Simple Nature - Light and Matter

t/A pan pipe is an asym-

metric air column, open at the top

and closed at the bottom.

u/A concert flute looks like

an asymmetric air column, open

at the mouth end and closed at

the other. However, its patterns of

vibration are symmetric, because

the embouchure hole acts like an

open end.

this, how many times greater will its frequency be? .Answer, p.

1057

The speed of sound example 13

We can get a rough and ready derivation of the equation for the

speed of sound by analyzing the standing waves in a cylindrical

air column as a special type of Helmholtz resonance (example

25 on page 344), in which the cavity happens to have the same

cross-sectional area as the neck. Roughly speaking, the regions

of maximum density variation act like the cavity. The regions of

minimum density variation, on the other hand, are the places

where the velocity of the air is varying the most; these regions

throttle back the speed of the vibration, because of the inertia of

the moving air. If the cylinder has cross-sectional areaA, then the

“cavity” and “neck” parts of the wave both have lengths of some-

thing likeλ/2, and the volume of the “cavity” is aboutAλ/2. We

getv =fλ= (...)

√

γPo/ρ, where the factor (...) represents nu-

merical stuff that we can’t possibly hope to have gotten right with

such a crude argument. The correct result is in factv=

√

γPo/ρ.

Isaac Newton attempted the same calculation, but didn’t under-

stand the thermodynamic effects involved, and therefore got a

result that didn’t have the correct factor of

√

γ.

This chapter is summarized on page 1078. Notation and terminology
are tabulated on pages 1066-1067.

6.2.5 ?Some technical aspects of reflection

In this section we address some technical details of the treatment

of reflection and transmission of waves.

Dependence of reflection on other variables besides velocity

In section 6.2.2 we derived the expressions for the transmitted

and reflected amplitudes by demanding that two things match up

on both sides of the boundary: the height of the wave and the

slope of the wave. These requirements were stated purely in terms

of kinematics (the description of how the wave moves) rather than

dynamics (the explanation for the wave motion in terms of Newton’s

laws). For this reason, the results depended only on the purely

kinematic quantityα=v 2 /v 1 , as can be seen more clearly if we

rewrite the expressions in the following form:

R=

α− 1

α+ 1

and T=

2 α

α+ 1

.

But this purely kinematical treatment only worked because of

a dynamical fact that we didn’t emphasize. We assumed equality

of the slopes,s 1 =s 2 , because waves don’t like to have kinks. The

underlying dynamical reason for this, in the case of a wave on a

string, is that a kink is pointlike, so the portion of the string at the

kink is infinitesimal in size, and therefore has essentially zero mass.

Section 6.2 Bounded waves 389