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.
Frequency
Matter waves have a broad range of frequency, from zero up to the current practical upper limit of about 1,000,000 cycles per sec (cps) in use in some ultrasonic-therapy and submarine-detection studies (Figure 3-1 (c)). The human ear is most sensitive from ~50 to ~10,000 cps; the range of man's ear, however, may be from 20 to 21,000 cps. This, then, is the auditory or sound range. Speech requires 60 to 500 cps. The piano ranges from 27.2 to 4138.4 cps. The great basso profondo, Italo Tajo, could reach a minimum of ~60 cps; the diminutive coloratura soprano, Lili Pons, could hit 1300 cps on a good day. Of course, these are the basic frequencies, and it is understood that a basic frequency generated by any physical vibrator will contain overtones, or harmonics, which are multiples (2x, 4x, even 8x) of the basic frequency. The quality of the tone is determined by the sum of all the components: the basic frequency plus its harmonics.
Training and youth combine to produce a receiver which can hear low power sound up to 12,000 cps. Some musicians can detect overtones from their instruments up to 14,000 cps, but these are few. Most of us can detect frequencies up to 18,000 from a signal generator, if the signal is intense enough, and the odd person can detect up to 21,000 cps. Dogs do it with ease. Porpoises have a phenomenal sonic system in their heads which can sweep frequencies repetitively from a few cycles to many thousands of cycles —both send and receive.
Below and overlapping the auditory range for man is the range (0 to 50 cps) of blast and shock waves, earth tremors, water waves, and the like. The masseur will use vibrations 1 to 50 cps; a ship will roll at 0.1 cps. An air hammer operates at ~ 15 cps, and we hear the overtones.
Above the range of sound, from 20,000 up to > 1,000,000, lies the important range of ultrasound, and the science and technology known as ultrasonics.
Velocity
The speed of matter waves depends sharply upon the medium, and in the case of a gas, its temperature and pressure. For instance, in air at 0°C and 1 atm pressure the speed is 331 meters/sec (mps) (730 miles/hr). In water and soft tissue it is 4 1 2 times higher than in air, and in solids it goes up to 5000 mps. The velocity of sound through fat is 1440, through muscle 1570, and through bone 3360 mps.
Velocity is independent of frequency; and it is probably just as well, otherwise the low tones of the organ might reach our ears later than the high tones of the same chord!
Amplitude and Intensity
There is a minimum pressure and power of matter waves below which the ear cannot detect the wave. This value is about 0.0002 dynes/cm2 , an extremely small value because the ear is very sensitive. The corresponding power or intensity limit is ~109 ergs/cm2 sec, i.e., ~10-16 w/cm2! This value places its sensitivity very close to the threshold of the power in heat motion, and thus very close to the minimum background agitation of matter in our environment. The maximum amplitude the eardrum can stand, without certain irreparable damage resulting, is ~200 dynes/cm2. Therefore, the range of sensitivity of the ear is phenomenally high, one to a million. It is the most sensitive at 1,000 cps.
The sense of touch, particularly on the fingers and tongue, is not nearly so sensitive, but responds down to much lower frequencies.
To our knowledge, man has no detection apparatus for frequencies above about 20,000 cps. However, there is some evidence that ultrasound can penetrate to the brain and cause psychological aberrations, which may or may not be a result of organic damage.
One of the most convenient ways of generating matter waves of controlled frequency is by means of the vibrating crystal. Certain crystals are piezoelectric—that is, they expand or contract if an electric voltage is applied to contacts with two different crystal faces (b). The amount of the
Piezoelectric Crystals scheme
Figure 3-2. About Piezoelectric Crystals: (a) Voltage difference is applied between two opposite faces, (b) The length changes as the applied voltage is changed, (c) Varying voltage, V, gives varying length, y. (d) Concave radiator concentrates matter waves on a target.
expansion or contraction increases with increasing applied voltage. Quartz and barium titanate are currently in wide use. If the applied voltage is varied, the crystal shape varies accordingly, or vibrates, and the matter wave so established is transmitted by contact with the medium. The amplitude of the vibration is higher the higher the vibrating voltage applied. The frequency of vibration follows that of the electrical signal, if the crystal is not too big. "The Piezoelectric Crystals scheme" illustrates these points.
Apparatus with output which ranges from a few to a million cycles per second, and from next to nothing up to a few hundred watts per square centimeter of crystal, has been built and used.
Constructed with a concave radiating surface (d), an array of piezoelectric crystals, if properly oriented, can be made to focus an intense beam of matter waves at a point a few centimeters from the radiating surface.
For example, in recent therapeutic work beams of 1 Mc (1,000,000 cps) were focused on a small target, and delivered energy at a rate (intensity) of 8 kw/cm2 of cross-section of the target!
Absorption
If waves are diverging, or being dissipated or scattered, the important general rule, called the "inverse square law," is obeyed. It says simply that the intensity, /, decreases as the distance from the source gets larger, in such a manner that if, for example, the distance between source and receiver is doubled, the intensity at the receiver falls to only one quarter. Quantitatively,
I(x)α 1 x2
where I(x) is the intensity at any distance, x, away from the source. See Figure 3-3.
If a parallel beam of matter waves is absorbed by the medium, the rate of absorption at a point is proportional to the intensity at that point; or
dl/dx = -kl
which integrates (see Chapter 1) to
I = I0e-kx
if I0 is the value of I where x = 0.
For the case in which the waves are diverging and also being absorbed, a linear combination of the inverse square law and the absorption law applies.
The energy absorbed from the matter-wave beam by the medium contributes to the thermal motion of the molecules of the medium. The absorption coefficient, k, is intimately related to several physical properties of the medium.
Inverse square law
Figure 3-3. Inverse Square Law. Radiation from source S diverges. Intensity (w/cm2) at distance, d, is four times the intensity at 2d because the same radiation is spread through four times the area by the time it reaches 2d.
However, there are two principal mechanisms of absorption of matter waves by tissue:
(a) Functional resistance:
The momentum of the propagation, which is directional (Fig. 3-1 (a)), is passed to the molecules of the tissue, which become momentarily polarized by the pulse of pressure. The directed energy thus received quickly decays into random, non-directional molecular motion.
This mechanism can be called "molecular absorption." It is important at medium and high frequencies.
(b) Elastic reactance of the bulk tissue:
Absorption occurs by movement of the bulk material; mass is displaced, and macro-oscillations result in sympathy with the impinging, oscillating pressure. Because the tissue is not perfectly elastic (i.e., the molecules will realign themselves so that they won't be polarized), the absorbed energy quickly dissipates in front of the pressure pulse as molecular motion or heat. This is the only method by which energy is absorbed at low frequencies—during earth tremors, train rumble, or massage, for example. This mechanism can be called "elastic absorption."
Reflection, due to the inertia of the tissue (its tendency to remain at rest unless forced to do otherwise—Newton's first law of motion), occurs at high frequencies for soft tissue and even at low frequencies for dense tissue such as bone. Truly elastic tissues simply reflect incident matter waves.
The absorption coefficient for molecular absorption (k) is well known for air and water:
where I is the frequency (cps) of the impinging wave, υ the velocity (cm/sec), ρ the density (g/cm3), η the viscosity (dyne sec/cm2), κ τ the heat conductivity (cal/sec deg cm), and the c's are the specific heats (cal/deg g) at constant pressure, P, and constant volume, V. Hence the energy absorbed per centimeter of penetration of the impinging wave increases linearly with the viscosity or "stickiness" of the medium and with its thermal conductivity; increases very rapidly with increasing frequency; but decreases with increasing density.
For water, which is a sufficiently good approximation to soft tissue for present purposes, k/f2 = 8.5 x 10-17 17 sec2/cm. For air the value is 1000 times higher, because although η is 50 times smaller for air than for water, υ is 41/2 times smaller and p is 1000 times smaller. For liquids only the first term (the frictional or viscous one) is important; for gases both are important.
Therefore it is useful to aerate a tissue before sonic therapy is applied, because absorption is higher.
Since reflection increases with increasing frequency, the method of application is important. In the absence of reflection, the above expressions describe the situation well. Direct application of the vibrator to the tissue assures this. However, if the sound is beamed through air, the situation is quite different: reflection occurs.
Quantitative studies on tissues are only recent. The general rule which has emerged is as follows: Beamed through air, sound of high frequency suffers little absorption and little damage results. The depth of penetration increases with increasing frequency. Most (>95 per cent) of the incident energy passes right through, or is reflected. Some of Von Gierke's figures (1950) are: 5 to 6 per cent absorbed at 100 cps; 0.2 to 4 per cent absorbed at 1000 cps; and <0.4 per cent absorbed at 10 kc. Beamed through liquid or solid, ultrasonic radiation is easily controlled and its absorption predicted. More will be said about this later, in the section on therapy.
