# Homework Help: Tone and musical note

1. May 20, 2004

### Johnny Leong

Illustrate the modes of vibration of the fundamental and the first two overtones when a musical note of fundamental frequency 256 Hz (cycles per second) is produced by air in a tube:
(i) open at both ends;
(ii) closed at one end.
Mark nodes and antinodes of displacement N and A respectively.
Ans: Please see my figure. Is my drawing right? Anythign wrong?

State the frequencies of each pair of overtones.
Ans: The frequencies of overtones are 3*256 Hz and 5*256 Hz respectively, right?

Why a musical note produced by an open tube or pipe more mellow than a note of the same fundamental frequency produced by a closed tube?
Ans: Because the antinodes are produced at both ends in an open tube, the sound is weakened.

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Last edited: May 21, 2004
2. May 21, 2004

### Mark

if its closed at both ends that the ends will both be anti nodes, forgot all about waves, just draw pictures

3. May 21, 2004

### Johnny Leong

I don't quite understand your meaning, Mark. How to explain the phenomenon of my last question?

4. May 21, 2004

### marcus

when you make a drawing like that it can either illustrate pressure (compared to normal) or velocity

let us suppose you mean to draw pressure (compared to normal)
in the both-end-open case the pressure must always be zero (normal) at each end
so the tube must be a half-wavelength long----about 2 feet

all possible overtones are produced in the 2 foot tube open at both ends
there is a pressure node (zeropoint) at each end'
and then there can be one extra halfway (for the 512 Hz overtone) and
there can be two extra nodes (1/3 and 2/3 of the way along) for the 3 x 256 Hz overtone

BUT THE closed at one end pipe is DIFFERENT
it has a pressurenode at the open end but not the closed end
at the closed end the pressure can go high and then low and then high etc.
so the tube must be a QUARTER wavelength------around 1 foot long

and this tube cannot produce the 512 Hz overtone. Try to draw it. It will not work!
it will only produce the 3x256
and the 5x256
and the 7x 256

the sound will not be so nice, I imagine, more like a sawtooth wave than a sinewave, I am not sure but it seems to me that the second tube (the closed at one end) is MISSING HALF THE OVERTONES and this will make
it not as nice a sound---however i did not think about this yet and i am not sure i can imagine the sound correctly

when you draw the picture for the closed at one end
and the 3x256 overtone
you should put the first node at the open end (always!)
and then the second node will go 2/3 of the way along the tube
(just 1/3 of the way from the closed end)

there will be no node at the closed end! it goes to max pressure to min pressure etc there at the closed end

if you want to draw air-velocity pictures a lot of this is reversed
so it can get very confusing. I advise you to draw pressure pictures.

5. May 22, 2004

### Staff: Mentor

The answer requires three diagrams for each kind of pipe. You only gave one diagram for each. For the closed pipe, you illustrated the second overtone; for the open pipe, you illustrated the first overtone.
For an open pipe (open-open, actually), the frequencies are given by mf, where f is the fundamental and m = 1, 2, 3, etc. So the two overtones are 2*256 Hz and 3*256 Hz.

For the closed pipe (closed-open), the frequencies are given by mf, but m = 1, 3, 5, etc. So the overtones are 3*256 Hz and 5*256 Hz.

No. As marcus stated, the open pipe can produce more harmonics (overtones) than the closed pipe. The more harmonics, the richer the sound. (I presume that's what is meant by "more mellow".)