234 DOC.

39

THEORY OF WATER WAVES

Doc.

39

Elementary Theory

of

Water

Waves

and

of

Flight

by

A.

Einstein,

Berlin-Wilmersdorf

What

accounts

for the

carrying capacity

of

the

wings*

of

our flying

machines and

of

the birds

soaring through

the air in their

flight?

There is

a widespread

lack

of

clarity

on

this

question.

I must

confess that

I

could not find

anywhere

in the

specialized

literature

even

the

simplest answer.

I

hope, therefore,

to

give

some

readers

pleasure

when

I

try

to

remedy

this

deficiency

with the

following

short consideration

on

the

theory

of

the motion

of

liquids.

Let

an

incompressible liquid

with

negligible

inner

friction stream

in

the direction

of the

arrows through a pipe

that

is

tapering

off

to

the

right (Fig.

1).

We ask for the

distribution of the

pressure

in the

pipe.

Since the

same

quantity

of fluid

per

second

Fig.

1.

must flow

through every

cross section,

the

velocity

of flow

q

must

be

the lowest at

the

largest

cross

sections and the fastest

at

the

smallest

cross

sections. The velocities

of

the

particles

of

the

liquid

will therefore

be smallest at

L

in

Fig.

1

and will

continuously

increase toward

R.

This acceleration

of

the

particles

of the

liquid can only

be effected

by

the

pressure

forces that act

upon

them. In order for the

cylindrical liquid particle

F

to execute

an

accelerated

movement

toward

the

right,

its

rear-end surface

A must

be

under

a higher pressure

than its front-end surface

B.

The

pressure

at

A

exceeds the

pressure

at

B. Repeating

this

conclusion,

it follows that the

pressure

within the

pipe

decreases

steadily

from

L

to

R.

The

same

distribution of

pressure (decrease

of

pressure

from L to

R)

is found

in

an

analogous

consideration

if

the direction

of

flow

of

the

liquid

is reversed.

In

generalizing,

we

can

state

the

following long-known

theorem

of

the

hydrodynamics

of fluids without friction.

When

we

follow the

path

of

a

liquid

particle

in

a

stationary flow,

we

find the

pressure p always

larger

where the

velocity

q

is

lower and vice

versa.

This theorem is

quantitatively expressed

for

noncompressible liquids by

the well-known

equation

*Translator's

note. Instead

of

the basic term

Auftrieb

=

wing lift,

Einstein

uses

the

extremely complex concept

of

Tragfähigkeit

der

Flügel

=

carrying capacity

of

wings,

which

linguistically

and

falsely suggests

in German

a

rather

elementary

quantity-at least to the

layman.

The R.

v.

Mises

lectures

in

Fluglehre,

which in

part

date

back

to

1913,

represent

the state

of

the art at the time

of

Einstein's

article;

they

were

later

incorporated

into the

Theory

of

Flight

(Dover

reprint,

1945 &

1959).

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