### Re: Hot Chocolate and rising pitch when tapping on bottom of cup

Area: Physics
Posted By: William J Bray, RESEARCH AND DEVELOPMENT
Date: Fri Jan 3 11:16:31 1997
Message:
Hi Mike,

It's very simple...

Imagine a steel cup.  The cup is narrower at the base than at
the top.  It gets broader at the top.  When you tap on the
bottom you start a wave which rides from the bottom to the
top.  To the wave, the cup is a multitidue of rings resting
on top of one another, the rings get larger at the top.  Each
ring resonates according to its characteristic frequency,
like a guitar string wrapped into a circle.  The rising pitch
you here is because the rings don't begin to resonate until
the wave is moving back from the lip of the cup toward the
bottom, having been muted by the liquid when you tapped on
the bottom, the outer most ring at the lip of the cup is free
to resonate when the wave reaches that point, since it is not
muted by liquid.  Now that it is resonating, the wave's
momentum carries it back down the length of the cup, the
pitch rising as the concentric rings get smaller and smaller.
The rings resonate now because the wave is organised.  The
organization of the wave occured at the lip of the cup when
it began to resonate.  An organized wave can make it back
and forth across the muted rings because the energy is
directed.

You don't hear the traveling wave in an empty steel cup
because it is not muted by the liquid, all of the tones come
out almost simultaneously - so it merely rings with a broad
spectrum 'dong.'  The viscosity of the liquid determines the
speed of which the wave will travel through the rims of the
cup, as the energy of the organized wave struggles against
the muting liquid.

I've actually though about that one breifly while consuming
intoxicating beverages of various viscosities in drinking
vessels made of different media, varying from glass to
ceramic.  For instance, a mud slide produces a significantly
slower wave in a crystal goblet than does red wine.  Scotch
is even faster yet.

As for stirring, and all of that - just try imagining the
effect that the motion of the liquid has on the energy of
the wave.  In some cases you can actually generate a
standing wave by the energy of the rotating liquid, but be
careful not to let the liquid 'wobble' inside the vessel, as
this will produce an additional muting effect.  In general,
if the liquid travels up the sides of the wall of the cup
while stirring, all but the highest part of the rim will be
muted, and a low pitch will result.  As the liquid slows, it
comes back down the inside of the cup, freeing the higher
pitched rings to resonate - a resulting rise in tone should
be heard.

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