The Logic Off Topic list

HI YA PEOPLE,

Hope you all well and being positive as all ways.

Does any one know how to calculate the harmonic series for an open pipe 
length 0.5m (ignoring end correction).

and the calculate the same if the pipe were closed at o ne end?

Does any one have any advice for me with this please?

Love

Debbie xx


[Non-text portions of this message have been removed]

Thoughts from the mind of DbbBrook@..., 13-11-2001:

>Does any one know how to calculate the harmonic series for an open pipe
>length 0.5m (ignoring end correction).
>
>and the calculate the same if the pipe were closed at one end?
>
>Does any one have any advice for me with this please?

I don't know what you already know about this stuff, so I'll assume a 
kind of worst case scenario :-).

If a pipe is closed (the simpler case), the wave it produces cannot 
"move" at the ends of the pipe.  Imagine the wave as a piece of rope 
being glued to the ends of the pipe.  That means that any wave 
"fitting" in the length of the pipe will have to have zero-crossing 
points at the ends.  I.e. you could have this (crude ascii drawing 
follows, please view in Courier)

  <------ pipe length ------->
  +----------######----------+
  |      ####      ####      |
  |   ###              ###   |
  |###                    ###|
  |                          |
  |                          |
  +--------------------------+

I.e. half a wave fitting in the pipe.
Note that these pictures are in fact freeze-frames of a dynamic 
movement -- a so-called standing wave.  Using the rope analogy, the 
above wave would move like the cord used in the childs-play "rope 
jumping" (or whatever it's called in English -- with the two kids 
rotating a rope and a 3rd jumping in the middle).

Obviously an entire wave (two "humps") of double frequency (half 
wave-length) would also fit:

  +-----###------------------+
  |   ##   ##                |
  | ##       ##              |
  |#           ##           #|
  |              ##       ## |
  |                ##   ##   |
  +----------------- ###-----+

Ditto for 1.5 waves (3 humps).  Etc.

Now if I remember my physics correctly, you calculate frequency as 
"speed of sound / wavelength", where speed is about 300 m/s (right?).

So with a 0.5 m closed pipe, the first wave fitting would be a 1 m 
wave (since half! the wave is 0.5 m).  So that's a 300/1 = 300 Hz 
tone.  The next wave fitting in the pipe (i.e. the first overtone) 
has a wavelength of 0.5 m (the entire wave fits in the pipe now). 
Frequency = 300/0.5 = 600 Hz.  Next one: 1.5 waves fit in 0.5 m, so 
wavelength is 2/3 * 0.5 = 1/3, and frequency = 300 / (1/3) = 900 Hz.
Clearly all waves fitting in the pipe form the harmonic series 300, 
600, 900, 1200, ...  I.e. all integer multiples of 300.

For open pipes (which are meant to be open on _one_ side only -- at 
least as far as I know (organ pipes) --- the story is similar.  Since 
one end is open, the "glued rope" is now attached to the other, 
closed, end only, and is allowed to "swing freely" at the open end. 
This means that at the closed end we'll find (again) a zero crossing, 
while at the open end we find a maximum.  Trying to ascii-draw again:

  +------------------########+
  |           #######
  |     ######
  |#####
  |
  |
  +--------------------------+

.. which is one quarter of a wave fitting in the pipe.  Next one 
fitting would be:

  +-------####---------------+
  |    ###    ###
  |  ##          ##
  |##              ##
  |                  ##
  |                    ###
  +-----------------------###+

i.e. 3/4 wave fitting in the pipe.  Next one is

  +----###-----------------##+
  |  ##   ##             ##
  | #       #           #
  |#         #         #
  |           #       #
  |            ##   ##
  +--------------###---------+

or 5/4 wave fitting in the pipe.

Clearly the series we get is 1/4, 3/4, 5/4, 7/4, ...

Making these into frequencies:
First one: 1/4 wave is 0.5 m. so entire wave is 2 m. Frequency = 
300/2 = 150 Hz.
Next: 3/4 wave is 0.5 m, so one wave is 2/3 m (since 3/4 * 2/3 = 
0.5).  Frequency = 300/(2/3) = 450 Hz.
Then: 5/4 wave is 0.5 m, so one wave is 0.4 m.  Freq = 300/0.4 = 750 Hz.

We thus get frequencies 150 (1*150), 450 (3*150), 750 (5*150), etc -- 
all _odd_ multiples of 150.

I.e. the difference between open and closed pipes is twofold: open 
pipes have a base frequency that's twice as low as closed pipes (one 
octave).  And open pipes only generate the odd harmonics, whereas 
closed pipes produce odd and even harmonics.

Now for a true harmonic series, the amplitudes (volumes) of these 
harmonics must be the inverse of their "order number".  I.e. if the 
base tone has a volume of one, the next harmonic has a volume of 1/2, 
the one after has volume 1/3, 1/4, 1/5, ...

So if a pipe produced a true harmonic series (which no real-world 
pipe does, I presume), a closed pipes would produce frequencies f, 
2f, 3f, 4f, 5f, ... with volumes 1, 1/2, 1/3, 1/4, 1/5...
An open pipe would then produce frequencies f, 3f, 5f, 7f, ... with 
volumes 1, 1/3, 1/5, 1/7, ...

Such a closed pipe would then (approx) generate a sawtooth wave, 
often associated with brass instruments (which exhibit the 
characteristics of closed pipes), and an open pipe would produce a 
square wave, often associated with woodwinds (which exhibit 
characteristics of open pipes).

Does this answer your question?


cheers,
HJ

-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html