A question of great interest to submariners is the degree of protection the Russian doublehull design provides against underwater explosions. It is certainly true that the farther one is from an explosion, the more likely one is to survive it. Yet it is not correct to impute a large degree of invulnerability to the fact that a torpedo warhead explodes a few feet away from a pressure hull, rather than in direct contact. This article offers some simple proofs of that statement.

In order to appreciate the problem, some knowledge of the explosion process is useful. High-explosives (HE) are oxygen-rich chemical compounds characterized by extremely rapid decomposition when suitably ignited. From the point of ignition, a detonation wave proceeds outward through the body of the material. It travels at a velocity greater than the speed of sound in the explosive. The significance of this fact is that since intelligence cannot be transmitted at a speed greater than sound in a solid, the unexploded material ahead of the detonation wave can have no knowledge of its approach, so to speak. (If it did, it would break up.) Behind the detonation wave, then, we have a mass of incandescent gas at high temperature and pressure; ahead of it, undisturbed explosive; and outside the explosive, undisturbed water.

At the explosive/water boundary, an enormous amount of energy just ••• well, just “appears.” “Enormous” is used advisedly. Temperatures are in the tens of thousands of degrees Kelvin, and pressures in the hundreds of thousands of psi. A shock wave is formed. This is a true shock with a rise time from zero to maximum pressure of less than a micro-second. For our purposes, we may safely ignore the physical chemistry that describes very high pressures in water, and just use the acoustical approximations. It happens that this is a conservative approach– i.e., any conclusions we may draw will always be on the safe side.

Empirically, we know that the peak shock wave pressure is a product of pounds of equivalent-TNT (modern HE’s have a TNT equivalence or about 1.5; i.e., 100# of modern HE = 150# of TNT) and standoff distance in feet. This product is adequately correct for charges ranging in weight from a few ounces to kilotons.

As a function of time, the peak pressure decays exponentially as shown in Figure 1.

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The “tail” of the curve will be discussed later.

Po is hydrostatic pressure

Pm is peak pressure

T1 is time of first bubble pulse

The actual pressure experienced by a submerged target from a reasonably-distant noncontact explosion (from a mine, a depth charge, an atomic depth bomb, etc.) is modified by the presence of the ocean surface. Figure 2 shows the geometry:

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The target “sees” the incident shock wave, P(t), shown in Figure 2. The shock that hits the surface, however, is reflected as a rarefaction, -P(t), which effectively cancels +P(t) after what is called the “cut-off time, time t” which is simply the interval between the arrival of +P(t) and -P(t) at the target and is measured in microseconds. The resultant shock wave history looks like this:

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This shows that a submarine is better off shallow than deep since cut-off time increases directly with depth. Barring other disadvantages, surfaced is best. We must note, however, that for the geometry we are considering here (explosion against an outer skin), the cut-off phenomenon is of only academic interest.

Now to the tail of Figure 1. The departure of the shock wave leaves behind a sphere of hot gas at very high pressure. It expands rapidly; so rapidly, in fact, that its momentum carries it past the point where its internal pressure equals the hydrostatic pressure. Naturally, it contracts; and again overshoots the hydrostatic pressure, P , emitting a pressure pulse — not a shock — at 0 time T 1 • This is called the “first bubble pulse,” and while its maximum pressure is typically 25J of P , it is significant that the area under the curWe, the “impulse,” may exceed the area under the shock wave itself. Except at very shallow charge depth, there is more than one bubble pulsation.

Finally, we know that the bubble migrates upward between pulsations a distance roughly equal to its maximum radius. This fact leads one immediately to the speculation that it should be possible to “tune” an under-keel warhead to a specific target. Specifically, one should be able to size the weight and to establish the charge’s depth below the keel in such a way that the first bubble pulse will be emitted practically at the target keel; and further, T 1 could be synchronous with the fundamental period of hull flexure. This is an absolutely devastating form af attack, against which no defense is known. Indeed, the notion has intrigued weapons designers for most of this century. It is quite possible to tune warheads in this way, and you might enjoy the exercise of doing it for a target with a draft of, say, 30 feet, and a fundamental period of 0.75 seconds. As a practical matter, of course, it would be unwise to carry a shipload of torpedoes, each tuned to a specific class or ship.

We come back now to the relative vulnerability or the double-hulled submarine. It is true that a torpedo warhead exploding a few feet from the pressure hull may not blow a hole in the hull. My statement, however, is that any respectable warhead a few feet away will leave the interior of the boat (including ship’s force) in a shambles.

To justify this statement, we present the term called “Shock Factor (SF).” SF is an interesting parameter. One way to regard it is as a measure of the energy density per square foot of pressure 112 hull; specifically, SF = Constant (Energy) ; but a more useful and informative way is to look at it as a measure of the velocity of the pressure bull due to the impact of the shock wave. This quantity is known as the “take-off velocity” or the “Taylor Plate Velocity,” after Sir Geoffrey Taylor, who published it a few decades ago. Calculation of this velocity, V , is too tedious for this article, but it involvei all the right things: the peak pressure; charge weight; standoff range; a time constant; and the mass per square foot of the hull. It is not surprising that this velocity is equal (very nearly) to some constant times the “shock factor” for a given hull thickness. For instance, for a 3″ hull, V = 90 x SF; for a 2″ hull, V = 108 x SF; and forma 1″ hull, V = 138 x SF, themsame for mild steel and HY-80. for a titanium hull of the same thickness, V is greater than it is for steel. Put anothir way, for a given charge geometry, there will be more shock damage inside a titanium hull than there will be inside a HY-80 hull of the same geometry. For two such 3″ hulls, V (or the effective Shock Factor) will be 28J hfgher for the titanium hull.

The next step in examining explosive damage to double-hulled subs is to propose and describe two different modes of material behavior. The first of these I will call “plastic” behavior, and HY-80 typifies it. If your boat has a test depth of, say, 1000 feet, and circumstances force you to 1300 feet, you are not in real danger. The second type of behavior I choose to call “brittle,” and a piece of blackboard chalk demonstrates what this is. If you bend a piece of chalk between your thumbs and forefingers, nothing happens until you get to a certain point. Then the chalk snaps; suddenly, completely, and without warning. Shock behavior is like that. Everything we know about equipment undergoing shock loading says that most of it is “brittle;” everything is fine up to a certain point. Just a little past that point, and things snap.

The Shock Factors to which we design submarine fittings and equipment are classified, but that need not deter us. Shock acceptance testing is controlled by a Navy MIL-Spec. It requires that equipment weighing over about 6000# be explosively tested in a floating shock test vehicle, the most severe test being the explosion of a 60# charge, depth 2~’• standoff range 20′. The MIL-Spec does not specify the material of the explosive. To be conservative, I have assumed it to be the modern 60# high-explosive to be the equivalent of 901 of TNT. For this explosive the Shock Factor is .47. If you accept shock loading as leading to “brittle” behavior, you will agree that at Shock Factors not very much higher than 0.47. undesirable things will happen.

The Table below needs some explanation. To get back from the theoretical to the real world, I have chosen two charges of nominal warhead size: 100# (150# TNT equivalent), and 500# (750# TNT), and two stand-off distances — 6 feet and 12 feet. These latter were picked because “Jane’s,” 1984, gives the separation between the inner and outer hulls as “possibly six feet” for the TYPHOON class, and as “ten or more feet” for the OSCAR class. The f irst two targets as tested, have a Shock Factor of 0.47. The next four targets represent two different warheads at two different stand-off distances.

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Now, it would be nice to be able to say that Targets 1 and 2 above. are “safe,” and Targets 36 are not; but nothing is that neat in the underwater explosion business. For one thing, only about 1S of the volume of Target #4 actually experiences a test Shock Factor as low as 0.47. For Targets 4-6, somewhere between 5 and 10% of the target volume experiences a Shock Factor greater than .47. These facts lend emphasis to the intuitive feeling that it is better to attack the Engine Room than the Crew’s Mess.

As a generalization, however, it is reasonable to say that Target #3 is going to need several minutes (at least) before that target is in any shape to return torpedo fire. Targets #4 and 5 are going to have trouble making it to the surface, and are very likely to be in need of a tow if they get there. Target #6 can be written off.

To sum up, it is correct that an outer hull affords some degree of protection; but it can be nullified by large warheads. In any event, it ought not to be exaggerated.

**VADH Robert Gooding, USN(Ret.)**