Sound Waves in Open and Closed Organ Pipe WITH Practice Question

7/29/2019 Sound Waves in Open and Closed Organ Pipe WITH Practice Question

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son51.htm

Lesson 51: Closed & Open Ended Pipes

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

This includes brass and woodwind instruments, as well asinstruments like pipe organs.

All instruments like this can be divided into two categories, open

ended or closed ended.

A pipe can be any tube, even if it has been bent into differentshapes or has holes cut into it.

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

An example would be an instrument like a trumpet. You blow in

through one end and the sound comes out the other end of thepipe.

The keys on the trumpet allow the air to move through the "pipe"in different ways so that different notes can be played.

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

An example would be an instrument like some organ pipes

(although in some designs they are open), or a flute.

Although you blow in through the mouth piece of a flute, theopening youre blowing into isnt at the end of the pipe, its alongthe side of the flute. The end of the pipe is closed off near the

mouth piece.

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

The diagrams that I will be drawing are based on the way the air

will move as a wave.
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Although the sound waves actually travel through the pipes aslongitudinal waves, I will be drawing transverse waves. This is justbecause they are easier to draw and recognize in the diagrams.

Identifying Fractions of Wavelengths

Before we look at the diagrams of the pipes, lets make sure that youknow what fractions of a wave look like. This will be important in theway you interpret the diagrams later.

Figure 1: One Wavelength

Figure 2: Three Quarters () of a Wavelength

Figure 3: One Half () of a Wavelength

Figure 4: One Quarter () of a Wavelength

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

When we are talking about the sounds that pipes can make, what

we are really concerned with is how much of the wave we can fitinto the pipe.

Different amounts of a wavelength in a pipe will result in a

different frequency being heard.

Because these are the frequencies of the waves that willnaturally resonate in the pipes, we call them the resonantfrequencies.

In music, you might have heard these referred to as harmonics.

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

If you have ever played a wind instrument, you know that with

the same keys pressed you can play different notes by controllinghow you blow into the instrument the difference between lets say

a low C and a high C note. The lowest note you can play (which is also the smallest part of

the wave that can fit inside the pipe) is usually called thefundamental.

Fitting in more of the wave produces different notes, differentharmonics.


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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 tobe reflected back.

There must also be an antinode where the opening is, since thatis 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 of awavelength.

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 wavelength I know that the length of this tube is of a wavelengththis leads to our first formula:

L =

L is the length of the tube in metres. On its own this formulareally doesnt help us much.

Instead, we have to solve this formula for and then combine it

with the formula v=f 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 its going to be reflected as aninverted wave (going from air to whatever the pipe is made of is apretty big change so this is what we would expect. It would look likeFigure 6.


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

What does the next harmonic look like? Its the 3rd Harmonic.

I know this name might seem a little confusing (Im the first to

agree with you!) but because of the actual notes produced and theway the waves fit in, musicians refer to the next step up in a closedend pipe instrument as the 3rd harmonic there is no such thing asa 2nd harmonic 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 (wherethe air is moving) and a node at the closed end (where the air cantmove). That means for the 3rd harmonic we get something like Figure7.

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

L =

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 youthe formulas for the fundamentals lengths. You need to rememberhow 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 5 th Harmonic (Figure9).


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

L = 5/4

And the note produced by the 5th

Harmonic is found using theformula

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 youre probably thinking that there couldnt possibly be anymore stuff to learn about this, but we still have to do open end pipes.Thankfully, theyre not that hard, and if you got the basics for closedpipes it should go pretty fast for you.

Figure 11: Fundamental

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

Thats why the smallest wave we can fit in is shown in Figure

11. This looks different than the 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 =

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 2nd harmonic.

Yep, open end pipes have a 2nd harmonic they can have any


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number harmonic they want, odd or even.

Again, it kind of looks weird, but trace it out and youll see that

there is exactly one wavelength here.

The length and frequency formulas are

L = 2/2

Figure 14: 2nd Harmonic with Reflection

Im not going to show you what the 3rd harmonic looks like. Instead, trydrawing it yourself and see what you get.

As a hint to help you, the formulas for the length and frequencyare

L = 3/2

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 youve just learned, although if you

wanted you could use v = f )

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 fundamentalnote would it be higher or lower frequency?

a)

L = = 2L = 2(3.6m) = 7.2 m

b)


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

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

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Sound Waves in Open and Closed Organ Pipe WITH Practice Question