1. Figures 1 and 2 are the polar plot of the sound radiation intensity given off by a vibrating diaphram in a wall, at various frequencies. what is the the relative intestity (relative to straight ahead) of the sound given off at 30 degrees for a frequency of twice the knee frequency?
The curve labled 2fx crosses the radial line labled 30 degrees at the cross which is about .52 (each circular radius here represents .2.)
At 60 degrees?
The x marks where it crosses the sixty degree line. It is a relative intensity of about .1
What are the relative intensities (compared to straight ahead) of the sound given off at 10 degrees and at 45 degrees for a frequency of eight times the knee frequency.
The two lines in the relative intensity graph are at 10 degrees and at 45 degrees.
Note that in this graph, the relative intensities are unmeasurable.
For such small values it is far better to look at the dB graph, which graphs the same
ratios, but in dB form.
On that graph the two lines at 10 degrees and 45 degrees are again plotted. In this case
each circle represents 10dB. The 10 degree line crosses at about -7dB, while the
45 degee line croses teh 8fx curve at about -24dB. (-7dB is about .5 relative intensity
-- -7dB= -10dB +3dB--> .1x2=.2 in relative intensity.
-24dB= -10 -10 -4 --> .1x.1x.25=.0025 relative intensity.
2. Deep sea divers can live underwater in a Helium-oxygen mixture, instead of air. (The nitrogen in the air dissolves in the blood under pressure, and is somewhat poisonous. It also comes out of solutions when the diver ascends, causing nitrogen bubbles in the blood, blocking the flow of blood- called the bends.) One diver likes to blow over coke bottles to annoy his mates. What would happen to the note that the bottle would produce in the helium-air mixture? Another diver brings down his guitar to use in the evenings. What effect would the different gas mixture have on his guitar? (Note that Helium-oxygen mixture is much lighter than air, although the compressibility is the same.)
The blowing over coke bottles excites the first mode of oscillation of the
air in the bottle. The Helium-Oxygen mixture is lighter than nitrogen oxygen, and thus
the mass of the air in the neck of the bottle will be less. However the compressibility
is essentially the same as ordinary air. Thus with the mass decreasing and the stiffness
remaining the same, the frequency will go up.
On the guitar, the pitch of the note is determined by the mass and tension in the
strings. Neither of those are changed by the helium mixture, and thus the pitch would
remain the same. Similarly the resonances of the wood in the guitar will be unchanged.
However the primary air resonance due to the hole in the body of the guitar and the
air space inside will have its frequency raised just as with the coke bottle. Thus the
lower notes of the guitar will no longer have the increased coupling to the air that
that they would have in air. The lower strings of the guitar will thus sound weak.
The higher strings will now have that increased efficiency and they will thus sound louder.
The balance of the guitar will thus be altered, even though all the pitches will
remain the same.
3. Consider a clarinet with just one finger hole. What would you expect to happen to the note produced by that finger hole as the hole is made smaller and smaller?
There are two limits here. Clearly if the finger hole has zero size, then the frequency of the clarinet, even with the finger hole of zero size open, will be the same as that of the finger hole closed. With the finger hole having a large size, the tube will act as though the tube ends at the finger hole, and the pitch will be that of a shorter pipe. If one goes between these two limits, the pitch will also lie between these two cases. The smaller the hole, the lower the note. In fact the hole has to be quite small befor the pitch begins to drop. At that point also the pressure in the instrument forces some of the air through the hole, damping the note as well, as for the register hole in the clarinet. Thus as the hole gets smaller the frequency slowly drops, then the note become harder to play (because of the increased damping) until finally the pitch drops to the one appropriate to the full length of the instrument.
4. Discussion:
The effect of the change in size of a tube on its resonances.
Given a tube fill of air ( either one end closed or not). If one changes the dimensions of the tube at some point, the modes and especially the frequencies of the modes will change. There is a law to how those frequencies change. If one widens [narrows] the tube at a point near where the pressure has an anti-node (i.e., has maximum oscillation and the velocity has a node) for a given mode, then the frequency of that mode will fall (rise). If one widens [narrows] the tube at a point where the mode has large velocity oscillations ( i.e., near the anti-node for the velocity, or node for the pressure), the frequency of the mode rises [falls]. Eg, for a coke bottle, one has narrowed the tube at the end where the velocity is high and pressure is near a node. The frequency of the fundamental ( and if fact of all of the modes) falls. Note that if one uniformly increases the diameter of the tube at all points by the same ratio, the two effects cancel, and the mode has the same frequency as before. (A wide pipe has the same resonant frequencies as a narrow tube to fist order.)
Question:
What is the dominant effect of the closed finger holes in a clarinet on the frequency of the lowest note of the clarinet? What is the effect of the bell at the end of the clarinet on the frequencies of the various modes? (remember that the walls of the clarinet, made of wood, have a substantial thickness).
Each of the closed finger holes still leaves a large dent in the inside of the tube. This large dent acts like an increase in the diameter ofthe tube at the location of that dent. If the closed finger hole is near the end of the tube, the effect will be to raise the note, since the velocity of the air is a maximum near the end of the tube while the pressure is a minimum. This is true of all of the finger holes in the bottom half of the tube. Similarly, the closed finger holes in the top half of the tube will cause the pitch of the lowest mode to go up. If an open finger hole were placed half way up the tube, the pitch would go up by an octave. Thus the number of finger holes between the half way point and the bell must be enough to play a full octave of notes (12 semitones). The finger holes in the top half of the tube need only cover the next perfect fifth which is seven semitones. Thus there are more fingerholes in the bottom half of the tube than in the top. this means that the effect of the closed fingerholes will be raise the lowest mode. Of course the maker of the instrument has to take this into account in building the instrument, and thus the placement of the finger holes differs from what a naive calculation would indicate.
5. What effect on the playing of a flute would the fact that it is made of gold or platinum rather than say plastic have?
a) The maker will spend a lot more time and effort on a platinum or gold flute
than a plastic flute, making the instruments sound better. Furthermore, the player, playing
such an expansive flute, is apt to take playing more seriously. (Whetehr this would cause
the player to choke rather than play better will depend on the player).
b) The sound pressure inside a flute is around 120dB. This is a pressure of about
.0002 of an atmosphere. This pressure may excite resonances in the flute body, but
the pressure is equal in all directions around the flute. Such radial expansion and contraction
of the flute body are very low in intensity and high in frequency (the flute body is very stiff
with respect to such radial expansion). Ie, there is little likelihood of these internal
pressures actually exciting resonances. But such resonances are the only effect that the
material which makes up the flute can have on the sound of the flute.
Ie, it is unlikely that the material which makes up the flute has any physical effect
on how the flute plays.
6. a) We are back to our deep sea divers again. Some of the divers take down their clarinet, trumpet and flute. Assume that the velocity of sound in the helium-oxygen mixture is twice that of air. What will the effect be on these instruments? Why does the trumpet player have difficulty in playing his instrument? (Remember the relation between the lips and the note the trumpet player wants to play). How will the flautist have to adjust her playing style in this mixture?
In all three cases the frequency of the instruments will increase because the mass
of the air decreases, but the stiffness does not. Thus all three will play higher.
for the Clarinet, the negative feedback is determined by the reed. The reed is very
light and responds almost instantly to the pressure on the reed
(the resonant frequenccy of the reed is about 3KHz). Thus the negative feedback of the
clarinet reed will behave just as in air even though the frequency is higher. The
instrument will sound higher, but will work almost the same as in air.
For the trumpet, the trumpet player must adjust his lip tension and mass so as to
tune his lips just beneath the note the trumpet is to sound. Since the trumpets modes
are much higher than for a trumpet in air, it is much harder to play, and the trumpeter will
be unable to play nearly as many notes as in air.
For the flute, the air flow across the blow hole must be adjusted so that the
time it takes for the air to flow across, is about 1/4 of a period of the note that
is being played. Since the flute sounds a much higher note, the flautist must blow much
more rapidly to play the same note. He/She can do so either by blowing harder, or by
making the distance across the blow hole shorter by rotating the flute under the
lips.
b) What happens to the knee frequency of a given sized "speaker" in such a mixture?
The knee frequency depends on the wavelength. (The wavelength equal to twice the diameter defines the knee frequency). Since the velocity of sound in helium is much higher than that of air (about twice), the knee frequency will also be higher.
7. Describe the similarities and differences between an oboe and a flute?
The modes of both are all of the multiples of the fundamental, and the fundamental of
both is the velocity of sound divided by twice the length. Both change the notes
by changing the length of the tubes by opening holes in the sides of the tube.
Both go up in register by playing the second mode of the air in the instrument.
The flutes negative resistance is obtained by the behaviour of the reed under the
pressure inside and outside the reed, while the flutes negative resistance arises out
of the timing of the air flow across the blowhole and the redirection of the
airflow by the velocity of the mode at the blowhole. The oboe is a cylindrical bore
instrument with the velocity of the air in the mode going to zero at the reed end,
while in the flute the velocity of air in the mode is a maximum at the open blow hole.
More minor differences-- flutes nowadays tend to be made of the metal,
usually silver or silver plated nickel, whole oboes are make of wood.
;''