How to draw diagrams for modes of vibrating air column in a tube closed at one end – Smarter Techniques

How to draw diagrams for modes of vibrating air column in a tube closed at one end – Smarter Techniques

“Harmonics in Closed Tube Explained with Diagram”

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1.Introduction:

   Understanding the vibration of an air column can be confusing for many students—especially when it comes to drawing the diagrams. Most of the time, students memorize the shapes without really knowing why they look that way.

But with a smart technique, you can draw these diagrams neatly on your A4 sheet within seconds.

In this blog, I will walk you through:

• What happens when air vibrates in a tube closed at one end
• Why only odd harmonics appear
• How to draw 1st, 3rd and 5th harmonics quickly
• A simple method from my “Smarter Diagram Techniques” series

Let’s begin.

    2. What Happens in a Tube Closed at One End?

    Imagine a tube, one end closed, and the other open, filled with air. Strike a tuning fork and hold it nears the open end, then, the air column inside vibrating, creating “standing waves.” These vibrations happen in specific ways known as “modes.”

Because one end is closed:

• The closed end is always a node (no vibration)
• The open end is always an antinode (maximum vibration)

This simple rule controls the entire pattern.

    3. Why Only Odd Harmonics?

        In a closed pipe, all allowed patterns must satisfy:

• Node at closed end
• Antinode at open end

This happens only in the:

• 1st harmonic (fundamental)
• 3rd harmonic
• 5th harmonic
• And so on…

That’s why a closed tube has only odd harmonics.

      4. Modes of vibration of the air column in the tube closed at one end:

Fundamental frequency↓ |  3 rd  Harmonics frequency↓    ↓   5 th  Harmonics frequency

Fig. A

   An air column of length L is formed in a tube closed at one end. This air is made to vibrate by a tuning fork of frequency n. Then the air column will vibrate in different modes.

   4.1. First mode (Fundamental/First harmonic):

     In this mode, there’s a node (point of no vibration) at the closed end and an antinode (point of maximum vibration) at the open end. Only one-quarter of a wave fits inside the tube for this lowest frequency vibration.

    Let  V be the velocity of sound waves in air and n fundamental frequency with λ wavelength because of one node and one antinode only. The tube length L covers half wavelength.

Thus L = λ/4

The velocity of sound waves in air V = n λ

Putting the value of λ, we get

V = 4nL

Then the fundamental frequency n is given by

n= V /4L

  4.2. Higher Modes (Third and Fifth Harmonics)

      4.2.1. Third harmonic (First overtone):

    Here, three-quarters of a wave fit inside the tube, including two nodes (one at the closed end and one inside the tube) and two antinodes. Let n₁ be third harmonic frequency with λ₁ wavelength

Thus L = 3λ₁/4

The velocity of sound waves in air V = n₁ λ₁

Putting the value of λ₁, we get

V = 4nL/3

Then the third harmonic frequency n₁ is given by

n₁= 3V /4L

 

    4.2,2. Fifth Harmonic (Second Overtone):

    Now, one and one-quarter wavelengths fit inside. Draw three antinodes and two additional nodes between the ends.

   Let n₂ be fifth harmonic frequency with λ₂ wavelength

Thus L = 5λ₂/4

The velocity of sound waves in air V = n₂ λ₂

Putting the value of λ₂, we get

V = 5nL/3

Then the fifth harmonic frequency n₂ is given by

n₂= 5V /4L

Next frequencies are 7,9 …… times of V/4L.

Thus, an air column in a tube closed at one end vibrates only with odd harmonics.

 4.3. Examples

• Bottle / glass bottle (blowing across the top acts like a closed–open tube)

• Tube with one end capped (PVC pipe with one end plugged)

• Didgeridoo

• Clarinet (approximately behaves like a closed–open tube because of the reed end)

• Organ pipe with a stopper (stopped pipe)

  5.  Smarter Technique: How to draw the diagrams easily:

     Students often draw wrong loops or uneven curves.
    Here is a simple method that never fails.

    See the figure Fig. A

✔ Step 1: Draw the Tube

• Draw a long vertical rectangle.
• Mark the closed end with a thick, dark line.
• Leave the top open.

This sets the boundary conditions.

✔ Step 2: Mark Node and Antinode

Always apply the rule:

• Bottom (closed) end → node
• Top (open) end → antinode

Keep this fixed for every pattern.

✔ Step 3: Add Loops Depending on Harmonic

Here is the trick:

One loop is formed by two nodes with antinode between them. The half loop consists of one node and one antinode.

• 1st harmonic → Half loop
• 3rd harmonic → One and half loops
• 5th harmonic → Two and half loops

Each loop represents half a wavelength.

✔️ Step 4: Draw the Wave Smoothly

• Start from a node at the closed end.
• End at an antinode at the open end.
• Use smooth, equal curves for each loop.

This gives a neat, textbook-perfect diagram.

 6. Video support:

  Do you want to know ‘ How to Draw Diagrams for Modes of vibrating Air column in a tube closed at One End?‘ using given guidelines.

Let us see from the following video for an actual smarter method of drawing a diagram.