PHYSICS OF A SUBMARINE

G. P. Harnwell*)

The modern submarine is in reality a diving boat. For most of the time it is in the surfaced condition and is capable of submerging only for a limited period. Its outlines, borrowed, as can be seen from Fig. 1, from a surface vessel, provide it with stability on the surface of the water. It is equipped with a large bridge for navigation, has twin screws, a diesel engine of approximately 6500 h.p. for surface running, and the usual radio, radar, and signaling equipment. This circumstance is very important, since it means that in designing submarines one must not only take into account the conditions of underwater navigation, but also make many compromises with the requirements of navigation in the surfaced condition. After submergence the displacement of the boat reaches 1500 tons, and in its operating conditions it resembles an airplane or an airship more than a surface vessel. Having “broken away” from the water surface, it moves in three dimensions, and its vertical control and stability are matters of constant concern. Its surfacing and submergence have many technical features in common with the takeoff and landing of an airplane. The bow and stern horizontal rudders play the same role in vertical maneuvering as elevators and ailerons. Mechanical damage constitutes, in both cases, accidents entailing consequences incomparably more severe than in travel over the surface of the earth or water.

The physics of the submarine is concerned chiefly with the differences between the physical properties of seawater and of air, which determine the profound difference between devices intended for motion in these two media. The physical laws are, of course, identical, but the differences in chemical composition, density, compressibility, and electrical conductivity are so great that they impose entirely different requirements on design and equipment.

*) American Journal of Physics 16, No. 3, 127–150 (1948). Abridged translation by M. Antokolsky.

PHYSICS OF THE SUBMARINE

Work in the depths of the sea, for us who are accustomed to living, moving, and communicating in air, requires complete renunciation of all habits and notions. As soon as the diving signal has sounded and the waves have closed over the boat, it enters a new and unfamiliar world. The slightly increased pressure on the ears tells us that all openings are hermetically sealed, and, at a sufficient distance from the surface of the water, any sensation of motion is lost. In the hull of the boat there is not a single window, since sunlight penetrates only to a very small depth, and the water of the ocean is too turbid for the use of a searchlight. This is reminiscent of flying in an airplane in a dense fog, with the difference that electric motors make almost no noise. All the usual means of contact with the external world are lost, and navigation and maneuvering are “blind” and are carried out by means of ingenious devices specially invented for submarines.

Fig. 1. Section of a typical submarine. 1 — stern torpedo room; 2 — steering room; 3 — motor room; 4 — engine room; 5 — forward torpedo room; 6 — mine compartment; 7 — room for the crew; 8 — wardroom; 9 — accumulator compartment; 10 — midshipmen’s mess; 11 — corridor; 12 — galley; 13 — officers’ compartment; 14 — radio room; 15 — central control station; 16 — conning tower; 17 — compressor station; 18 — pumps; 19 — an oil-filtering unit; 20 — officers’ messroom; 21 — officers’ quarters; 22 — new accumulator; 23 — tool storage; 24 — diving torpedo compartment; 25 — diving compartment; 26 — rudder storage room.

The unwillingness to break away completely from the environment natural to us and to our mechanical devices, as well as the fact that the opaque ocean, even with very slight submergence, provides a reliable refuge, led to the bold decision to retain two surface organs: the periscope and the “snorkel.” The periscope is a triumph of the opticians’ art, which has supplied the submariner with a high-quality telescope allowing him to survey the surface of the water and the air above it,

projecting a tube no more than an inch in diameter several feet above the water. The snorkel, which is a double intake-and-exhaust pipe for the diesel, rises from the roof of the boat like a snail’s horn. Although it reveals the boat to a somewhat greater degree than a periscope, it nevertheless allows the boat to move under diesel power without expending battery charge. It rises only a few feet above the surface of the ocean and is fitted with a floating valve that prevents water from entering the pipe when the latter is submerged under water by waves. But these two auxiliary devices, reflecting the present desire to make the submarine remain a surface vessel, do not belong among the details that characterize the submarine proper.

Fig. 2. Compression of the submarine hull.

BUOYANCY AND VERTICAL MANEUVERABILITY

A submarine moves in a medium almost 800 times denser than air. Immersion by each 10 m adds one atmosphere of pressure, so that at a depth of 200 m the excess pressure is about 20 kg/cm².

The pressure hull of a submarine is given the shape of an elongated ellipsoid so that the steel plates, owing to their curvature, can more easily withstand this pressure. If the hydrostatic pressure is equal to $p$, then the force per unit length pressing together the two halves separated in Fig. 2 by the dotted line is equal to $2Rp$. This pressure must be resisted by a wall of thickness $t$, so that the circumferential stress in the latter is

$$ \frac{pR}{t}. $$

If the diameter of the boat is about 6 m, and the wall thickness is 3 cm, then the stress in the steel is 100 times greater than the pressure of the water. Thus a depth of 200 m corresponds to a compression in the steel equal to 200 kg/cm². To obtain an adequate margin of strength, steel with a temporary resistance of 10,000 kg/cm² is therefore required. All openings and hatches that lead to an increase in stresses must be considerably reinforced, and every superfluous opening in the hull must be regarded as extremely undesirable.

The conning tower, which remains habitable even when the boat is submerged, has the shape of a cylinder parallel to the pressure hull, to the upper part of which it is welded, but shorter than it and smaller in diameter. Since the radius of curvature of the tower is smaller than that of the hull, the stresses in the steel, with the same wall thickness, are smaller. The difficulty-

of fabrication and welding of steel plates having a thickness exceeding several centimeters limits the diameter of the hull; and, in order to increase the cross-sectional area of the boat, recourse is sometimes had to welding together two ordinary hulls, forming a transverse section in the shape of a figure eight.

When a submarine is in motion, the stern and bow rudders give it maneuverability, but its equilibrium depends on careful adjustment of the distribution of weight. A surface vessel has a large excess of buoyancy and, generally speaking, possesses great vertical stability.

In contrast, a submerged submarine, although it does not experience rolling, is in unstable equilibrium, and only with a definite weight and a strictly fixed distribution of it is it maintained at a constant depth and in the proper position. The positive buoyancy necessary for surfacing is provided by the main ballast tanks, located outside the strong hull, between it and the thin outer plating. These tanks are blown with compressed air, which expels water through valve closures located in the upper part, and are filled for submergence by opening kingston valves at the top and bottom of the tanks. Since the pressure inside the tanks is at all times equal to the external pressure, they require no structural reinforcement. An entirely independent trimming system consists of tanks in the bow, amidships, and stern of the vessel, and serves to correct the neutral buoyancy, which changes as fuel, ammunition, and other stores are consumed, and to maintain the proper moment of buoyancy. Maintaining transverse stability presents no particular difficulty, owing to the relatively short arms of the rotating forces; but in the longitudinal direction the great length of the vessel makes the inclination of the keel highly sensitive to small inequalities in the weight of the bow and stern. Since these tanks are filled only partially, they must withstand external pressure. The trimming itself is carried out by taking in or pumping out water, or by transferring it from one tank to another.

For the best control of the boat, nearly neutral buoyancy is necessary, but observing this requirement may prove rather difficult, since both the volume of displaced water and its density vary with the depth of submergence. Both the submarine and the water are compressible and occupy a smaller volume at great depths. Likewise, the temperature of the water changes with depth—usually the water is warmer near the surface—and the thermal expansion of water also affects buoyancy.

Assuming that the boat is in thermal equilibrium with the water at a given depth, one can carry out the following analysis of the problem of equilibrium. The total submerging force \(F\) is equal to \(Mg - V\rho g\), where \(M\) is the mass of the boat, \(V\) is its volume, \(\rho\) is the density

water, and \(g\) is the acceleration of gravity. At equilibrium this force must be equal to zero, and for the stability of the equilibrium it is necessary that an upward displacement should produce an increase of this force, and conversely. Taking the positive \(x\)-axis downward in the direction of \(F\), and recalling that \(\rho\) and \(V\) are functions of \(x\) by virtue of their dependence on the pressure \(p\) and the temperature \(T\), we can compute the first variation of \(F\). Since

\[ F=Mg-\rho Vg, \]

we have

\[ \delta F=-(\rho\,\delta V+V\,\delta\rho)g = \left[ \rho\left(\frac{\partial V}{\partial p}\frac{dp}{dx} +\frac{\partial V}{\partial T}\frac{dT}{dx}\right) + V\left(\frac{\partial\rho}{\partial p}\frac{dp}{dx} +\frac{\partial\rho}{\partial T}\frac{dT}{dx}\right) \right]g\delta x = -\left[ (\alpha_{\mathrm{w}}-\alpha_{\mathrm{l}})\frac{dp}{dx} - (\beta_{\mathrm{w}}-\beta_{\mathrm{l}})\frac{dT}{dx} \right]Mg\delta x . \]

Here

\[ \alpha=-\frac{1}{V}\frac{\partial V}{\partial p}, \qquad \beta=\frac{1}{V}\frac{\partial V}{\partial T}, \]

and the subscripts “w” and “l” denote, respectively, water and the submarine. For the stability of equilibrium the expression in brackets must be positive, and since

\[ \frac{dp}{dx}\simeq \rho g, \]

the stability condition is

\[ (\alpha_{\mathrm{w}}-\alpha_{\mathrm{l}})\rho g> (\beta_{\mathrm{w}}-\beta_{\mathrm{l}})\frac{dT}{dx}. \]

Fig. 3. Equilibration of a submarine.

For a zero temperature gradient this means that the compressibility of water must exceed the compressibility of the submarine. But this is practically unattainable, since \(\alpha_{\mathrm{w}}\simeq 5\cdot10^{-11}\ \mathrm{dyn}^{-1}\), while \(\alpha_{\mathrm{l}}\) for a thin hull, as is easily shown, is approximately \(5\cdot10^{-10}\ \mathrm{dyn}^{-1}\). If the lower edges of the ballast tank are open, then water can enter the tank, compressing the air in it, and we obtain a very unstable equilibrium. Since, moreover, \(\beta_{\mathrm{w}}>\beta_{\mathrm{l}}\), \(\dfrac{dT}{dx}\) must be negative and greater in absolute value than

\[ \rho g\,\frac{\alpha_{\mathrm{w}}-\alpha_{\mathrm{l}}}{\beta_{\mathrm{w}}-\beta_{\mathrm{l}}}. \]

Substituting the typical values \(\beta_{\mathrm{w}}\simeq 10^{-3}\) and \(\beta_{\mathrm{l}}\simeq 4\cdot10^{-5}\ (\mathrm{deg.\ C})^{-1}\), and the above values of \(\alpha_{\mathrm{w}}\) and \(\alpha_{\mathrm{l}}\), we find that the temperature gradient for stable equilibrium must exceed approximately \(5\cdot10^{-2}\ ^\circ\mathrm{C}\) per meter of depth. Such gradients are often found at depths of less than \(100\ \mathrm{m}\), and in such regions, which have received the name “thermoclines” (Fig. 3), a submarine can maintain stable equilibrium with its engine not operating. One more circumstance deserves mention—

this is that the density of water increases with its salinity. This can prove very dangerous for a submarine moving in the direction of a river mouth: in the absence of careful observation of the manometer it may suddenly find itself on the bottom. In view of this variety of factors affecting buoyancy, boats are equipped not only with manometers indicating depth, but also with self-recording thermometers, and sometimes also with salinity meters.

PROPULSION OF THE BOAT AND ITS SPEED

The chief difference between air and water, from the point of view of a self-propelled vessel, is that only the former contains the necessary component for the chemical reaction that serves as the source of energy for motion. An airplane, for its engine to operate at great altitudes, may require the use of a compressor, but so long as it remains in the earth’s atmosphere it has oxygen for burning fuel. A submerged submarine has no free oxygen at its disposal, and the sometimes proposed electrolytic decomposition of water and combustion of the oxygen and hydrogen thereby obtained contradicts the principle of conservation of energy. Consequently, in a submarine all the energy needed for its motion must be stored in a convenient form.

If the energy is stored in chemical form, the fuel may be the same as on any other vessel, but the difference lies in the supply of oxygen. Technology offers two possible solutions to this problem: compressed oxygen in cylinders and hydrogen peroxide. Both methods are expensive and not entirely safe. The cheapest and most convenient fuels are hydrocarbons, as a result of which the product of combustion is carbon dioxide, a large quantity of which must be pumped out of the boat against the pressure of the outside water, which considerably lowers the coefficient of efficiency. Thus, the use of internal-combustion engines presents serious difficulties. They are partly resolved by such motors as the Walter engine, although the cost of the fuel is extraordinarily high. The only practical method of storing energy at the present time remains the storage battery, which, in comparison with chemical methods, possesses all the advantages except only a considerably smaller specific reserve of energy per 1 kg of weight. The battery has a high coefficient of efficiency, is quite safe when properly installed, develops little heat, makes no noise, has a relatively long service life, and converts energy into mechanical energy by means of convenient motors. As a result, batteries are the generally used source of energy for boats when submerged; however, in contrast, for example, to oil, which has a heat of combustion of \(10\ \mathrm{kWh/kg}\), they store energy only in the amount of

0.1 kWh/kg. Therefore, a submarine in the surfaced position has a cruising endurance measured in months, whereas in the submerged position it can move at an appreciable speed only for several hours, after which it must surface in order to charge its batteries with the aid of its diesels. A uranium high-temperature reactor might prove to be an ideal engine for submarines, since the mass of the substance consumed is insignificant and there is no problem of exhaust gases.

However, such engines do not yet exist.

The density of water is the principal factor determining the difference in the speeds attained by airplanes and submarines. The speeds at which viscous forces play an essential role lie below the speeds of submarines. Forces caused by compressibility become essential only at speeds comparable with the speed of sound. This effect is important for airplanes, but not for submarines, since the speed of sound in water is about 1.5 km/sec. For submarines, as also for airplanes at subsonic speed, there is an intermediate case in which the resistance force is associated with the turbulent motion of the fluid. A force of this origin depends on the density, speed, and linear dimensions of the moving body (Fig. 4). Quantitatively it is proportional to the quantity \(a \rho v^{2}\), where \(a\) is a coefficient characterizing the vessel and called the effective frontal area, \(v\) is the speed of the vessel, and \(\rho\) is the density of the fluid. Thus, for streamlined bodies having the same frontal area, the resistance forces are proportional to the density of the medium in which they move and to the square of the speed.

Fig. 4. Fluid resistance.

For water and air, comparison in terms of power is more indicative. Denoting by \(P_c\) and \(P_l\) the power required for an airplane and a submarine, and recalling that \(P = Fv\), we obtain:

\[ \frac{P_c}{P_l}=\frac{\rho_{\text{air}} v_c^{3}}{\rho_{\text{water}} v_l^{3}} \]

or

\[ \frac{v_c}{v_l}=\left(\frac{P_c \rho_{\text{water}}}{P_l \rho_{\text{air}}}\right)^{1/3}. \]

Thus, an airplane with an 8000 h.p. engine, as might be expected, will move 20 times faster than a 1000 h.p. boat in water, which is 1000 times denser than air. This illustrates the difference between an airplane traveling at 300 miles per hour and the 15-knot speed of a submarine. Obviously, this fundamental circumstance limits the magnitude of the speeds attainable by submarines.

It is interesting to note that one of the sources of resistance to the motion of a surface vessel is absent in submarines—namely, the surface wave that spreads, when a surface vessel moves, from each of its sides. The amount of energy expended in creating this wave can be computed, to a very close approximation, as follows. Consider the system of standing waves shown in Fig. 5 at the moment of maximum. At this moment the surface of the liquid is at rest and, by assumption, has the form of a sinusoid. The amount of energy per unit width and over a length equal to the wavelength \(\lambda\) (at the moment of maximum all the energy is in the form of potential energy) is given by the equation

Fig. 5. Potential energy of a surface wave.

\[ \int_0^\lambda \rho g y\, dx\, \frac{1}{2}y = \frac{\lambda}{4\pi}\lambda \rho g \int_0^{2\pi} A'^2 \sin \frac{2\pi x}{\lambda}\, d\left(\frac{2\pi x}{\lambda}\right) = \frac{1}{4}\rho g \lambda A'^2, \]

where \(A'\) is the amplitude of the standing wave. This wave may be represented as consisting of two traveling waves of amplitude \(A=\frac{1}{2}A'\), propagating in opposite directions. Hence the energy per unit area in a traveling wave is equal to \(\frac{1}{2}\rho g A^2\). Thus, if the wave created by a ship has an amplitude of \(1\) m, a width in the direction normal to the crest of about \(5\) m, and propagates with a speed of \(10\) m/sec, then the power required to create two such waves, one on each side of the vessel, is approximately \(700\) h.p. This constitutes a very considerable part of the total power of a small vessel and noticeably reduces its speed. But the surface disturbance decreases exponentially with depth, so that this resistance is absent for a sufficiently submerged submarine. Thus, if the vessel is given a well-streamlined form, then, with the same expenditure of power, it can move faster in the submerged position than on the surface.

UNDERWATER MEANS OF COMMUNICATION

It may be that the greatest difference between the conditions existing under water and those familiar to us lies in the turbidity of ocean water and its opacity to electromagnetic waves.

Most of the most accurate information we obtain by means of vision; meanwhile the conditions of visibility beneath the surface of the water are very unfavorable. The usual greenish-blue color of ocean water corresponds to the greatest optical transparency of the water, but even for such regions the maximum radius of visibility does not exceed several hundred feet. Intense light is detected at a greater distance as well, but in that case all details become invisible, and only a general increase in illumination is detected, just as the sun is seen through a dense fog.

[Figure labels: “Amplitude of the electromagnetic vector”; “Air”; “Sea”; \(d(1/10)\); \(d(1/1000)\); “depth of attenuation of the electromagnetic vector to \(1/n\) of its value at the surface”; \(d(1/1000)\); “depth of attenuation to \(1/1000\)”; “Depth in meters (logarithmic scale)”; “Conductivity of seawater taken as 5 ohm\(^{-1}\) per meter”; “Wavelength in meters (logarithmic scale)”; \(1\), \(10\), \(100\), \(1000\), \(10000\), \(100000\); \(1\) mm, \(1\) cm, \(10\) cm, \(1\) m, \(10\) m.]

Fig. 6. Absorption of electromagnetic waves.

The application of radio communication and radar is connected with the question of the propagation of the longer waves of the electromagnetic spectrum. One may expect that classical electromagnetic theory will prove quite suitable for describing the propagation of these waves, which have a relatively low frequency. The electrical conductivity of seawater \(\sigma\) is very great, of the order of \(5\ \mathrm{mho/cm}\). The amplitude of electromagnetic waves, when propagating in a conducting medium, decreases exponentially, and the exponent for 10 is equal to \(14.6\,x\sqrt{\frac{\sigma}{\lambda_0}}\), where \(x\) is the distance traversed and \(\lambda_0\) is the wavelength in vacuum, both expressed in meters. Thus, in seawater the amplitude of the electric or magnetic field decreases by a factor of 10 over a path equal to 0.03 of the square root of the wavelength in vacuum, if the lengths are expressed in meters (Fig. 6).

The factor of exponential attenuation is so predominant that the simplest conclusions can be drawn without any allowance for such factors as the angle of incidence of the radiation on the sea surface and the reflection coefficient. Long radio waves, say of \(10\,000\) m, are attenuated by a factor of 10 for every 3 m of path. Assuming that the signal inten-

intensity, a thousand times less than its intensity at the surface of the water, could still be received by a submarine, we conclude that its antenna may be at a maximum depth of 9 m. It is important to note that this refers to the reception of a signal, but by no means to anything resembling vision: such radio waves cannot be used to detect the contours of any object whose dimensions are less than several hundred kilometers. The waves used in radars are capable of detecting an object having the dimensions of a ship, but the small penetration of these waves into water makes them unsuitable in the case of submarines. It is of interest, for completeness, to check whether it is possible to use electromagnetic waves of frequency higher than that of visible light. In the region of X-rays, γ-rays, and cosmic rays, the mechanisms responsible for absorption of energy in a beam of rays are the quantum processes of photoionization, Compton scattering, and the production of electron–positron pairs. The absorption curves in water for radiation of high energies show that the absorption coefficient changes little with energy, although it passes through a minimum at several million electron-volts. But even this most penetrating radiation is attenuated to one tenth of its intensity over a path of 15 cm. Thus, this radiation is also of little use for underwater communication or signaling.

Since electromagnetic waves prove to be so unsuitable for purposes of communication or navigation, the only alternative for replacing them turns out to be sound. For the use of sound, the properties of pure water are extremely favorable. The attenuation of sound waves in water is much less than in air. Thermal conductivity and viscosity are not capable of explaining the observed magnitudes of the sound absorption coefficient, and the true mechanism of energy loss still remains, to some extent, obscure*). However, the attenuation of the intensity of a plane sound wave can be approximately expressed by the empirical formula \(10^{-10^{-12}\nu^2 x}\) for the frequency range \(\nu\) from 50 to 50,000 cycles/sec., where \(x\) is expressed in meters. Thus, a sound wave in seawater is attenuated to one tenth of its intensity over a path \(x = 10^{12}\nu^{-2}\) m for \(\nu = 100\) cycles/sec. This is a very large distance, and even for \(\nu = 10^4\) cycles/sec., \(x = 10\) km.

It follows from this that absorption plays a very small role, except at the very highest frequencies, and the attenuation of sound waves in water is caused mainly by the divergence of the spherical wave from its source. The speed of propagation of sound in water is approximately five times higher than in air, dispersion is practically absent, and Doppler effects caused by flow in the medium are negligible, since here high speeds of sound are combined with the usually low velocities of currents.

*) For a discussion of this interesting question, see F. B. Fox and G. D. Rock, Physical Review 70, 68 (1946).

G. P. GARNWELL

ACOUSTIC PROPERTIES OF THE SEA

Although, thus, sound represents the best means for transmitting information under water, its use is associated with all sorts of difficulties, and submarines up to the present time have been forced to operate with incomparably less knowledge of the situation than aircraft. In this respect the difference is as great as it is in speed. Some of the difficulties with which the use of sound is associated will become clear from a brief characterization of the acoustic properties of the sea.

The equation expressing the displacement \(\xi\) of a particle in one dimension in an ideally elastic medium is the equation of motion of a plane sound wave

\[ \frac{\partial^2 \xi}{\partial x^2}=\rho\alpha\,\frac{\partial^2 \xi}{\partial t^2}, \]

where \(t\) is time, and the other notations were indicated above. A solution of this equation is any function of the form \(f\left(t\pm \frac{x}{v}\right)\), where \(v=\)

\[ =\frac{1}{\sqrt{\rho\alpha}}. \]

From the above-given values of these quantities, \(v\) proves to be close to \(1500\ \mathrm{m/sec}\). This is much lower than the speed of electromagnetic waves, so that information about distant events is noticeably delayed. The equation of continuity

\[ \rho\alpha=-\frac{\partial \xi}{\partial x} \]

relates pressure and displacement. The amplitude value of the pressure in a wave is limited approximately by the hydrostatic pressure, owing to the tendency toward cavitation that appears at negative pressures. This limits the intensity of a sound wave to the magnitude

\[ \frac{1}{2}p^2\sqrt{\frac{\alpha}{\rho}}. \]

Near the surface of the water, where \(p\) does not greatly exceed atmospheric pressure, the intensity of a harmonic wave of long wavelength cannot exceed \(1\ \mathrm{W/cm^2}\). With a practically permissible radiator area of the order of \(1000\ \mathrm{cm^2}\), this radiation is quite weak in comparison with powerful radio transmitters.

It is interesting to note that the ratio of pressure to particle velocity in a sound wave, known as the acoustic impedance, is determined by the formula \(\sqrt{\frac{\rho}{\alpha}}\) or \(\rho v\). For water this quantity is about \(1.5\cdot 10^5\ \mathrm{dyn\ sec/cm^3}\), and for air, \(36\ \mathrm{dyn\ sec/cm^3}\). Thus, a sound wave in water, in comparison with a wave in air, is characterized by high pressures and small displacements. This completely changes the character of microphones and radiators. Magnetostrictive and piezoelectric transducers are used in them, since they give the optimal impedance ratio and

at the same time the most advantageous conditions for conversion between electrical and acoustic energies. For the same reason, sound is transmitted very weakly through the water–air surface: the ratio of impedances here is so unfavorable that no more than \(10^{-4}\) of the intensity of the incident wave passes into the other medium. Thus, the sea surface is an almost ideal reflector of sound waves; however, because of its oscillating and uneven shape it gives regular reflection only for very long waves or for oblique incidence.

Near the sea surface there are usually small bubbles. They arise as a result of wave motion and other causes, such as, for example, the movement of ships in a rough sea. The influence that they exert on the propagation of sound is by no means proportional to the small volume that they occupy. Water containing a small number of small bubbles differs little in density from pure water, but because the air in the bubbles is easily compressible, the quantity \(a\) in it is considerably larger. Indeed, the effective value of \(a\), as is easy to see, is equal to

\[ a_{\text{water}}\left(1+\frac{a_{\text{air}}}{a_{\text{water}}}\frac{V_{\text{air}}}{V_{\text{water}}}\right). \]

If, for example, one bubble of diameter \(1\ \mathrm{mm}\) occurs on average in \(1\ \mathrm{m^3}\) of water, then the expression written gives, for the compressibility of water with such a bubble content, the value \(20a_{\text{water}}\); here \(a_{\text{air}}\), equal to \(\frac{1}{p}\), is taken for a depth of about \(10\ \mathrm{m}\). This reduces the propagation speed by approximately 4.5 times and leads to a considerable disturbance of impedance matching both for radiators and for receivers. Bubbles, animal and plant plankton, and fish act as scattering centers for energy concentrated in some limited sound beam. The ocean contains a large quantity of such scattering material, and some fish and crustaceans themselves produce sounds by vibration of the swim bladder or by clicking their claws. Obviously, these diverse factors create great interference when sound is used to obtain accurate information underwater.

Diffraction phenomena in the sea differ from the same phenomena in air only quantitatively. Since the wavelength in water is 5 times greater than in air, a frequency 5 times higher is needed in order to preserve the same ratio between the wavelength and the linear dimensions of receivers, radiators, and other equipment. In a first approximation, the beam given by a radiator may be regarded as the same as the diffraction beam produced by a plane wave passing through an aperture of the same shape as the surface of the radiator or receiver. Thus, the intensity of the emitted

sound, or the sensitivity of the receiver in different directions, decreases with increasing angle formed with the central axis, and reaches a minimum at an angle \(\theta\), determined for a square aperture whose linear aperture in the plane \(\theta\) is equal to \(d\), by the equation \(d \sin \theta = \lambda\). With a further increase in the angle, there occur

Figure annotations: Directional action of the radiating piston for different ratios of wavelength to piston diameter. Circular radiating diaphragm. The length of the radius vector \(R\) is proportional to the acoustic intensity in the given direction. \(\lambda = D\). \(7\lambda = D\). \(10\lambda = D\).

Fig. 7. Directionality curves of a piston radiator.

small maxima. For a circular aperture an additional numerical factor appears, and the beam width \(\varphi = 2\theta\) between the first minima is determined by the expression \(\varphi = 2 \arcsin 1.22\,\lambda/d\) (Fig. 7). Decreasing \(\lambda\) in order to achieve greater directivity requires increasing the frequency. If the limiting frequency audible to the ear proves to be exceeded, then the sound effect can be converted into a visual one, for example on the screen of a cathode-ray tube, or else, by means of heterodyning, the signal can be translated into the audible range. These technical methods are well developed and introduce no difficulties into the use of ultrasonic frequency.

Refraction of sound rays due to the variable refractive index of water is, in all probability, the principal factor distorting the underwater picture “seen” with the aid of sound. The chief parameters on which the speed of sound depends are temperature, pressure, and salinity. Salinity is of significance only near shores, where rivers deliver appreciable quantities of fresh water. In the deep ocean only the distribution of temperatures and pressures determines the speed and, consequently, the refraction of sound. The influence of pressure is of an entirely constant character and therefore does not constitute a special hindrance; temperature conditions, however, in the first hundred meters below sea level are highly variable, depending on the season, the time of day, cloudiness, wind speed, and other meteorological conditions. With the exception of regions of steady currents or those close to them, the temperature in general decreases monotonically, though irregularly, from \(10—20^\circ\mathrm{C}\) to the temperature of maximum density (\(4^\circ\mathrm{C}\)) at great depths. The use of sound by surface vessels and submarines at small depths is strongly dependent on the large temperature differences and sharp gradients that often exist in the first 10 meters below the sea surface. The very gradients that allow a boat to maintain equilibrium may so distort the path of sound rays that regions with a large temperature gradient prove almost impenetrable to sound signals.

The dependence of the speed of sound on temperature and pressure in the temperature interval \(6^\circ\mathrm{C}<t<17^\circ\mathrm{C}\) is given by the expression \(v=1410+4.21t-0.037t^2+0.0175d\ \text{m/sec}\), where \(d\) is the depth in meters (Fig. 8). Two special cases are of particular interest. The first is a uniform decrease of temperature near the surface by several tenths of a degree per meter. Taking the surface temperature to be \(15^\circ\mathrm{C}\) and the temperature gradient \(\gamma\ \dfrac{\text{degrees C}}{\text{meter}}\), we obtain \(v=v_0(1-2.12\cdot10^{-3}\gamma d)\), where \(v_0\) denotes the speed at the surface. In a medium in which the refractive index varies in one dimension, the path of a sound ray emerging from the surface at an angle \(\theta_0\) may be computed by Snell’s law, according to which the product of the refractive index and the cosine of the angle formed by the ray with the normal to the direction of variation of the refractive index is constant. Thus,

\[ \frac{\cos\theta}{v}=\frac{\cos\theta_0}{v_0} \quad\text{or}\quad \frac{\cos\theta}{\cos\theta_0}=1-2.12\cdot10^{-3}\gamma d \]

and, as is seen from Fig. 9, the sound ray is an arc of a circle making an angle \(\theta_0\) with the surface, whose radius is equal to

\[ \frac{1}{2.12\cdot10^{-3}\gamma\cos\theta_0}. \]

The rays are bent downward in the form of arcs of circles with radius

at several thousand meters with a gradient of several tenths of a degree per meter. If the gradient is nonuniform, the curvature of the rays changes, and if the gradient fluctuates in time, which actually

Fig. 8. Parameters affecting the speed of sound in the sea.

Labels in the figure:

• The speed of sound in the sea depends on temperature, pressure, and salinity.
• High temperature of the water surface under the hot sun.
• Pressure is equal to atmospheric at the surface and increases linearly with depth.
• Low salinity at the mouth of a river.

\[ V = 1410 + 4.21t - 0.037t^{2} + 0.0175a + 1.14s \]

where \(V\) is the speed in m/sec; \(t\) is the temperature in degrees Celsius, \(0^\circ C < t < 17^\circ C\); \(a\) is the depth in meters; \(s\) is the salinity in per mille.

takes place, the sound rays are bent and shift, resembling the pattern of optical rays in the air above a hot plate.

\[ d = R(\cos\theta_s - \cos\theta) \quad \text{or} \quad \cos\theta = \cos\theta_s\left(1 - \frac{d}{R\cos\theta_s}\right) \]

Fig. 9. Curvature of a sound ray downward under a linear temperature gradient.

The second interesting case corresponds to great depth, where the temperature slowly falls, tending toward the temperature of maximum density of water, and the conditions are very stable and constant throughout the whole year. The result depends little on special assumptions; we may assume that the temperature increases according to a quadratic law from \(4^\circ\) at a depth of \(2000\ \text{m}\) to \(20^\circ\) at the surface, or

\[ t = 4\left[1 + (2-d)^2\right], \]

where \(d\) is the depth in kilometers. Substituting this quantity into the expression for the speed of sound, we obtain approximately

\[ v = 1.46 - 17.5(2-d) + 15.7(2-d)^2\ \text{km/sec}. \]

This expression has a minimum at the depth

\[ d_m = 1.45\ \text{km}, \]

and, measuring vertically the height \(y\)

from this level, we find \(v = 1.45\,(1 + 1.1 \cdot 10^{-2} y^2)\). Since the speed of sound increases for depths both greater and smaller than this depth, the rays bend toward this level and form what is called a “deep sound channel”*) (Fig. 10). Rays only slightly inclined to the horizon are drawn into this channel, and, since attenuation may practically be neglected, the sound intensity falls proportionally not to the square, but to the first power of the distance.

Figure annotations:
Depth in meters; Temperature; Change of temperature and speed with depth; Deep sound channel; Sound rays bend toward regions of decreasing speed; Speed of sound in m/sec.

Fig. 10. Formation of a deep sound channel.

Sound signals from small explosions, propagating through such a channel across the ocean from Dakar (West Africa), were heard in the Bahama Islands, taking about an hour to cover this distance. The use of deep sound channels to determine the position of a vessel by means of small explosions and by determining the differences in the travel times of sound to several observation stations promises to play a major role in rescue operations at sea. This method is known by the abbreviated name “Sofar” (Sound Fixing and Ranging).

Snell’s law may also be used to calculate the sound ray in a sound channel for small angles between the vector of sound propagation and the horizontal. With sufficient approximation,

\[ \cos \theta = 1 - \frac{1}{2}\operatorname{tg}^{2}\theta, \]

and since

\[ \operatorname{tg}\theta = \frac{dy}{dx}, \]

the equation

\[ \frac{\cos \theta}{v} = c \]

passes into

\[ \frac{dy}{dx} = \sqrt{(1 - c v_m) - c v_m c y^2}, \]

) Another explanation and an exact calculation of the phenomenon of the “sound channel” have recently been given by L. M. Brekhovskikh, DAN, vol. 62, no. 4 (1948). (Translator’s note.*)

where \(c=\dfrac{\cos\theta'}{v'}\) (primes indicate initial conditions) and \(v=v_m(1+\varepsilon y^2)\). Direct integration gives

\[ y=\sqrt{\frac{1-cv_m}{cv_m\varepsilon}}\, \sin\{\sqrt{2cv_m\varepsilon}\,(x+\varphi)\}, \]

where \(\varphi\) is the constant of integration. If the initial displacement of the ray and its inclination are small, the equation becomes, approximately,

\[ y=\sqrt{y'^2+\frac{\theta'^2}{2\varepsilon}}\, \sin \sqrt{2\varepsilon}\,(x+\varphi). \]

Thus the trajectories are sinusoids with period \(2\pi/\sqrt{2\varepsilon}\), as shown in Fig. 11. Using the value of \(\varepsilon\) given above, we find that the wavelength is \(42.5\) km. It is interesting to note that for a ray coinciding with the horizontal at the depth of minimum velocity, more time is required to cover a given distance than for a ray traveling along a sinusoid. The time required to traverse the path \(S\) is

Fig. 11. Ray in a sound channel.

\[ \tau=\int_0^S \frac{ds}{v} =\int_0^{2\pi/\sqrt{2\varepsilon}} \frac{dx}{cv_m(1+\varepsilon y^2)^2} =\frac{2\pi}{\sqrt{2\varepsilon}\,v_m} \left[1-\frac12\left(\varepsilon y'^2+\frac12\theta'^2\right)\right]. \]

If \(y'\) and \(\theta'\) do not vanish, then the sound has a shorter travel time than if they are equal to zero. This is clearly observed when receiving sonar signals, which last up to 10 sec and end with a sharp clap when the slowest wave arrives, traveling along the central part of the channel.

USE OF SOUND BY SUBMARINES

The physical properties of the sea determine the acoustic technology of the submarine—the so-called “Sonar” (an abbreviation of “Sound Navigation and Ranging”). From the preceding it is already clear that the sea presents exceptional difficulties for the operation of acoustic means. The sea is full of sounds of all possible origins and, what is most serious, is filled with sounds emitted by the vessel itself. The source of these sounds is so located as to do the greatest harm: in the immediate vicinity of the transducer. If the ship’s noise is strong, then the effective use of acoustic

means is almost impossible, since the signal is almost completely masked by the background. The first rule in sonar operation is the possibility of reducing the ship’s noise. Certain natural noises, such as, for example, the noise of surf and the sounds emitted by crustaceans, may be useful for locating the shore and shoals; in general, however, sea noises prove useful only for masking the vessel’s own noises from observation by the enemy.

The attenuation of sound in the sea is small, and the sea is filled with objects that scatter sound. It is bounded above by an oscillating reflecting surface, and below by a bottom that is irregular in shape and variable in its reflecting conditions. In addition, sound rays are curved by temperature inhomogeneities. On the whole, one must be surprised at the very possibility of effectively using acoustic means. A rough optical analogy would be the conditions of visibility in a room bounded by oscillating mirrors and filled with clouds of vapor and luminous points. However, fundamental study of the properties of the sea and painstaking development, improvement, and testing of equipment have led to the creation of sonar apparatus satisfying the most urgent needs of submarine navigation. Most of the practical problems that had to be solved belong to the fields of electronics and engineering and thus do not pertain to the physics of the submarine. Nevertheless, the present essay would be incomplete without a brief description of devices for listening and location—two extensive fields of application of acoustics in naval affairs.

Listening, as the name indicates, is the passive reception of sounds arriving at the submarine from any sounding objects. The intensity of the sound received on the submarine is, in general, inversely proportional to the square of the distance from the source, but this law is modified by local conditions of refraction. Since high frequencies are noticeably suppressed during propagation, the low-frequency components are selected and the harmonics are weakened. If the signal received by the hydrophone is much weaker than random noises, then no degree of subsequent electrical amplification makes it possible to extract useful information from such a signal. In the effort to increase effectiveness, the principal efforts are directed toward reducing the background of random noises and increasing angular and frequency selectivity, so that sounds arriving from an undesirable direction and possessing an undesirable frequency can be excluded. The remarks made above about diffraction and the shape of beams show that angular and frequency selectivity are closely connected with each other: low frequencies give a broader diffraction pattern and less directivity than high ones. Moreover, in listening, the width of the band selected by filters cannot be reduced too greatly. Otherwise the characteristic features will be destroyed—

of each sound, as a result of which valuable information may be lost.

A magnetostrictive hydrophone is an example of a modern listening device (Fig. 12), possessing high sensitivity and mechanical strength. It consists of a nickel tube closed at the ends and having inside it a longitudinal partition assembled from steel sheets, around which a winding is placed on two wooden cores. The nickel, closing the magnetic flux between the ends of the steel plates,

Fig. 12. Diagram of a magnetostrictive hydrophone.

is magnetized by the current passed through the winding. Owing to the magnetostrictive properties of nickel, changes of pressure on the tube cause changes in the magnetic flux penetrating the winding, and thus sound waves are converted into an alternating potential difference between the ends of the winding, which can be amplified, filtered, and detected.

The tube is about \(1.5\ \mathrm{m}\) long. For wavelengths shorter than \(0.75\ \mathrm{m}\), the half-angle divergence of the beam may, with an error not exceeding a few percent, be taken equal to \(\lambda/1.5\). For 15,000 periods per second the half-angle divergence of the beam is about \(3^\circ\), and for 2000 periods—about \(30^\circ\). The diameter of the nickel tube is several centimeters, and the tube has practically no directivity in the plane perpendicular to its axis. A special baffle eliminates its sensitivity to waves arriving from behind. The selected beam thus acquires the shape of an orange segment, and rotating the tube about an axis perpendicular to its own axis makes it possible to determine the azimuth of the noise.

In sound location the transmitter sends a short sound pulse, for which purpose a brief burst of high-frequency current is passed through its circuit; immediately thereafter an automatic switch-over to the receiving device is made, and the transmitter is converted into a hydrophone receiving the echo, if one arrives. Transmitters for location are in many respects similar to hydrophones for listening, the only difference being that, as transmitters, they must deliver the greatest possible output without cavitation arising. The selectivity in listening may, however, be much greater, since only a narrow band of the spectrum need be received and amplified, namely the one that was emitted. This

allows undesirable noise to be suppressed considerably. The distance to the object that produced the echo is immediately determined by the formula \(\tau/2v\), where \(\tau\) is the time interval between the sending of the pulse and its reception, and \(v\) is the speed of sound. The factor 2 is due to the fact that during the time \(\tau\) the sound travels the path twice.

Fig. 13 shows the scheme of a typical emitter for location. A large number of nickel tubes are welded to a steel plate having a narrow flange for fastening it to the ship’s hull or to a special supporting structure. The length of the tubes and the thickness of the plate are chosen so that the system has resonance at the selected frequency, with antinodes at the free ends of the tubes and at the front surface of the plate, and a node at the fastening flanges. A high-frequency current pulse passes through the windings of the nickel tubes, and the magnetostrictive effect sets the system into vibration. Matching of the impedances between the plate and the water produces radiation of sound energy in this direction, while the mismatch with the impedance of air prevents loss of energy into acoustic radiation. In a similar way, a wave incident on the plate from the water sets the system into vibration, and the electromotive force induced in the coils is amplified and recorded by the receiver.

Fig. 13. Emitter scheme.

One of the first applications of this method was the echo sounder, used for measuring depth. In this device the emitter is fastened directly to the bottom of the ship and gives a broad sound beam directed vertically downward. The time interval until the arrival of the echo gives a direct measure of the depth of the sea. An automatic key switches on the current periodically, and the reading on a scale or on an automatic record directly gives the depth in units of length.

The echo arriving from the sea floor is usually observed easily; however, by this method one can also detect and determine the location of much smaller objects, giving correspondingly weaker echoes, provided only that the conditions are not too unfavorable. The hulls of submarines and surface vessels, whales and schools of fish, thickets of seaweed, sharp inhomogeneities of temperature and salinity produce a characteristic echo. A typical locating emitter for detecting submarines is mounted so that the normal to its surface is vertical (Fig. 14), and is provided with a device for rotation about the vertical axis in order to determine the azimuth of the target. It resembles an acoustic searchlight, capable of changing the azimuth and sometimes also the inclination of the transmitted beam. If the emitter is stabilized relative to the motions of the ship, it can give sufficiently accurate data on the direction of the object. The distance to it is de-

is calculated from the travel time, just as in the case of the echo sounder. The difficulties that arise in identifying the object are easy to understand; however, an experienced operator can judge a great deal from the character of the echo. If there is relative motion along the line connecting the ship and the object, then the Doppler effect, amplified by heterodyning, often helps in identification, although whales sometimes move even faster than submarines. Determining the depth of the object being reconnoitered is often made very difficult by its proximity to the surface of the water, which gives its own reflection, and also by the bending of the sound beam as a result of changes in temperature conditions.

Fig. 14. Sound location of a submarine.

Physicists and engineers did a great deal of work to create sonar, yet over the long years of the naval war more schools of fish and thickets of seaweed were bombed with depth charges than submarines. And nevertheless sonar is the method on which submarine reconnaissance must be based. With the best modern equipment, the vigilance, persistence, and skill of the sonar operator work wonders even in an environment that presents as many difficulties as the ocean.