Question

What harmonics are present in open pipe and closed pipe

Answer 1

Closed Ended Pipes



Figure 5: The Fundamental

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

The air at the closed end of the pipe must be a node (not moving), since the air is not free to move there and must be able to be reflected back.

There must also be an antinode where the opening is, since that is where there is maximum movement of the air.

The simplest, smallest wave that I can possibly fit in a closed end pipe is shown in Figure 5.

Notice how even though it has been flipped left-to-right and it looks squished and stretched a bit to fit, this is still ¼ of a wavelength.

Since this is the smallest stable piece of a wave I can fit in this pipe, this is the Fundamental, or 1st Harmonic.

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

“L” is the length of the tube in metres. On it’s own this formula really doesn’t help us much.

Instead, we have to solve this formula for λ and then combine it with the formula v=fλ to get a more useful formula:



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.

This does not change the length of the wave in our formula, since we are only seeing the reflection of the wave that already exists in the pipe.

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

I know this name might seem a little confusing (I’m the first to agree with you!) but because of the actual notes produced and the way the waves fit in, musicians refer to the next step up in a closed end pipe instrument as the 3rdharmonic… there is no such thing as a 2ndharmonic for closed end pipes.

In fact, all of the harmonics in closed end pipes are going to be odd numbers.



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

This is the third harmonic of the closed end pipe. The formula for the frequency of the note we will hear is…



Do you notice a pattern forming in the formulas? Hopefully, because for both open and closed end pipes, we will only give you the formulas for the fundamentals lengths. You need to remember how to get the rest.

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).

There is one full wavelength in there (4/4) plus an extra ¼ of a wavelength for a total of 5/4. The length of the pipe is…

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 Harmonicfor 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

The fundamental(first harmonic) for an open end pipe needs to be an antinode at both ends, since the air can move at both ends.

That’s why the smallest wave we can fit in is shown in Figure 11.

This looks different than the ½ wavelength that I showed you in Figure 3, but it is still half of a full wavelength.

That means the length of the tube and frequency formula are…

L = ½ λ



The whole thing after it reflects at the other end looks like Figure 12.



Figure 12: Fundamental with Reflection



Figure 13: 2nd Harmonic

The next note we can play is the 2ndharmonic.

Yep, open end pipes have a 2ndharmonic… they can have any number harmonic they want, odd or even.

Again, it kind of looks weird, but trace it out and you’ll see that there is exactly one wavelength here.

The length and frequency formulas are…

L = 2/2 λ





Figure 14: 2nd Harmonic with Reflection

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

As a hint to help you, the formulas for the length and frequency are…

L = 3/2 λ

please make brainlist

Answer 2

Even as well as odd and odd harmonics are present in opened pipe and closed pipe respectively.

Closed pipe:

The air is blown in the open end of the closed pipe.

Antinode is present in the open end and node is present in the closed end.

The length of the pipe is given as:

l = λ₁/4 or λ₁ = 4l

The fundamental frequency is given as:

n₁ = v/λ₁ = v/4l

The frequency of the overtone is considered in harmonics which occurs when the air is blown strongly.

l = 3λ₃/4 or λ₃ = 4l/3

n₃ = v/λ₃ = 3v/4l = 3n₁

This is third harmonic or first overtone.

l = 5λ₅/4 or λ₅ = 4l/5

n₅ = v/λ₅ = 5v/4l = 5n₁

This is fifth harmonic or second overtone.

The ratio of frequency of harmonics is:

n₁ : 3n₁ : 5n₁ ⇒ 1 : 3 : 5

Since, all the numbers are odd number, this harmonics is odd harmonics.

Opened pipe:

The air is blown in the closed pipe.

The length of the pipe is given as:

l = λ₁/2 or λ₁ = 2l

n₁ = v/2l

Similarly, next node is:

l = 2λ₂/2 or l = λ₂

v = n₂λ₂ = n₂l

n₂ = v/l = 2n₁

Thus, above obtained is first overtone.

Similarly, next node is:

l = 3λ₃/2 or λ₃ = 2l/3

v = n₃λ₃ ⇒ n₃ = v/λ3 ⇒ v = n₃(2l/3)

n₃ = 3v/2l = 3n₁

Thus, above obtained is second overtone.

The ratio of frequency of harmonics is:

n₁ : 2n₁ : 3n₁ ⇒ 1 : 2 : 3

Since, the ratio has natural numbers, this harmonics is even as well as odd harmonics.