MHT-CET Physics · Superposition of Waves

Organ Pipes, the Resonance Tube and the Doppler Effect

An open pipe has antinodes at both ends and sounds every harmonic of v/2L; a closed pipe has a node at the closed end and sounds only the odd harmonics of v/4L; end corrections lengthen the air column, and a moving source or listener shifts the frequency heard.

All Superposition of Waves notes MHT-CET Physics strategy

Why this matters

23 PYQs, five HARD. Three shapes: the frequencies and overtones of open and closed pipes (including a pipe dipped in water, which becomes closed), end corrections and the resonance tube, and the Doppler formula.

Concept 1 of 2: Open and Closed Pipes

An open end is free, so it is an antinode; a closed end is blocked, so it is a node. An open pipe fits half a wavelength and all its multiples; a closed pipe fits only a quarter-wavelength and its odd multiples — so it is an octave lower and missing every even harmonic.

Definition

Open: $n_p = \dfrac{pv}{2L}$ , all harmonics; $p$ th overtone = $(p + 1)$ th harmonic.
Closed: $n = \dfrac{(2p - 1)v}{4L}$ , odd harmonics only; the $p$ th overtone is the $(2p + 1)$ th harmonic. Its fundamental is half the open pipe's of the same length.
A pipe dipped in water becomes closed with the air length above the water: 80% immersed makes $n_2 = \dfrac{v}{4(0.2L)}$ .
End correction $e = 0.6r$ : open pipe length $l + 2e$ , closed $l + e$ . Resonance tube: $l_1 + e = \dfrac{\lambda}{4}$ , $l_2 + e = \dfrac{3\lambda}{4}$ .
A closed pipe's second overtone (5th harmonic) has 3 nodes and 3 antinodes.

Pipe frequencies

n_{\text{open}} = \frac{pv}{2L}, \qquad n_{\text{closed}} = \frac{(2p - 1)v}{4L}

Worked example

A 0.5 m pipe, sound at 340 m/s. Fundamental if open, and the first three frequencies if one end is closed?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 10th May Shift 1 · Q14Easy

Example 1 · Superposition of Waves · Pipes, Resonance Tube, and Doppler Effect

The pipe open at both ends and pipe closed at one end have same length and both are vibrating in fundamental mode. Air column vibrating in open pipe has resonance frequency

n_1

and air column vibrating in closed pipe has resonance frequency

n_2

, then

Numbering a closed pipe's overtones like an open one's

A closed pipe has no even harmonics, so its first overtone is the 3RD harmonic and its second the 5th. Counting 2nd, 3rd as for an open pipe gives the wrong frequency every time.

One end correction for an open pipe

An open pipe has TWO open ends, so its effective length is

l + 2e

; a closed pipe has one,

l + e

. Using

l + e

for an open pipe is the planted wrong wavelength.

Drill 20 more on open and closed pipes

Concept 2 of 2: The Doppler Effect

A listener moving toward the source meets the wave crests sooner; a source moving toward the listener crowds its crests together. Either way the pitch rises. A car sounding its horn at a wall is both: the wall hears a moving source, and the driver hears the echo as a moving listener.

Definition

$n = n_0\dfrac{v \pm v_L}{v \mp v_S}$ : upper signs when they approach each other.
Car at speed $u$ toward a wall, hearing its own echo: $n = n_0\dfrac{v + u}{v - u}$ .
Moving apart: both signs flip, and the pitch falls.

Doppler effect

n = n_0\,\frac{v + v_L}{v - v_S} \quad (\text{approaching})

Worked example

A 340 Hz source moves at 20 m/s toward a stationary listener; sound travels at 340 m/s. Frequency heard?

Practice this conceptself-check · 1 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 15th May Shift 1 · Q4Moderate

Example 2 · Superposition of Waves · Pipes, Resonance Tube, and Doppler Effect

The driver of a car travelling with a speed

V_{1}

m/s towards a wall sounds a siren of frequency

n

Hz. If the velocity of sound in air is

V

m/s, then the frequency of sound reflected from the wall and as heard by the driver, in Hz, is

Putting the source's speed in the numerator

The LISTENER's speed goes on top, the SOURCE's underneath. Approaching means plus on top and minus below — both push the pitch up.

Drill 1 more on the doppler effect

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
$n_{\text{open}} = \frac{pv}{2L}, \qquad n_{\text{closed}} = \frac{(2p - 1)v}{4L}$
The Doppler Effect
Doppler effect
$n = n_0\,\frac{v + v_L}{v - v_S} \quad (\text{approaching})$

Watch out for (3)

Numbering a closed pipe's overtones like an open one's
→ Open and Closed Pipes
One end correction for an open pipe
→ Open and Closed Pipes
Putting the source's speed in the numerator
→ The Doppler Effect

Test yourself on Superposition of Waves

20 past MHT-CET questions from this chapter, timed at 18 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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