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4 Where Does the Wave Equation Come From?
can understand on the basis of Newtonian mechanics, from its linear dimension L and the
quotient of the force constant D of the elastic springs and the inertia m of both masses.
We will formulate this differently: The motion behaviour of elastic coupled masses
is determined only by Newton’s equations. Wave phenomena occur with it. These can
be verified and proven using only our system of two elastic coupled masses. These
wave phenomena become characteristic for systems made up of many elastic coupled
masses. In the limiting case of infinitely many masses, these wave phenomena occur
in their pure form. This means that the Newtonian motion of such a system is described
by the wave equation. This we will prove in Chap. 5.
Next, we will illustrate using the model of a so-called linear chain, how we arrive
at the well-known elastic oscillations of a medium from the oscillations of single
coupled elastic masses.
We distribute in equal distances along a length L a total mass m as equal point
masses m = m/N in such a manner that the N th mass is at the end of the length
L. Then at the beginning of L, there is no mass. We also consider a spring of the
length L with the force constant D. This spring is then segmented into exactly the
same number of partial springs as there are N masses to be coupled, i.e. N partial
springs of the length L/N with N masses. We then assure ourselves that the force
constant D n , belonging to the spring length L/N , has the value D n = N D. The
force constant D is the relationship of a force F to the absolute deflection s of the
spring, D = F/s. If one hangs the same weight on two identical springs suspended
above each other, then the deflection s is doubled. This can easily be verified by
anyone using two springs. The directional force D is halved when the length of the
spring is doubled, and the force is doubled when the length of the spring is halved.
Up to now, we have represented this in Fig. 4.5.
Now the two end points of the length L are merged together to form one point.
One can imagine this by joining the length L at both of its ends together and forming
a circle. Of course, this would only make sense if N ≥ 3. This is, however, no
restriction for us, because we are interested in a very large value for N . We get
Fig. 4.6. The second representation of this arrangement can be shown by lining up
identical copies of the length L with its N masses in both directions. As a result,
there is now also a mass at the beginning of the length L which is identical to the
mass at the end of the neighbouring length, i.e. their motions and forces are identical.
This enables us to have an infinite number of masses, however the (k + N )th mass
is identical to the kth mass. Such a periodical arrangement of coupled elastic masses
Fig. 4.5 Subdivision of a spring of the length L into eight identical parts
4 Where Does the Wave Equation Come From?
can understand on the basis of Newtonian mechanics, from its linear dimension L and the
quotient of the force constant D of the elastic springs and the inertia m of both masses.
We will formulate this differently: The motion behaviour of elastic coupled masses
is determined only by Newton’s equations. Wave phenomena occur with it. These can
be verified and proven using only our system of two elastic coupled masses. These
wave phenomena become characteristic for systems made up of many elastic coupled
masses. In the limiting case of infinitely many masses, these wave phenomena occur
in their pure form. This means that the Newtonian motion of such a system is described
by the wave equation. This we will prove in Chap. 5.
Next, we will illustrate using the model of a so-called linear chain, how we arrive
at the well-known elastic oscillations of a medium from the oscillations of single
coupled elastic masses.
We distribute in equal distances along a length L a total mass m as equal point
masses m = m/N in such a manner that the N th mass is at the end of the length
L. Then at the beginning of L, there is no mass. We also consider a spring of the
length L with the force constant D. This spring is then segmented into exactly the
same number of partial springs as there are N masses to be coupled, i.e. N partial
springs of the length L/N with N masses. We then assure ourselves that the force
constant D n , belonging to the spring length L/N , has the value D n = N D. The
force constant D is the relationship of a force F to the absolute deflection s of the
spring, D = F/s. If one hangs the same weight on two identical springs suspended
above each other, then the deflection s is doubled. This can easily be verified by
anyone using two springs. The directional force D is halved when the length of the
spring is doubled, and the force is doubled when the length of the spring is halved.
Up to now, we have represented this in Fig. 4.5.
Now the two end points of the length L are merged together to form one point.
One can imagine this by joining the length L at both of its ends together and forming
a circle. Of course, this would only make sense if N ≥ 3. This is, however, no
restriction for us, because we are interested in a very large value for N . We get
Fig. 4.6. The second representation of this arrangement can be shown by lining up
identical copies of the length L with its N masses in both directions. As a result,
there is now also a mass at the beginning of the length L which is identical to the
mass at the end of the neighbouring length, i.e. their motions and forces are identical.
This enables us to have an infinite number of masses, however the (k + N )th mass
is identical to the kth mass. Such a periodical arrangement of coupled elastic masses
Fig. 4.5 Subdivision of a spring of the length L into eight identical parts