Matter waves are of two types, which differ only in the direction of the vibration relative to the direction of propagation. In transverse waves the vibration is perpendicular to the direction of propagation (a plucked violin string, for example). In longitudinal waves the vibration is parallel to the direction of propagation (the pressure waves from a blast, or in front of a piston, for example). Most of the matter waves which are of interest here are, like water waves, a combination of both.

The two basic properties are the pressure (force/unit area) of the wave and its rate of change with time. The former is usually called the amplitude, ψ (dynes/cm2). The latter is usually expressed as the number of times the value of ψ cycles back and forth per second, i.e., as the frequency (cycles/sec).

All matter waves, no matter what the shape, can be expressed as a superposition of simple, sinusoidal waves, of the type discussed in the systems concept and ten useful pillars of mathematical expression.

There are traveling waves and standing waves ( traveling waves and standing waves scheme). A sound wave moving through air travels from its source and imparts an energy to the receiver. This energy is primarily in the direction of propagation, but with scattering some of it becomes transverse.

*traveling waves and standing waves scheme*

By contrast, the standing wave can impart no longitudinal energy—it has none. But it can impart transverse energy to the medium. The generation of the sound by the vibrating violin string is an example.

The intensity, I, of the matter wave is the power delivered by it per unit area. In. other words, I is the rate at which the wave expends energy. All traveling waves move at a certain velocity, υ (cm/sec). Hence the product of amplitude (a pressure) times distance is the energy expended per unit area:

ω = ψ d (dynes/cm2 x cm = ergs/cm2)

The product of amplitude and velocity is the power expended per unit area:

I = ψ υ (dynes/cm2 x cm/ sec = ergs/cm2 sec)

The intensity or power expended per unit area by the traveling wave, is highest for those media having molecules with the greatest number of degrees of freedom in which energy can be stored—gases for example. Both the range and speed of sound are highest in solids, somewhat less in liquids, far less in gases. However, for any medium of constant density, p, the velocity has a fixed value. This fact results in another useful relationship, that between amplitude (pressure) and intensity (power):

I = ψ 2 / υ p

which says simply that power delivered per unit area to any medium is proportional to the pressure squared, if velocity and density are held constant.*

This (I α ψ 2) is a very useful rule-of-thumb, applicable, it turns out, to all field phenomena.

Useful also is the fact that, although low-frequency waves are easily reflected and diffracted by air and hence are non-directional (or will go around corners), high-frequency waves are only slightly scattered by air. Therefore, the latter can be beamed in a preferred direction from a source, and even focused on a particular spot by proper (saucer-like) design of the vibrating source.