harmonics only. Count overtones carefully: the pth overtone of an open pipe is harmonic p + 1; of a closed pipe, harmonic 2p + 1. Matching two pipes means setting these equal. End correction adds about 0.6r per open end, so it grows with the pipe's width. A tube dipped so only a quarter of it is above water becomes a closed pipe of that quarter length.

Definition

Open: $f_n = \dfrac{nv}{2L}$ , all harmonics; pth overtone = harmonic $p + 1$ .
Closed: $f = \dfrac{(2n - 1)v}{4L}$ , odd harmonics; pth overtone = harmonic $2p + 1$ .
Same length: $f_{\text{open}} = 2f_{\text{closed}}$ . First overtones equal ⇒ $L_c : L_o = 3 : 4$ .
End correction ≈ 0.6r per open end: open pipe $f = \dfrac{v}{2(l + 1.2r)}$ .
Closed pipe, second overtone (5th harmonic): 3 nodes and 3 antinodes.
Two open pipes joined: $\dfrac{1}{f} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$ .

Pipe frequencies

f_{\text{open}} = \frac{nv}{2L}, \qquad f_{\text{closed}} = \frac{(2n - 1)v}{4L}

Worked example

The first overtone of a closed pipe equals the fundamental of an open pipe 50 cm long. Length of the closed pipe?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 11th May Shift 2 · Q44Hard

Example 1 · Sound · Pipes, Resonance, Overtones, and Beats

The frequency of the third overtone of a pipe of length '

L_c

', closed at one end is same as the frequency of the sixth overtone of a pipe of length '

L_0

', open at both ends. Then the ratio

L_c:L_0

Numbering overtones as harmonics

The first overtone is the SECOND harmonic of an open pipe but the THIRD of a closed pipe. Convert every overtone to its harmonic before setting frequencies equal.

Allowing even harmonics in a closed pipe

A closed pipe has only odd harmonics: v/4L, 3v/4L, 5v/4L. There is no 2v/4L.

Adding one end correction to an open pipe

An open pipe has two open ends, so its effective length is l + 1.2r (0.6r at each end); a closed pipe adds only 0.6r.

Drill 13 more on open and closed pipes

Concept 2 of 2: Resonance Tubes, Strings and Beats

A resonance tube is a closed pipe whose length you set by pouring in water: it resonates when the air column is λ/4, 3λ/4, 5λ/4…; the least water needed is the tube height minus the longest resonant length that fits. A string of length l has frequencies (n/2l)√(T/μ), where μ is mass per unit LENGTH, so the mass is μl. Its frequency goes as √T and, for wires of one material, as 1/(r l). Two frequencies f₁ and f₂ together beat |f₁ − f₂| times a second. If the beat count stays the same while the string's frequency moves from one side of the fork to the other, the fork lies exactly between them.

Definition

Resonance tube: air column $\dfrac{(2n - 1)\lambda}{4}$ ; least water = height − longest fitting column (λ = 1 m in 1.5 m ⇒ 25 cm).
String: $f_n = \dfrac{n}{2l}\sqrt{\dfrac{T}{\mu}}$ , mass = μl; same material: $f \propto \dfrac{\sqrt{T}}{rl}$ .
Beats = $|f_1 - f_2|$ ; forks rising by x each: last = first + (N − 1)x.
Same beats at T₁ and T₂ ⇒ fork between: $\dfrac{f - b}{f + b} = \sqrt{\dfrac{T_1}{T_2}}$ (225 N, 256 N, 6 beats ⇒ 186 Hz).

Strings and beats

f_n = \frac{n}{2l}\sqrt{\frac{T}{\mu}}, \qquad f_{\text{beat}} = |f_1 - f_2|

Worked example

A 340 Hz fork is held over a 1 m tube (v = 340 m/s). Heights of water at which resonance occurs?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 2nd May Shift 1 · Q26Hard

Example 2 · Sound · Pipes, Resonance, Overtones, and Beats

A wire under tension 225 N produces 6 beats per second when it is tuned with a fork. When the tension changes to 256 N, it is again tuned with the same tuning fork, the number of beats remain unchanged. The frequency of tuning fork will be

Reporting mass per unit length as the mass

√(T/μ) gives μ in kg/m. A 0.5 m string with μ = 0.02 kg/m has mass 10 g, not 20 g.

Adding frequencies to get beats

Two notes beat at the DIFFERENCE of their frequencies. 256 Hz and 260 Hz give 4 beats a second.

Taking the first resonance for the least water

The least water gives the LONGEST air column that still resonates. In a 1.5 m tube with λ = 1 m, that is 1.25 m of air and 25 cm of water.

Drill 7 more on resonance tubes, strings and beats

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Open and Closed Pipes
Pipe frequencies
$f_{\text{open}} = \frac{nv}{2L}, \qquad f_{\text{closed}} = \frac{(2n - 1)v}{4L}$
Resonance Tubes, Strings and Beats
Strings and beats
$f_n = \frac{n}{2l}\sqrt{\frac{T}{\mu}}, \qquad f_{\text{beat}} = |f_1 - f_2|$

Watch out for (6)

Numbering overtones as harmonics
→ Open and Closed Pipes
Allowing even harmonics in a closed pipe
→ Open and Closed Pipes
Adding one end correction to an open pipe
→ Open and Closed Pipes
Reporting mass per unit length as the mass
→ Resonance Tubes, Strings and Beats
Adding frequencies to get beats
→ Resonance Tubes, Strings and Beats
Taking the first resonance for the least water
→ Resonance Tubes, Strings and Beats

Test yourself on Sound

15 past MHT-CET questions from this chapter, timed at 14 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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