4 Where Does the Wave Equation Come From?
25
Fig. 4.4 Two oscillating masses m/2
We therefore receive the equations for the motion of both masses,
m
2
¨
s 1 = −
5
4
D s 1 +
3
4
D s 2 ,
m
2
¨
s 2 = −
5
4
D s 2 +
3
4
D s 1 .
⎫
⎪ ⎬
⎪ ⎭
(17)
The motions s 1 and s 2 of both masses m/2 can be reduced to two elementary oscillations, the additive displacement a and the relative displacement r of the masses in
this system according to
a =
1
2
(s 2 + s 1 ) , r =
1
2
(s 2 − s 1 )
and
s 1 = a − r ,
s 2 = a + r .
⎫
⎪ ⎬
⎪ ⎭
(18)
Using addition or subtraction of Eq. (17), we find for a and r
m ¨
a = −D a ,
m ¨
r = − 4D r .
(19)
The system of Eq. (17) is thus separated. In other words, we have for our system
containing two masses, two independent harmonic equations of oscillation, each
equation having the form of a harmonic equation of oscillation of a single mass (13).
Our system can therefore oscillate harmonically in phase if alone the solution of the
first Eq. (19) differs from zero,
a(t) = a o cos(ωt + φ) ,
r (t) = 0 ,
←→
s 1 = a o cos(ωt + φ) ,
s 2 = a o cos(ωt + φ) ,
(20)