Klarinet Archive - Posting 000079.txt from 1996/05

From: Teri Herel <EnsHerel@-----.COM>
Subj: Re: A question about acoustics
Date: Sat, 4 May 1996 23:45:03 -0400

Excellent physics! Now what we need is an equally clear and concise
description of why this is not the case for the ideal closed-at-one-end
*conical* tube, i.e., the oboe, bassoon, and saxophone.

In a message dated 96-05-04 00:40:49 EDT, cohler@-----.NET (Jonathan Cohler)
writes:

>Dan,
>
>Others have cited good sources. My favorite is Benade's "Fundamentals of
>Musical Acoustics" (Dover). The new edition of this is in paperback and
>contains many corrections over the old.
>
>The reason that an ideal cylindrical tube closed at one end and open at the
>other supports only odd harmonics is as follows.
>
>The open end is open to the atmosphere and therefore the airpressure at
>that point does not vary. In other words, it is a pressure node (always at
>0, which represents atmospheric pressure).
>
>The closed end allows for build up and decline of air pressure as the wave
>"comes in" and "recedes" (think of a water wave).
>
>Now here's the simplest explanation that I know for why the fundamental
>wavelength of such a system (node at one end, antinode at the other) is
>four times the length (and not two times the length as with a flute, which
>has a node at both ends).
>
>Imagine a pulse (a very short, high pressure spike) traveling down the tube
>to the open end. As the pulse "hits" the open end, the pressure must
>always add to zero. Therefore, a negative pulse (a low pressure spike) is
>reflected back (the negative pulse and the positive pulse sum to zero at
>the node).
>
>Now, when the negative pulse "hits" the closed end, it just bounces off the
>"wall" and comes back as a negative pulse.
>
>It travels back to the open end, where it is now reflected back as a
>positive pulse and finally we have made one complete round trip. We are
>back to where we started, and the process starts over again.
>
>This is why the fundamental wavelength is four times the length of the
>instrument. This same logic can be used to show why the fundamental
>wavelength of a string instrument (node at both ends) or a flute (node at
>both ends) is two times the length of the vibrating element.
>
>Now, here's why we get only odd harmonics in the closed-open
>(antinode-node) case. Draw a quarter of one cycle of a sine wave on a
>piece of paper. The sine wave starts at zero (the node point) and rises to
>its maximum (the anti-node point) one quarter of the way through the cycle
>(one traversal of the lenght of the instrument).
>
>Continue drawing the sine wave. Where is the next point where the sine
>wave reaches a maximum (positive or negative)? It is three quarters of the
>cycle. That represents the next possible mode, because all possible modes
>must have an antinode at that end.
>
>Keep drawing and you'll see that the next possible mode traverses is 5/4 of
>a cycle in one crossing of the length.
>
>State this mathematically and you get that the possible wave lengths are:
>
> 4L, 4L/3, 4L/5 etc.
>
>Or, in other words, the possible frequencies are:
>
> v/4L, 3v/4L, 5v/4L, etc. (only odd harmonics!)
>
>Do the same drawing excercise with a node at both ends and you will quickly
>discover that the allowed wavelengths and frequencies are respectively:
>
> 2L, L, L/2, L/3, etc. (wavelengths)
> v/2L, 2v/2L, 3v/2L, etc. (all harmonics)
>
>I hope this helps without getting too technical.
>
>----------------------
>Jonathan Cohler
>cohler@-----.net
>
>