Lesson 51: Closed & Open Ended Pipes

Many musical instruments depend on the musician in some way moving air through the instrument.

An open ended instrument has both ends open to the air.

A closed ended instrument has one end closed off, and the other end open.

The frequencies of sounds made by these two types of instruments are different because of the different ways that air will move at a closed or open end of the pipe.

Identifying Fractions of Wavelengths

Before we look at the diagrams of the pipes, let’s make sure that you know what fractions of a wave look like. This will be important in the way you interpret the diagrams later.

Figure 1: One Wavelength
Figure 2: Three Quarters (¾) of a Wavelength
Figure 3: One Half (½) of a Wavelength
Figure 4: One Quarter (¼) of a Wavelength

These wave fractions might appear upside down, flipped over, turned around, etc., but they will still represent portions of a wave.

Although the actual length of the pipe remains the same, different notes are played.

Closed Ended Pipes

Figure 5: The Fundamental

Remember that it is actually air that is doing the vibrating as a wave here.

Since the length of the tube is the same as the length of the ¼ wavelength I know that the length of this tube is ¼ of a wavelength… this leads to our first formula:

L = ¼ λ

f = frequency of sound (Hz)
v = velocity of sound in air (m/s)
L = length of tube (m)

Figure 6: Fundamental with Reflection

When the wave reaches the closed end it’s going to be reflected as an inverted wave (going from air to whatever the pipe is made of is a pretty big change so this is what we would expect. It would look like Figure 6.

What does the next harmonic look like? It’s the 3rd Harmonic.

Figure 7: Third Harmonic

Remember that we have to have an antinode at the opening (where the air is moving) and a node at the closed end (where the air can’t move). That means for the 3rd harmonic we get something like Figure 7.

This is ¾ of a wavelength fit into the tube, so the length of the tube is…

L = ¾ λ

If we drew in the reflection of the third harmonic it would look like Figure 8.

Figure 8: Third Harmonic with Reflection
Figure 9: The Fifth Harmonic

One more to make sure you see the pattern. The 5th Harmonic (Figure 9).

L = 5/4 λ

And the note produced by the 5th Harmonic is found using the formula…

Figure 10 shows the reflection of a 5th Harmonic for a closed end pipe.

Figure 10: The Fifth Harmonic with Reflection

Open End Pipes

I know you’re probably thinking that there couldn’t possibly be any more stuff to learn about this, but we still have to do open end pipes. Thankfully, they’re not that hard, and if you got the basics for closed pipes it should go pretty fast for you.

Figure 11: Fundamental
Figure 12: Fundamental with Reflection
Figure 13: 2nd Harmonic

The next note we can play is the 2nd harmonic.

L = 2/2 λ

Figure 14: 2nd Harmonic with Reflection

I’m not going to show you what the 3rd harmonic looks like. Instead, try drawing it yourself and see what you get.

L = 3/2 λ

Example 1: An open ended organ pipe is 3.6m long.

a) What is the wavelength of the fundamental played by this pipe?

b) What is the frequency of this note if the speed of sound is 346m/s? (Calculate it using the formulas you’ve just learned, although if you wanted you could use v = f λ)

c) What note could be played as the third harmonic on that pipe?

d) If we made the pipe longer, what would happen to the fundamental note… would it be higher or lower frequency?


L = ½ λ
λ = 2L
λ = 2(3.6m)
λ = 7.2 m


c) Notice that the third harmonic is three times bigger than the first harmonic.

d) If we made the pipe longer, the wavelength would be bigger (just look at the formula in part "a" of this example), and since wavelength and frequency are inversely related, that means the frequency would be smaller.