How to draw diagrams for modes of vibrating air column in a tube open at both ends? (Smarter Techniques):

How to draw diagrams for modes of vibrating air column in a tube open at both ends? (Smarter Techniques):

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 is difficult for many students — especially when it comes to drawing the diagrams correctly. Most students try to copy the curves from memory without understanding why the shapes look the way they do.

     With a simple, 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 open at both ends
• Why all harmonics (1st, 2nd, 3rd, …) appear
• How to draw the 1st, 2nd and 3rd harmonics quickly
• A simple method from my “Smarter Diagram Techniques” series

Let’s begin.

  2. What happens in a tube open at both ends?

    Consider a tube of length L open at both ends and containing air at room temperature. When a tuning fork of suitable frequency is held near one end, longitudinal waves travel along the air column and are reflected at the ends. At certain frequencies, the incident and reflected waves superpose to form stationary (standing) waves in the air column.

    At an open end of a tube, the air is free to move, so the displacement is maximum and the pressure variation is minimum. Hence each open end is a displacement antinode (A) and a pressure node. Between two successive antinodes there must be at least one displacement node (N). Only those patterns which satisfy “A at both open ends and N–A–N… sequence inside” are possible modes of vibration.

Because both ends are open:

• Each open end is always an antinode (maximum vibration)
• A certain number of nodes (no vibration) form inside the tube

This simple rule controls every standing-wave pattern in an open–open tube.

 3.Why all harmonics occur in an open tube?

   In an open–open pipe, the allowed patterns must satisfy:

• Antinode at the top
• Antinode at the bottom

These conditions are satisfied by:

• 1st harmonic (fundamental)
• 2nd harmonic
• 3rd harmonic
• 4th harmonic…and so on

   That’s why all harmonics are present in a tube open at both ends.

  4.Modes of vibration of the air Column in a tube open at both ends:

1st harmonic ↓ (fundamental)	↓   2nd harmonic	↓   3 rd harmonic

   An air column of length L is formed in the tube. When a tuning fork excites it, the air column vibrates in different modes, as shown in the figure above .

  4.1.  First mode (Fundamental / First Harmonic):

• Antinode (A) at top
• Antinode (A) at bottom
• One node (N) exactly at the centre
• Half a wavelength fits inside the tube

     Let V be the velocity of sound in air. For an open tube, the allowed frequencies form a harmonic    series. Let n be the frequency of 1^st harmonics and λ be the correspond wavelength.

L = λ/2

Velocity of sound:
V = nλ

Substituting:
n = V/2L

This is the lowest  (Fundamental) frequency of the vibrating air column.

  4.2. Second mode (Second Harmonic):

• Antinodes at both open ends
• Two internal nodes
• One complete wavelength fits in the tube
• Let n₁ be the frequency of 2^nd harmonics and λ₁ be the correspond

L = λ₁

Velocity of sound:
V = n₁ λ ₁

Substituting:
n₁ =  2.V/2L

      This is 2nd harmonic frequency of the vibrating air column.

  4.3. Third mode (Third Harmonic):

• Antinodes at both open ends
• Three internal nodes
• One and a half wavelengths fit inside

• Let n₂ be the frequency of 3^rd harmonics and λ₂ be the correspond

L = 3.λ₂ /2

Velocity of sound:
V = n₂ λ₂

Substituting:
n₂ =3.V/2L

  This is 2nd harmonic frequency of the vibrating air column.

 Therefore, the frequencies are:

n =V/2L, n₁ = 2.V/2L, n₂ = 3.V/2L ……..so on

Thus, a tube open at both ends vibrates with all harmonics.

  4.4. Examples

• Flute (embouchure hole acts as an open end)
• Recorder
• Open metal or PVC pipe (both ends not covered)
• Organ pipe without a stopper (open pipe)
• Paper straw
• Wind chimes (hollow tubes open at both ends)

  5. Smarter Technique: How to Draw the Diagrams Easily:

      Students often draw uneven curves or place nodes at the wrong positions.
Here is the simple method that never fails . (See the diagram given above)

✔ Step 1: Draw the Tube

• Draw two long, perfectly parallel vertical lines of length 6 cm.
• Keep both ends open.
• Mark the length L on the left for the first harmonic diagram.

✔ Step 2: Mark Antinodes and Nodes

    Apply the rule for every mode:

• Top → Antinode (A)
• Bottom → Antinode (A)
• Internal nodes (N) depend on the harmonic number.
• For 1st harmonic N at 3 cm.
• For 2 nd harmonic one at 1.5 cm second one at 4.5 cm.
• For3 rd harmonic one at 1cm second one at 3 cm and third at 5 cm.

✔ Step 3: Add Loops According to Harmonic

Each loop represents half a wavelength.

• 1st harmonic → 1 loop (½ λ)
• 2nd harmonic → 2 loops (1 λ)
• 3rd harmonic → 3 loops (1.5 λ)

    Mark A’s at the ends and N’s at equal distances inside.

✔ Step 4: Draw Smooth Curves

• Connect A → N → A with a smooth curve.
• Repeat for each loop.
• Ensure symmetry on both sides of the tube
• This gives you neat, clear, textbook-quality diagrams every single time.

5. Conclusion:

       Understanding the rules of nodes and antinodes makes drawing these diagrams extremely simple. Using the Smarter Diagram Technique, you can draw the 1st, 2nd, 3rd… harmonics of an open–open air column quickly and neatly — exactly as shown in the figure.

 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.