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			CONCEPTS IN CONTEXT

	AN ALLEGED BLUNDER IN THE STEALTH BOMBER'S DESIGN

[This discussion is based on W. Biddle, "Skeleton Alleged in the Stealth
Bomber's Closet," _Science_ (May 12, 1989)]

Managerial economics is of great use in the aerospace industry, but this
does not mean that errors sometimes will not occur. The B-2 "Stealth" bomber
has cost billions to develop. According to Joseph Foa, an emeritus professor
of engineering at George Washington University, its design is fundamentally
flawed because two aerodynamicists made a mistake: they mistook a minimum
point for a maximum point.

The B-2 is basically a jet-powered "flying wing" aircraft. In a secret study
for the Air Force, the two aerodynamicists, William Sears and Irving Ashkenas
(then at the Northrop Corporation), used mathematical formulas to determine
how an aircraft's volume should be proportioned between wing and fuselage
in order to maximize its range. Taking the derivative of range with respect
to volume, they found that this derivative equaled zero when the total volume
was almost all in the wing. Hence, they concluded that a "flying wing" design
would maximize range.

But, in a subsequent analysis, Foa showed that the second derivative was 
positive, not negative, under these circumstances. Thus, the "flying wing"
design minimized range; it didn't maximize it. In Foa's words, "The flying
wing was the aerodynamically worst possible choice of configuration."

This is a very interesting example of how important it is to look at the 
second derivative to make sure that you do not confuse a maximation point
with a minimization point. While the backers of the B-2 bomber claim that
it is a good plane despite this error, no one denies that the error is an
embarrassment.

============================================================================

[Article]

	SKELETON ALLEGED IN THE STEALTH BOMBER'S CLOSET

A mathematical error in an appendix to a secret report completed 43 years
ago has come back to haunt its authors--and, perhaps the Air Foce's contro-
versial B-2 "Stealth" bomber program.

Joseph V. Foa, an emeritus professor of engineering at George Washington
University, has charged in a memorandum he circulated recently among scientific
organizations and members of Congress that there had been "an unrelenting 
effort to conceal the facts and to obfuscate the record" about what he claims
is the inferior range of jet powered "flying wing" aircraft such as the B-2.

Foa, whose 40-year research career included a decade as chairman of the 
department of aeronautical engineering and astronautics at Rensselaer Poly-
technic Institute, traces what he labels a "cover-up" of the flying wing's
alleged deficiency to an embarrassing error in research performed for the
Air Force in 1945 by two Northrop Corporation aerodynamicists. Northrop at
the time was the prime contracter for an all-wing bomber that is generally
viewed as a precursor of the B-2, which is also being built by Northrop.

The paper by William R. Sears and Irving L. Ashkenas was part of a secret
assessment of promising military technologies undertaken in the immediate
aftermath of World War II. A team of prominent scientists and engineers 
directed by Theodore Von Karman surveyed such fields as aviation, rocketry,
electronics, and psychology to identify the most fertile areas for future
development. Sears, a former student of the illustrious Von Karman, was 
Northrop's chief of aerodynamics and Ashkenas was his assistant. Von Karman
and Sears are widely credited as the guiding lights behind the Northrop 
XB-35 and YB-49 flying-wing bombers of the 1940's, which were scrapped by
the Air Force after the production of 15 airframes.

In their paper, which was delivered to General Henry H. "Hap" Arnold at the
beginning of 1946 and the circulated among top Air Force officials, Sears
and Ashkenas made frequent reference to the promise of all-wing aircraft.
Only two technical appendices were attached to the general discussion, one
that analyzed some of the still-novel flight characteristics of rockets and
another that claimed to prove mathematically that, for best range, an 
airplane's volume should be contained almost entirely in the wing. At the
time, Northrop was in a fierce competition for a contract to build a strategic
bomber for the Air Force. It was developing the exotic XB-35--a visually 
stunning tailless span of 172 feet, powered by four 3000-horsepower propeller
engines.

The XB-35's test program turned out to be plagued by mechanical problems
with the prop engine assemblies, as well as questions about stability. In 
the early months of 1947, while Northrop was converting the XB-35 to a 
turbojet version tagged YB-49, Foa headed a research group at the Cornell
Aeronautical Laboratory in Buffalo, New York, that was studying the range
and aerodynamics of an unmanned jet-propelled aircraft called Hermes. Sears
had recently left Northrop to become chairman of Cornell's graduate school
of aeronautical engineering in Ithaca. During his theoretical work, Foa 
came upon what he thought was a remarkable contradiction of conventional
wisdom, namely that--based on aerodynamic considerations alone--the range
of Hermes would be considerably lower with an all-wing configuration than 
with a traditional wing-fuselage shape. As he looked into the problem further,
he convinved himself that the result was universally true for jet-propelled
aircraft: an all-wing shape would always have range inferior to a wing-
fuselage shape, with other specifications being equal.

In April 1947, Foa brought this finding to the attention of Sears and suggested
in a letter to the Laboratory's heirarchy that they submit a proposal to the
Air Force for more research into its implications--"especially in view of
the large sums of money that are now being spent to maintain leadership in
a race which is apparently running on the wrong track [i.e., the Northrop
YB-49 project]," he wrote.

Foa does not recall encountering a wellspring of enthusiasm. "Sears responded
by stating, in effect, that what I was claiming was absurd, that he and 
Irving Ashkenas had rigorously proven in a Northrop report (of which he 
could not provide me a copy) that the optimum configuration for range in 
the case of the YB-49 was indeed a flying wing and that we should definitely
not proceed with the submission of the proposal I had suggested," Foa writes
in his current memorandum. "It was not until 3 months later that I was able
to see the Sears-Ashkenas report," which turned out to be the appendix to
the secret 1945 Air Force study.

In that mathematical exercise, Sears and Ashkenas had written formulas  
involving such standard parameters as wieght, flying speed, thruse, fuel
consumption, drag, lift, and air density that could be manipulated to reveal
how an aircraft's volume should be proportioned between wing and fuselage 
for best range. The formulas were valid, Foa found, but when Sears and
Ashkenas calculated the maximum and minimum values for the ratio of total
volume to wing volume they reversed the correct answers.

In the jargon of calculus, the vanishing of the first derivative of range
with respect to volume had given two solutions, one where the total volume
was almost all in the wing and another where the wing volume was much less
than the total. Sears and Ashkenas then simply stated that "it can be ascer-
tained that the former gives a maximum range while the latter gives a minimum."
But with his own contrary research in the back of his mind, Foa examined the
calculations and found that the Sears-Ashkenas "maximum" was in fact a 
minimum. "In other words, the flying wing was the aerodynamically worst 
possible configuration for the YB-49," Foa writes in his recent memo.

On 15 July 1947, Foa sent a letter to Sears pointing out the error. Sears
replied on 17 July, saying "As you can imagine the error is embarrassing to
Irv and me--although I hardly suppose anyone has taken serious action as a
result." It certainly appeared that the optimum configuration was not a 
flying wing, Sears admitted. Nonetheless, he ended his letter by stating
his opinion that the Laboratory should not undertake further study of the
problem for the Air Force. "I suspect you will find that they have enough
studies from airplane manufacturers so that they wouldn't be particularly
excited about our proposal," Sears wrote.

"I found his response shocking," Foa recalls, but he says he felt "that 
the only responsible and honorable way out of it was for Sears himself to
disclose the truth." Foa says he asked the Laboratory's director to tell 
Sears that if he or Ashkenas would do so, I would not find it necessary 
to make any public statement on my own, and would agree to remain silent."
Meanwhile, the YB-49 began test flights in October 1947.

The reaction to Foa's ultimatum was a paper written by Ashkenas titled 
"Range performance of turbojet airplanes" published in the February 1948
issue of the _Journal of Aeronautical Sciences_. In it, Ashkenas engaged
in a far more abstruse parametric study than the one appended to the 1945
Air Force report. A crucial graph showed that for certain values of a 
dimensionless factor he called the "geometric shape parameter," the all-wing
configuration gave best range. Foa's interpretation of the text indicated,
however, that these values would produce a wing impractically thick.

Foa sent a critique of Ashkenas's paper to the _Journal_ on 13 December 
1948. It was published in the magazine's April 1949 edition along with a 
contentious reply from Ashkenas, with no further debate on the record.

A month after Foa's critique was sent in--on 11 January 1949, to be precise--
the Air Force canceled Northrop's YB-49 contract. The offical reason was
budget limitations, but aviation historians still argue about the technical
and political aspects of the decision. According to Northrop publications,
the YB-49 achieved a range of 3155 miles with a 16000-pound bomb load. In
Von Karman's introduction to the secret 1945 Air Force study, he had stated
the range goal as 3500 miles with 20000 pounds of bombs. Later in 1949, the
Air Force told the House Armed Services Committee that "the YB-49 showed
considerable promise in speed and altitude but had inadequate range."

There the matter rested until it was revealed that the B-2 would be a flying
wing like the YB-49.

Sears and Ashkenas, while acknowledging their old error, do not take Foa's
underlying concerns seriously. Today, Sears is emeritus professor of 
aerospace and mechanical engineering at the University of Arizona, where 
he moved in 1974 after a distinguished career at Cornell that spanned 
nearly three decades. Reached by telephone in Tucson, he declined an 
opportunity to respond in detail to Foa's memorandum. "Of course we were
embarrassed by it," he said of the 1945 error, but "we never agreed with 
Foa about his conclusions." Engineers "make these parameter studies to get
general trends," he noted, and they are of limited value in the real world
of aircraft constuction. "It never seemed very important," he added. "It
didn't change anything."

Irving Ashkenas, now vice president of Systems Technology Inc.--a consulting
firm in Northrop Corp.'s hometown of Hawthorne, California--remembers the 
appendix to the 1945 Air Force report as "just a little simple exercise 
that I thought was cute, and that backfired on me." He says that no 
correction was ever issued (the report was not declassified until 1977).
"It was a small part of my contribution and I don't recall worrying about it."

Both Sears and Ashkenas contend that other advantages to the all-wing design
have been established in recent years that should more than make up for the
aerodynamic penalty on range expounded by Foa. Chief among these is a theory
called "span-loading," whereby the airplane's weight is distributed along
the wing, resulting in relatively small bending movements and therefore a 
lighter structure. In a 1987 _Aerospace America_ article, Sears went so 
far as to refer to the YB-49 project as "the world's first serious effort
to prove or disprove the span-load theory for designing big airplanes."

If weight saving was hoped for in the B-2 design, then there are indications
that Northrop has had difficulty obtaining it. In 1984, the B-2 underwent
a major redesign costing at least $1 billion that changed the aircraft's
wing structure and decreased its weight. The Air Force's publicaly stated
reason for such massive rework was to give the B-2 a capability to fly low-
altitude earth hugger missions, in addition to the high-altitude attacks
initially posited for the bomber. But terrain-following flight, which 
requires continuous use of radar altimeters to maintain proper distance
above ground, would negate some of the stealthiness that was the primary 
reason for selecting a flying-wing design in the first place.

Resolution of the Foa-Sears debate awaits the B-2 flight test program, which
has yet to record its first hop, though the bomber was rolled out last 
November. Anyone outside the tight Air Force security circle can only
speculate about related technical issues, but last week, House Armed Services
Committee chairman Les Aspin (D-WI) noted that the aircraft is "new
technology with perhaps fundamental problems."

						# WAYNE BIDDLE

	Wayne Biddle is a Washington, D.C.-based journalist who
	is writing a book about the aerospace weapons industry.

Article Posted by:

Kevin Johnson
kmj@bilbo.baylor.edu 
Department of Mathematics
Baylor University