5 The Wave Equation and the Third Axiom
41
We will now illustrate how to arrive at the mechanics for a continuum from
Newtonian point mechanics. Our special attention will be turned to the derivation of
the wave equation (1), which in Chap. 2 we took as granted from the field of acoustics,
from the Newtonian system of point mechanics.
We will presume that interaction forces, i.e. internal forces F B A (= force of mass
M B on mass M A ) and further external forces F A (= external forces on M A ) for
a system of N masses M A , A = 1, 2, . . . , N are present. For the positions X A of
the masses M A in an inertial system (which we always presume), the following
Newtonian equations are valid:
d
dt
M A
d
dt
X A
=
N
B =A
F B A + F A ,
F AB = −F B A .
⎫
⎪ ⎬
⎪ ⎭
(53)
(B = A means that the force from M A does not influence itself). The second equation
of (53) is Newton’s reaction axiom, the Third Axiom. It can however be (and in most
cases usually is) that X A do not represent the positions of physical particles, but
represent the centres of inertia of n A masses m a at the position x a , a = 1, 2, . . . , n A ,
so that
M A =
n A
a=1
m a ,
X A =
n A
a=1
m a x a
M A
,
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(54)
see Fig. 5.2. However, because of the validity of Newton’s mechanics between all
n masses m a (n =
N
A=1
n A ), internal forces f ba that suffice the Third Axiom as well
as other external forces f a apply, so that once again the Newtonian equations are
applicable,
d
dt
m a
d
dt
x a
=
n
b =a
f ba + f a ,
f ab = −f ba .
⎫
⎪ ⎬
⎪ ⎭
(55)
The supposed forces F B A and F A are defined by the forces f ba and f a as
F B A =
n B
b=1
n A
a=1
f ba ,
F A =
n A
a=1
f a .
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
(56)
41
We will now illustrate how to arrive at the mechanics for a continuum from
Newtonian point mechanics. Our special attention will be turned to the derivation of
the wave equation (1), which in Chap. 2 we took as granted from the field of acoustics,
from the Newtonian system of point mechanics.
We will presume that interaction forces, i.e. internal forces F B A (= force of mass
M B on mass M A ) and further external forces F A (= external forces on M A ) for
a system of N masses M A , A = 1, 2, . . . , N are present. For the positions X A of
the masses M A in an inertial system (which we always presume), the following
Newtonian equations are valid:
d
dt
M A
d
dt
X A
=
N
B =A
F B A + F A ,
F AB = −F B A .
⎫
⎪ ⎬
⎪ ⎭
(53)
(B = A means that the force from M A does not influence itself). The second equation
of (53) is Newton’s reaction axiom, the Third Axiom. It can however be (and in most
cases usually is) that X A do not represent the positions of physical particles, but
represent the centres of inertia of n A masses m a at the position x a , a = 1, 2, . . . , n A ,
so that
M A =
n A
a=1
m a ,
X A =
n A
a=1
m a x a
M A
,
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(54)
see Fig. 5.2. However, because of the validity of Newton’s mechanics between all
n masses m a (n =
N
A=1
n A ), internal forces f ba that suffice the Third Axiom as well
as other external forces f a apply, so that once again the Newtonian equations are
applicable,
d
dt
m a
d
dt
x a
=
n
b =a
f ba + f a ,
f ab = −f ba .
⎫
⎪ ⎬
⎪ ⎭
(55)
The supposed forces F B A and F A are defined by the forces f ba and f a as
F B A =
n B
b=1
n A
a=1
f ba ,
F A =
n A
a=1
f a .
⎫
⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎭
(56)