# The Physics of Bells

Quite some time ago, Dick Crawford posted a long message to the Handbell-L reporting on a forum on handbell physics/acoustics held at the 1996 Director's Seminar in Albuquerque. With his permission, I am repeating it here, with hopefully only a few minor redactions.

Dick also tells me this material will be expanded in a forthcoming series of articles in Overtones in early 1998.

The general format of the following is as such:

• All text presented is the opinion of Jake Malta of Malmark, and should be treated as such.
• Occasional comments are inserted from Dick Crawford [DC:...] or Rusty Sanders [RS:...] who attended a different session. A few comments here and there from me, [PK:...].
• Jack Kearns introduced Jake and laid down the ground rule of "No bell wars": no opinions about whose bells are better than whose were expressed. Jake was extremely fair to everyone -- his objectivity, honesty, integrity, professionalism and courtesy were admirable in every way.

### Overtones

A struck handbell vibrates in an infinite number of ways ("modes") but only a few of these cause the surrounding air to vibrate strongly enough to be audible. The simplest mode gives rise to the lowest frequency vibration (the lowest pitch) which is called the "fundamental".

[DC:The form of the oscillation is a bit hard to describe in words, but I'll try. If you look at the bell end-on, the clapper pushes the bell outwards at the strike point (which I'll call the "top" of the bell in this short description). Because there is a fixed amount of metal in the bell, the sides must move inward. This in turn squeezes the bottom of the bell, pushing it outward. The net result is that the circular shape of the bell is turned into an ellipse. The stresses caused by this distortion causes the bell to return to its circular shape, but at the instant the circle is attained the metal is moving. This inertia leads to another ellipse, this time with the top and bottom inside the circle and the sides outside -- another ellipse, but rotated 90 degrees from the first one. There are four places where these two ellipses intersect. These points are motionless, and are called "nodes". They are actually lines that extend down the full length of the bell. A pair of nodal lines that are directly opposite each other defines a geometric plane, which includes the axis of the bell. Hence the fundamental is said to consist of two "nodal planes" separated by an angle of 90 degrees.]

The fundamental comes primarily from the lip of the bell.

The overtone that essentially makes a handbell sound different from a tuning fork is the twelfth. [DC:Some books call this the second overtone (assuming the first overtone to be the fundamental); others the first (assuming the fundamental to be zero). This creates a lot of confusion.] The twelfth arises from an oscillation that has three nodal planes, separated by an angle of 60 degrees.

The twelfth is emitted most strongly from the curved part of the bell -- about a third of the distance from the lip to the top.

It is only the fundamental and the twelfth that are tuned. The twelfth ends up being an octave and a fifth above the fundamental.

Both the fundamental and the twelfth are radial vibrations -- the motion of the metal is more-or-less directly towards or away from the axis of the bell [DC:"more-or-less" because I'd guess it's actually perpendicular to the bell surface, which isn't quite parallel to the axis]. The air outside the bell is caused to vibrate radially as well, so the note will be loudest radially outward. It's also louder if a point between nodes (an "antinode") is pointed towards the listener. But the antinodes for the fundamental and the twelfth don't all coincide. Obviously the ones at the strikepoint line up, but the next antinode for the fundamental is at 90 degrees, which is a node for the twelfth.

Since the clapper drives an antinode at both the strike point and the point 180 degrees around the bell, there isn't much difference in which point the clapper actually strikes.

According to Prof Thomas Rossing of Northern Illinois University (hereafter "TR") there should be cancellation of these vibrations within the bell, so there should be no sound radiated from within the bell at these frequencies (unless something else causes it). (It wasn't obvious that Jake shares this opinion...)

Handbells do not produce a "hum tone" an octave below the fundamental. That's a characteristic of tower bells.

[PK: This is probably where the confusion in numbering the overtone series comes from. In tower bells, the hum tone is overtone 0, fundamental overtone 1, and the 12th is overtone 2. In handbells, there is no hum tone, so you can either use the same numbering as the tower bells (with no zero), or start afresh with the fundamental as zero.]

[DC:The octave is also generated by a "secondary" mechanism described by TR in his 1981 and 1983 articles reprinted in "Overtones 1955-1986". The change between circular and elliptical cross-sections for the bell produces a volume change as well. Air is pushed out of the bell when it becomes elliptical and is sucked back in when it returns to circular. Since this happens twice for each fundamental vibration, it produces the octave. Similarly, it produces the octave above the twelfth. These tones, of course, are emitted from the end of the bell instead of the sides.

I asked Jake about this effect, but apparently didn't make myself clear. I discussed it further with him after the fact. The upshot was that, since there's nothing that can be done about it and the octaves are exactly in tune, he doesn't worry about them at all.

During several concerts I was seated where a C3 would occasionally be rung with its axis pointed directly at me. I think I was able to hear the octave, but it may have been my imagination.]

### Materials

Bell bronze (about 80% copper and 20% tin) is hard and brittle, which is why it supports high-frequency vibrations. The only other metal used to make handbells is Aluminum (the low Malmarks). Aluminum produces a stronger fundamental and a weaker twelfth, compared to bronze. TR's belief is that this is due to Aluminum's having a higher sound speed at low frequencies and a lower sound speed at higher frequencies. Aluminum bells are much larger than bronze at the same pitch because Aluminum is not as strong, and therefore the bells must be thicker. Since the pitch of a bell is determined by the diameter-to-thickness ratio... [PK:...the necessary thickness of the bell, to compensate for the softer metal, makes for a wider bell at any given pitch. The thickness of the aluminum bells is also one reason why the aluminum bells are louder than the comparable bronze bells)].

Malmark and Schulmerich use the same alloy -- Jake didn't know about other manufacturers.

Bells are cast at the foundry (Malmark and Schulmerich use the same one, [PK: although M & S each provide their own molds to use for the casting process]) and then tuned at the individual factories. Tuning is done by removing metal from inside the bell. Every bell does not have its own raw casting; instead, typically two bells share the same mold. Since the lower-pitched bell will need to have a larger diameter, it will need to have more metal removed from the inside and thus will weigh less than its higher-pitched moldmate. Wall thickness determines the loudness of a bell. (That's why tower bells are so loud -- they're thick.) [RS: When asked if this meant that the sharps were louder than the naturals below them, Jake thought about it and said something like "I guess so -- I never thought about it, but they should be". But they wouldn't be much louder, as they aren't too much thicker.]

There might be some small improvement in tone possible if each bell had its own mold, but the increase in expense would be immense.

Over the long term Malmark rejects about a quarter of the castings from the foundry. (Whitechapel's experience is similar.) However occasionally virtually all castings are rejected. Since when this happens it's usually spring or autumn, it's most likely weather-dependent. But neither the foundry nor the factory has been completely able to figure out what's going on.

The largest problem is center-line porosity. The bronze is poured at 2000-1850 degrees into damp sand molds. Of course, the first bronze to freeze is the "skin". As the cooling progresses into the center, the cooler skin contracts. Since this occurs on both surface of the bell, the result is a "crack" in the middle of the casting body. Often this crack doesn't show up until most of the way through the tuning process, as the lathe uncovers it. It's an expensive problem to find, as much of the labor is complete before it turns up.

The casting process also effects sound quality. Sometimes the alloys aren't homogenous and fully mixed, so some portions of the bell will have more or less tin relative to copper. Likewise, some portions of the bell may cool at a different rate from others, which causes different grain sizes in the frozen metal. These differences are what cause beat and wow. [DC:They are also the reason that the strike point is important -- since the bell is no longer chemically symmetric, there may be different compositions at different antinodes.]

Sometimes there are problems with pouring the castings down the line, and the bronze may get poured at or below the critical 1850 degrees where it starts to set up. Also, the castings that are poured first tend to have more time in the molds before they are turned out, giving them a longer cooling time. Malmark has tried letting the casting sit in the molds overnight before being turned out. The bells were were smoother and mellower, but the yield [of usable bells] didn't improve.

Bells tarnish over time [no kidding]. Theoretically this changes the chemical composition of the bell and thus could have some effect on its tone. [DC:In the Whitechapel display in the HIC room was a bell that hadn't been cleaned for perhaps a century -- it's a dirty dark gray. It's sound is only marginally different from a refurbished bell from the same set.]

[RS: Jake's impression "first service" was slightly different, but he hadn't thought about it then. We asked if corrosion from handling (i.e. finger prints et al) could effect tone. He said that oxidized bronze was harder, and thus should theoretically have a different tone. If a bell was unequally oxidized (again, as in the case of hand or finger prints) that would be similar to unequal tin/bronze or grain homogeneity, and thus could cause a different wow or beat. But he wasn't sure.

In [Dick's] case the question was about a tarnished bell. In our case we asked about a "grundgy" bell (like SO MANY we saw at the seminar). Again, he may have changed his mind after the class, but during class he thought this might have an effect on the tone and quality of the sound.]

Bells also "normalize" over time -- stresses that were "frozen in" as the casting cooled are gradually able to relax as the bell is used and as the temperature fluctuates. Irregularities in the crystalline structure of the metal also tend to disappear with time. These tend to reduce the pitch, since regions of high stress vibrate with higher frequencies. They also make the bell mellower and louder. Thus after a bell is a decade or so old it may need to be revoiced -- which usually means nothing more than rotating the clapper assembly to a different strike point. [DC:The strike point is chosen to produce the "best" sound. Were the bell perfectly homogeneous in composition and shape, all possible strike points would sound the same. But it ain't.]

Thermal expansion in bronze is such that a temperature rise of 2 degrees Fahrenheit produces a pitch decrease of 1 cent (one hundredth of a semitone). At 20 degrees the vibrations themselves will warm the bell enough to prevent damage, but at zero...[PK: the pitch will be 10 cents flat, which is a noticable difference from "normal" pitch. Or maybe Dick means that the bronze becomes dangerously brittle at 0 degrees F.]

### Clappers

Since the clapper must drive both the fundamental and the twelfth, which are primarily produced at two different distances from the lip of the bell, the proper length of the clapper shaft is a bit of a compromise. In general, the shorter the clapper shaft, the more twelfth is driven. Malmark also tries to keep the mechanism short enough that the clapper assembly is entirely inside the bell so that the bell can be stood on end. [DC:So does Schulmerich.]

Clapper heads used to be made of various hardnesses of rubber; now they are polyethylene or, for the highest bells, nylon or delrin (sp?). Where in the chromatic scale to change from one clapper head to another is hard to determine and, indeed, may vary from set to set.

The old rubber heads harden and crack with time, which leads to a deadened tone. The new plastic heads can also crack over time (because the elastomers which keep the material soft evaporate), but the tone does not seem to be as seriously affected. Felt squashes, of course, so it needs to be replaced occasionally.

Someone in the first class asked if altitude would make a difference to clapper lifetime (he rings near the continental divide at >7000 ft). After some thought between Jake and the others it was decided that yes, high altitude and low humidity could lead to premature clapper failure (which is evidenced by harsher or deader sound, depending on the failure mode).

Most of the materials used for clappers these days probably wouldn't be hurt by temperature extremes, but the action probably will be sluggish if very cold.

### Tang

Some metal is left attached to the top of the bell in the casting process. This gives the factory something to grab ahold of while tuning and polishing the bell. This extra metal is called "tang". There is some controversy among the manufacturers about what to do with the tang. Some manufacturers leave a bit of it to attach the handle and handguard; others remove it completely. TR feels that it has no effect on the tone quality of the bell, so its presence or absence is purely a matter of convenience, weight, and aesthetics.

### Manufacturers

[DC:Yes, we really did avoid bell wars.] The major differences between bells produced by the various makers are in the casting profile and the clapper mechanism. The profile (the shape of the bell) is determined empirically, since the mathematics is such that exact analytic solutions are impossible to compute. They are based upon years of trial-and-error, and are closely-held trade secrets. The composition and profile determine the sound of the bell; the clapper mechanism determines things like ease of maintenance and durability.

Malmark chooses to try to push undesired overtones to frequencies as high as possible. [DC:I think this is what Jake said, but I suspect he meant that they try to reduce the amplitude of the overtones (other than the fundamental and twelfth, of course) as much as possible.] This produces a thinner, drier tone than otherwise. Other manufacturers aim for a different tone quality.

### Mallets

Mallets should strike the bell at the same distance from the lip as does the clapper. Many people strike the bell much too far down the casting, which means that not all of the energy they put into the bell goes into the fundamental. Also, the casting is thickest at the clapper strike point, and thinnest towards the waist of the bell. So not only do you have to strike it harder to get sound, you're striking the casting at its weakest point. More bells are cracked due to improper malleting than by martellato.

### Chimes

Malmark uses standard square-cross-section Aluminum tubing -- the same stuff used for making table and chair legs (but with the increased demand for chimes, they are investigating custom stock which should further improve the tone). Suzuki rounds the edges of the tubes, which introduces discordant [Jake's word] overtones. [DC:Of course, the rounded edges are *much* more comfortable to hold and play.]

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