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.