What You Need

Either

Large container, at least 2-l size

Air bleed valve to fit the container (or make your own; see instructions)

Water

Flexible tubing

Or

Aquarium air pump

Flexible tubing

Tubes of different sizes to fit the ends of the flexible tubing

Water

What You Do

Although you can simply tip water out of a wine bottle to hear musical glugging,

the effect is only momentary. One way to extend the glugging time is to find a

large container and fit it with a small neck. The pitch of the glugging still changes

quite rapidly, however. Instrumenting the container with a pressure gauge will

give you a clue as to why this happens: the static pressure in the bottle changes.

This pressure change does not in itself change the frequency of the glugging

SOUNDS I N T E R E S T I N G52

much, but the fall in static pressure changes the size of the bubbles being pro-

duced, as well as the rate at which they are produced. The smaller bubbles yield

higher frequencies, which explains why the pitch of the glugging rises as you

pour out the liquid.

You can better control of the size of the bubbles if you provide separate

paths for the water going out and the air coming in. Installing an air pipe will

control the bubble size much better than just relying on the air space at the top

of the neck of a bottle. However, the pipe should exit underneath the water in

the container, otherwise glugging won’t happen at all. Choosing a large container

also helps because it gives you more time to test out the bubble frequency and

see what is going on.

With the largish container suggested, you will find that the pitch of the glug-

ging changes only slowly with time. The container should ideally be fitted with

a proper air bleed valve and water outlet nozzle. If you look around, you may

find a tap or faucet that does exactly what you need. There are “no-drip” taps

for barrels and other containers that have a small plastic assembly with an air

inlet, a water outlet, and a lever that cuts off both flow paths. Containers with

this kind of faucet are sold for putting in a refrigerator to dispense cool water,

for example. But if you can’t find one of these, just glue two tubes into the neck

of your container, the lower, larger pipe for water, and the upper, smaller pipe for

air. The water outlet could be 8–10 mm in inner diameter, and the inlet 3–4 mm

in inner diameter, for example. The air inlet should end a short way, say, 5 cm,

inside the container.

Another simple technique is to abandon the container altogether and pump

air at a suitable rate into a tube of a suitable diameter. Changing the tube diame-

ter at the same volumetric flow rate, which is what I am suggesting here, changes

the pitch of the bubbles. You can use a simple aquarium oscillating-diaphragm

pump. These pumps are inexpensive and readily available, but they do have cer-

tain disadvantages. They produce a pulsatile air flow—pulsating at the domestic

electricity frequency of 50 or 60 Hz—rather than a smooth flow. A reservoir

vessel will smooth out these pulses.

How It Works

Glugging occurs when newly formed bubbles compress and expand rhythmically.

This rhythmic vibration is just loud enough to escape the liquid and just the right

frequency for us to hear it. Smaller bubbles compress and expand more rapidly

MUS I C A L G L UGG I N G 53

than larger ones, and the frequency of oscillation is inversely proportional to

bubble size.

The pump setup simply blows bubbles into the water. In the container setup,

the glugging occurs when the pressure in the container falls below the ambient

air pressure. The water runs out but air is not getting back in, so the pressure in

the air space falls as the air expands. The lower pressure means that ambient air

can blow bubbles from the air bleed nozzle inside the neck of the container.

Whatever their source, the bubbles will oscillate a little after they are formed,

creating the musical sound.

SOUNDS I N T E R E S T I N G54

THE SC I ENCE AND THE MATH

We can estimate the oscillation frequency of a

bubble as follows. Suppose a bubble of volume V0(4/3πr3) expands to volume V0 + ∆V, undergoing acorresponding change in its radius, ∆r. The mass,M, of the surrounding liquid that is displaced by the

expansion is related to the product of the bubble

volume and the density of the liquid. The force, F,

required to achieve this expansion is given by

F = M(d2∆r/dt2) = V0ρ(d2∆r/dt2).

This force derives from a change in pressure inside

the bubble, F = A∆P, where ∆P is the change in

pressure caused by the change in volume, ∆V, and Ais the surface area of the bubble. If we assume that

the bubble is spherical (A = 4πr2) and that we haveapproximately isothermal “ideal gas” conditions,

we can use the ideal gas law,

PV = nRT,

where n is the number of moles of gas, R the gas

constant, and T the absolute temperature in Kelvin,

to determine the pressure change:

(P0 + ∆P)(V0 + ∆V) = nRT,

where P0 is the ambient pressure. Solving for small

pressure changes, ∆P, from P0 and small volume

changes, ∆V, from V0 gives

P0 = nRT/V0

and

P0 + ∆P = nRT/(V0 + ∆V).

So

∆P = –(nRT/V0)(∆V/V0).

Because F = A∆P, we arrive at

V0ρ(d2∆r/dt2) = –A(nRT/V0)(A∆r)/V0

and

d2∆r/dt2 = –(A/V0)2(P0 /ρ)∆r.

This equation is rather like the equation for

simple harmonic motion for a mass M on a spring

of rate K:

d2x/dt2 = –(K /M)x,

where x is the displacement of the spring. Inte-

gration gives x = sin(2πft), which has frequency f =(1/2π)√(K/M). Because A/V0 is just 3/r and because

(K/M) = (A/V0)2(P0/ρ), our bubble frequency, f, is

expected to be approximately

f = [3/(2πr)]√(P0/ρ).

In practice, our assumption of isothermal con-

ditions is likely to be incorrect. At the fast bubble

oscillations we are talking about, heat does not have

time to diffuse from the middle of the bubble to the

edges. In other words, the gas expansion is adia-

batic, so that the ideal gas equation does not apply.

The ratio γ of the specific heats at constant pressureand constant volume for the gas might enter the

equation to compensate for this, as often happens

when adiabatic processes occur in gases. In addi-

tion, our simplistic assumption of the mass of liquid

that is moved by the bubble expansion is not quite

right. In fact, Marcel Minnaert first estimated the

oscillation frequency of a bubble to be

f = [√3γ /(2πr)]√(P0 /ρ).

For monatomic gases such as argon (γ = 5/3), oursimple formula is in fact remarkably accurate: Min-

naert’s equation gives √3γ/2π (= 2.9/2π) for the con-stant multiplier, as opposed to our value of 3/2πfor the equivalent numerical factor in our formula.

However, air contains only 1 percent argon, being

mainly nitrogen and oxygen, and has a γ value of 7/5,which makes our result less spectacularly accurate.

And now for some example numbers: for a 5-mm

air bubble near the surface of water (at atmos-

pheric pressure), the frequency should be 770 Hz, a

musical tone roughly equivalent to the A# above

middle C.

Our analysis so far deals with the sound emitted

by oscillating bubbles, but it does not deal specifi-

cally with the glugging sound that occurs when liq-

uid is poured from a container. As you pour liquid

from a container, the pressure in the air space above

the liquid (the ullage space) falls below the ambient

air pressure because air is not getting back in. If we

start with an initial volume Vi at an initial pressure

Pi , Boyle’s law tells us that

PiVi = PfVf ,

where Pf and Vf are the pressure and volume after

some liquid has been poured out. This means that if

Vf > Vi, then Pf < Pi . If, for example, we startedwith a 100-ml volume at the top of the bottle and

poured out 10 ml of liquid (thus increasing the

ullage volume to 110 ml), then the pressure should

drop from 1,000 mbara (atmospheric pressure) to

909 mbara, a drop of 91 mbar. I measured the

pressure in the ullage space in a 2-l container as it

glugged. Depending upon the angle of pouring and

how much water was left, the pressure varied from

about –3 mbar to –15 mbar.

The pressure in the water is higher at the level of

the air bleed nozzle; the pressure, P, varies with water

depth, H, according to the following equation:

P = ρgH,

where ρ is the water’s density. For the kind of waterdepths we are talking about in our experimental

arrangements, we can take a value of about 10 cm.

Now 1 cm of water exerts about 1 mbar of pressure.

So we might expect to need pressure on the order of

a few millibars to push bubbles of air back into the

bottle.

Our negative Boyle’s law pressure needs to exceed

the hydrostatic pressure in order for glugging to

take place—a minimum of a few millibars with the

container nearly horizontal or 15 mbar or so with

the container nearly vertical. Once glugging starts,

the air coming in reduces the negative pressure until

an equilibrium is reached with air coming in at

roughly the same rate as water is leaving through

the main spout, with a fairly steady pressure in the

ullage space, and with the pressure in the bottle

never exceeding 10–20 mbar.

MUS I C A L G L UGG I N G 55