268
XVI
FLOATING ELASTIC PLATES
C is equal to 57721.. The rows are so arranged that each
horizontal row by itself represents a particular integral of the
given differential equation. This form shows that z remains
finite when p = 0; further, the integral over a small circular
area surrounding the origin is
¹zdw
2πLt
[v'adw = 2# [po'zip - 2#14 ( p 20³½
=
др /о S
P.
Hence the integral considered is the required one; at the
same time the form (3) is one very suitable for the numerical
calculation of z for small values of p. For large values we
use a semi-convergent series which one gets from (2) by
expanding the root and integrating the separate terms, and
whose first terms are
2=
a²P
пе
- αρνί
2πTS 2 √αp
N
-
1
8p
Jap
sin (ap √++)
8/
(4).
sin (ap √ +3)+...}
8
The solution can be expressed in several additional forms.
We shall interpret the above in the following remarks.
=
1. At the place where the weight is put, the indentation
of the plate has its greatest value z = zo a²P/8s. The plate
rises from there in all directions towards the level zero; at
first slowly, then faster, then again more slowly. At the
distance pa, z=646%; for p = 2a, z = •25820; for p =
3a,
z='066% Near the distance p = √2a = 3·887a, z changes
sign and thus the plate appears raised into a ridge round the
central depression. But it is extraordinary that at further
distances from the origin ridges and hollows follow each other
with the period T√2. a. The plate is thrown into a series of
circular waves; it is true that they diminish so rapidly in
height as we go outwards that we need not wonder at being
unable to see them without special arrangements. The
quantity a, which is characteristic of the system of waves, is a
length. To calculate it for the case of ice floating on water,
we notice that s = 10-kg/mm³; μ can be taken as ; and,