Real physical bodies, unlike theoretical point bodies, have not only mass *m* but also extension,
measured by their volume
.
Nevertheless, to understand the mechanical behavior of extended bodies, we can think of them as made by
an infinite number of point bodies, each of them having infinitesimal mass *dm* and infinitesimal volume
d, so that their density *ρ* is

that is

If *ρ* is constant, that is the body is homogeneous,

If the infinitesimal mass *dm*, placed at the point O, interacts with contiguous infinitesimal masses
in a way such that a displacement from O is countered by a force proportional and opposite to the displacement,
that is by an elastic force, the mass *dm*, when displaced, swings with harmonic motion. This motion displaces
the contiguous masses which begin to swing with identical harmonic motion, but delayed by a time depending on
their distance from O and on the internal structure of the body.

This is what happens, for example, when a stone hits a point O on the still surface of a pond. The point O, displaced from its position of balance by the impact of the stone, after the sinking of the stone, is subject to forces opposite to the displacement, due to surface tension forces and intermolecular forces, and begins to swing with harmonic motion. After a time, every other point P of the surface of the pond fluctuate with harmonic motion, as can be verified by observing the motion of any small object floating on the surface of the pond.

The phenomenon of oscillation that invests all points of the surface of the pond is called **sine wave**.
The surface of the pond is the **propagation medium** of the wave.
The sets of the contiguous points of the surface that swing in phase, that is the points that at every instant have
the same displacement, are circumferences called **wavefronts**.
The smallest distance between two wavefronts that oscillate in phase is called **wavelength**.
The point O, from which the wavefronts start, is the **source of the wave**.

In the example, the harmonic motion of O is described by the temporal trend of its displacement *z _{O}*
from the still surface of the pond

where *A*, the amplitude, is the greatest displacement from the untroubled surface.

If we admit that O can keep its amplitude for a long time and that the medium is perfectly elastic, every other
point P of the surface, at a distance *r* from O, after a time *Δt = r/v*, where *v*
is the constant speed *v* at which the disruption spreads, starts swinging with same amplitude and angular
frequency as O. The height *z* of a point P, with respect to *r* and *t* is

In general, what happens to the height of a point on the water surface can happen to many other
scalar or vector physical quantities whose values (magnitude and/or direction, if vectors) depend on position.
Such quantities are said **fields**. In the proposed example, the water surface is a field of altitudes.

When the value of a field Ψ at the point P with position * r(x,y,z)* changes with the time like
the altitude of a point on the pond surface, we say that there is a wave

The (3.6) is said **wave equation**.

With the value of *ω* given in (2.10) we have

The product *vT* represents the distance covered by a wavefront in one period *T*: this is the
**wavelength λ**

We can write

You can see some maybe useful animations in Waves.

If we now remember the expression (2.13) of the total energy of a harmonic oscillator and apply such equation to a
mass *dm* of a material one-dimensional medium harmonically swinging with amplitude *A*
and angular frequency *ω*, we have

Let the infinitesimal volume
d of the considered mass be that of a
spherical shell with center in the source O, radius *r* and thickness *dr*

From (3.8) we have

If we divide both the sides by *dt* we have

The first side of (3.10) shows that the spherical surfaces of the shell are crossed by energy. But the mass
*dm* does not move. It oscillates, but its mean position is constant.

Therefore the equation (3.10) highlights that **when a wave spreads within a medium there is propagation of
energy without motion of mass**.

By comparing this result and that obtained in section 1, we see that the energy can propagate from a source to the surrounding space in two fundamental ways:

**by a flux of particles**; in this case, the energy, concentred within the particles, propagates by discrete, granular amounts;**by waves in a continuous elastic medium**; in this case the energy, after a time, is spread throughout the medium and flows without interruptions.

These two ways differ greatly from each other for a fundamental reason: two waves, emitted from two different
sources, ** may overlap without reciprocal interaction** in the same point of the medium.
In this case the mechanical behavior of the point is forced simultaneously by both the waves, but the waves
go on without any change to their nature, as they were alone. This phenomenon is said

Physicists in 18th and 19th centuries, faced with phenomena of propagation of energy like, for example, emission of light from a gas or of radiation from a cathode, asked themselves whether such phenomena were carried by waves or particles, considering incompatible, mutually exclusive, the two modes.

Light was reckoned to be an ondulatory phenomenon, according to the opinion of Huygens and in opposition to the
opinion of Newton, because Young and other physicists observed that two beams of light can interfere to enhance or
reduce the light intensity. Cathodic rays instead were interpreted as formed by particles called *electrons*.