The Tacoma Bridge disaster — an application of differential equations

tacomaBridgeOn July 1, 1940, the Tacoma Narrows Bridge at Puget Sound in the state of Washington was completed and opened to traffic. From the day of its opening the bridge began undergoing vertical oscillations, and it soon was nicknamed “Galloping Gertie”. Strange as it may seem, traffic on the bridge increased tremendously as a result of its novel behaviour. People came from hundreds of miles in their cars to  enjoy the curious thrill of riding over a galloping, rolling bridge. For four months, the bridge did a thriving business. As each day passed, the authorities in charge became more and more confident of the safety of the bridge — so much so, in fact, that they were planning to cancel the insurance policy on the bridge.

Starting at about 7.00 on the morning of November 7, 1940, the bridge began undulating persistently for  three hours. Segments of the span were heaving periodically up and down as much as three feet. At about 10.00am, something seemed to snap and the bridge began oscillating wildly. At one moment, one edge of the roadway was twenty-eight feet higher than the other edge; the next moment it was 28 feet lower than the other edge. At 10.30am, the bridge began cracking, and finally, at 11.10am, the entire bridge came crashing down. Fortunately, only one car was on the bridge at the time of its failure. It belonged to a newspaper reporter who  had to abandon the car and its sole remaining occupant, a pet dog, when the bridge began its violent twisting motion. The reporter reached safety, torn and bleeding, by crawling on hands and knees, desperately clutching the curb of the bridge. His dog went down with the car and the span — the only life lost in the disaster.

There were many humorous and ironic incidents associated with the collapse of the Tacoma Bridge. When the bridge began heaving violently, the authorities notified Professor F. B. Farquharson of the University of Washington. Professor Farquharson had conducted numerous tests on a simulated model of the bridge and had assured everyone of its stability. The professor was the last man on the bridge. Even when the span was tilting more than 28  feet up  and down, he was making scientific observations with little or no anticipation of the imminent collapse of the bridge. When the motion increased in violence, he made his way to safety by scientifically following the yellow line in the middle of the roadway. The professor was one of the most surprised men when the span crashed into the water.

One of  the insurance policies covering the bridge had been written by a local travel agent who  had pocketed the premium and had neglected to report the policy, in the amount of US $800,000 to the company. When he later received his prison sentence, he ironically pointed out that his embezzlement would never have been discovered if  the bridge had only remained up for another week, at which time the bridge officials had planned to cancel all of the policies!

A large sign near the bridge approach advertised a local bank with the slogan “as safe as the Tacoma bridge.” Immediately following the collapse of the bridge, several representatives of  the bank rushed out to  remove  the billboard.

After the collapse of the Tacoma bridge, the governor of the state of Washington made an emotional speech, in which he declared “We are going to build the exact same bridge, exactly as before.” Upon hearing this the noted engineer Von Karman sent a telegram to  the governor stating “If you build the exact same bridge exactly as before, it will fall into  the exact same river exactly as before.”

The collapse of the Tacoma Bridge was due to an aerodynamical phenomenon known as stall flutter. This can  be explained very briefly in the following manner. If there is an obstacle in a stream of air, or liquid, then a “vortex street” is formed behind the obstacle, with the vortices flowing off at a definite periodicity, which depends on the shape and dimension of the structure as well as on the velocity of the stream (see Fig. 1 download attachment). As a result of the vortices separating alternately from either side of the obstacle, it is acted upon by a periodic force perpendiculat to the direction of the stream, and of magnitude

F_{0}cos (\omega t). The coefficient F_{0} depends on the shape of the structure. The poorer the streamlining of the structure; the larger the coefficient F_{0} and hence, the amplitude of the force. For example, flow around an airplane wing at small angles of attack is very smooth so that the vortex street is not well defined and the coefficient F_{0} is very small. The poorly streamlined structure of a suspension bridge is another matter, and it is natural to expect that a force of large amplitude will be set up. Thus, a structure suspended in an air stream experiences the effect of this force and hence goes into a state of forced vibrations. The amount of danger from this type of motion depends on how close the natural frequency of the structure (remember that bridges are made of steel, a highly elastic material) is to the frequency of  the driving force. If the two frequencies are the same, resonance occurs, and the oscillations will be destructive if the system does not have a sufficient amount of damping. It has now been established that the oscillations of this type were responsible for the collapse of the Tacoma Bridge. In addition, resonances produced by the separation of vortices have been observed in steel factory chimneys, and in the periscopes of submarines.

The phenomenon of resonance was also responsible for  the collapse of the Boughton suspension bridge near Manchester, England in 1831. This occurred when a column of soldiers marched in cadence over the bridge, thereby setting up a periodic force of rather large amplitude. The frequency of this force was equal to the natural frequency of the bridge. Thus, very large oscillations were induced, and the bridge collapsed. It is for this reason that soldiers are ordered to break cadence when crossing a bridge.

More later…

Nalin Pithwa



A surprising application of Differential Equations — The Van Meegeren art forgeries Part II

However, we can still use equation 3 to distinguish between a 17th century painting and a modern forgery. The basis for this statement  is the simple observation if  the paint is very old compared to the 22 year half-life of lead, then the amount of radioactivity from the lead-210 in the paint will be nearly equal to  the amount of radioactivity from the radium in the paint. On the other hand, if the painting is modern (approximately 20 years old or so), then the amount of radioactivity from the lead-210 will be much greater than the amount of radioactivity from the radium.

We make this argument precise in the following manner. Let us assume that the painting in question is either very new or about 300 years old. Set t-t_{0}=300 years in equation 5. Then, after some simple algebra, we see that

\lambda y_{0}=\lambda y(t)e^{3000}-r(e^{3000}-1) equation VI

If the painting is indeed a modern forgery, then \lambda y_{0} will be absurdly large. To determine what is an absurdly high disintegration rate we observe that if the lead-210 decayed originally (at the time of manufacture) at the rate of 100 disintegrations per minute per gram of white lead, then the ore from which it was extracted had a uranium content of approximately 0.014 percent. This is a fairly high concentration of uranium since the average amount of uranium in rocks of earth’s crust is about 2.7 parts per million. On the other hand, there are some very rare ores in the Western Hemisphere whose uranium content is 2-3 percent. To be on the safe side, we will say that a disintegration rate of lead-210 is certainly absurd if it exceeds 30000 disintegrations per minute per gram of white lead.

To evaluate \lambda y_{0}, we must evaluate the present disintegration rate, \lambda y(t) of the lead-210, the disintegration rate r of the radium-226, and e^{3000}. Since the disintegration rate of polonium-210 equals that of lead-210 after several years, and since it is easier to measure the disintegration rate of polonium-210, we substitute these values for those of lead-210. To compute e^{3000}, we observe from equation 3 that \lambda = (ln 2/22). Hence,,

e^{300 \lambda}=e^{(300/22)ln 2}=2^{(150/11)}

The disintegration rates of polonium-210 and radium-226 were measured for the “Disciples at Emmaus” and various other alleged forgeries have been calculated. For the “Disciples at Emmaus”, the disintegration rate of polonium-210 is 8.5 per minute per gram of white lead and the disintegration rate of radium-226 is 0.8 per minute per gram of white lead.

If we now evaluate \lambda y_{0} from equation 6 for the white lead in the painting “Disciples at Emmaus”we obtain that

\lambda y_{0}=(8.5)2^{150/11}-0.8(2^{150/11}-1)=98.050

which is unnecessarily large.

Thus, this painting must be a modern forgery.

More later…

— Nalin Pithwa

A surprising application of Differential Equations — The Van Meegeren art forgeries part I

Normally, differential equations are associated with Newton’s laws of motion; Maxwell’s equations are PDE’s. Most people are under the impression that differential equations are used in physics/engineering and of course, what salivates many are their applications to finance/econometrics/stock-market algorithms.

My aim, in the present blog, is to raise the awareness that any problem which can be mathematically modelled (under certain assumptions) can be solved mathematically !! The trick or validity of the  mathematical model lies on your knowledge of Math, both the width and depth, and the knowledge of other areas of the problem at hand.

Let us look at an application of differential equations to arts! (This is a classic example which I picked up  from available literature on differential equations and their applications and I would like to share with you) It was proved that the beautiful painting “Disciples at Emmaus”‘ which was bought by the Rembrandt Society of Belgium for $170,000 was a modern forgery. The story is as given below:

After the liberation of Belgium in World War II, the Dutch Field Security began its hunt for Nazi collaborators. They discovered, in the records of  a firm which had acted as an intermediary in the sale to Goering of the painting “Woman Taken in Adultery” by the famed 17th century Dutch painter Jan Vermeer. The banker in turn revealed that he was acting on behalf of  a third rate Dutch painter H. A. Van Meegeren, and on May 29, 1945 Van Meegeren was arrested on the charge of collaborating with the enemy. On July 12, 1945 Van Meegeren startled the world by announcing from his prison cell that he had never sold “Woman Taken in Adultery” to Goering. Moreover, he stated that this painting and the very famous and beautiful “Disciples at Emmaus”, as well as four other presumed Vermeers and two de Hooghs (a 17th century Dutch painter) were his  own work. Many people, however, thought  that Van Meegeren was only lying to  save himself from the charge of treason. To prove his point, Van Meegeren began, while in prison, to forge the Vermeer painting “Jesus Amongst the Doctors” to demonstrate to the skeptics just how good a forger of Vermeer he was. The work was completed when Van Meegeren learned that a charge of forgery had been substituted for that of collaboration. He, therefore, refused to finish and age the painting so that hopefully investigators would not  uncover his secret of aging his forgeries. To settle the question an international panel of distinguished chemists, physicists and art historians was appointed to  investigate  the matter. The panel took x-rays of  the paintings to determine whether other paintings were underneath them. In addition, they analyzed the pigments (coloring material) used in the paint, and examined the paintings for  certain signs of old age.

Now, Van Meegeren was well aware of these materials. To avoid detection, he scraped the paint from old paintings that were not  worth much, just to get the canvas, and he tried to use pigments that Vermeer would have used. Van Meegeren also knew that old paint was extremely hard, and impossible to dissolve. Therefore, he cleverly mixed a chemical, phenoformaldehyde, into the paint, and this hardened into bakelite when the finished painting was heated in an oven.

However, Van Meegeren was careless with several of  his  forgeries, and the panel of  experts found traces of  the  modern pigment cobalt blue. In addition, they also detected the phenoformaldehyde, which was not discovered until the  turn of  the 19th century, in several of the paintings. On the basis of this evidence Van Meegeren was convicted of forgery, on October 12, 1947 and sentenced one year in prison. While in prison, he suffered a heart attack and died on December 30, 1947.

However, even following the evidence gathered by the panel of  experts, many people still refused to believe that the famed “Disciples at Emmaus” was forged by Van Meegeren. Their contention was based on the fact that the other alleged forgeries and Van Meegeren’s nearly completed “Jesus Amongst the Doctors” were of a very inferior quality. Surely, they said, the creator of the beautiful “Disciples at Emmaus” could not produce such inferior pictures. Indeed, the “Disciples at Emmaus” was certified as an authentic Vermeer by the noted art historian A. Bredius and was bought by the Rembrandt Society for $170,000. The answer of the panel to these skeptics was that because Van Meegeren was keenly disappointed by his lack of status in the art world, he worked on the “Disciples at Emmaus” with the fierce determination of proving that he was better than a third rate painter. After producing such a masterpiece, his determination was gone. Moreover, after seeing how easy it was to dispose of the “Disciples at Emmaus” he devoted less effort to his  subsequent forgeries. This explanation failed to satisfy the skeptics. They demanded a thoroughly scientific and conclusive proof that the “Disciples at Emmaus” was indeed a forgery. This was done in 1967 by scientists at Carnegie Mellon University, and we would now like to describe that work.

The key to the dating of paintings and other materials such as rocks and fossils lies in the phenomenon of radioactivity discovered at the turn of the century. The physicist Rutherford and his colleagues showed that the atoms of certain “radioactive” elements are unstable and that within a given time period a fixed proportion of the atoms spontaneously disintegrates to form atoms of a new element. Because radioactivity is a property of  the atom, Rutherford showed that the radioactivity of a substance is directly proportional to  the number of atoms of the substance present. Thus, if N(t) denotes the number of atoms present at time t, then

\frac {dN}{dt}=-\lambda N equation I

The constant \lambda which is positive is known as the decay constant of the substance. The larger

\lambda is, of course, the faster the substance decays. One measure of the rate of disintegration of a substance is its half-life which is defined as the time required for half of a given quantity of radioactive atoms to decay. To compute the half-life of a substance in terms of \lambda, assume that at time t_{0}, N(t_{0})=N_{0}. Then, the solution of the initial-value problem

\frac {dN}{dt}=-\lambda N_{0} if N(t_{0})=N_{0} is

N(t) =N_{0}exp(-\lambda \int_{t_{0}}^{t}ds)=N_{0}e^{-\lambda (t-t_{0})}

or \frac {N}{N_{0}} = exp (-\lambda (t-t_{0})).

Taking logarithm of both sides we obtain that

-\lambda (t-t_{0})= ln \frac {N}{N_{0}} equation II

Now, if \frac {N}{N_{0}}=1/2 then -\lambda (t-t_{0})= ln (1/2) so that

(t-t_{0})= \frac {ln 2}{\lambda} = 0.6931/(\lambda) equation III

Thus, the half-life of a substance is ln 2 divided by the decay constant \lambda . The dimension of \lambda, which we suppress for  simplicity of  writing is reciprocal time. If t is measured in years, then \lambda has the dimension of reciprocal years, and if t is measured in minutes, then \lambda has the dimension of reciprocal minutes. The half-life of many substances have been determined and recorded. For example, the half-life of carbon-14 is 5568 years and the half-life of uranium-238 is 4.5 billion years.}

Now, the basis of “radioactive dating” is essentially the following. From equation II, we can solve for

(t-t_{0})=(1/\lambda) ln (\frac {N}{N_{0}}).

If t_{0} is the time the substance was initially formed or  manufactured, then the age of the substance is

(1/{\lambda}) ln (\frac {N_{0}}{N}).

The decay constant \lambda is known or can be computed in most instances. Moreover, we can usually evaluate N quite easily. Thus, if we knew N_{0} we could determine the age of the substance. But, this is the real difficulty of course since we usually do not know N_{0}. In some instances though, we can either determine N_{0} and such is the case for the  forgeries of Van Meegeren.

We begin with the following well-known facts of elementary chemistry. Almost all rocks in the earth’s crust contain a small quantity of uranium. The  uranium in the rock decays to another radioactive element, and that one decays to  another and another, and so forth in a series of elements that results in lead, which is not radioactive. The uranium (whose half-life is over four billion years) keeps feeding the elements following it in the series, so that as fast as the they decay, they are replaced by the elements before them.

Now, all paintings contain a small amount of the radioactive element lead-210 (Pb^{210}), and even a smaller amount of radium-226 (Ra^{226}), since these elements are contained in white lead (lead oxide), which is a pigment that artists have  used for over 2000 years. For the analysis which follows, in turn, is extracted from a rock called lead ore, in a process called smelting. In this process, the lead-210 in the ore goes along with the lead metal. However, 90-95% of  the radium and its descendants are removed with other waste products in a material called slag. Thus, most of  the supply of lead-210 is cut off and it begins to decay very rapidly, with a half-life of 22 years. This process continues until the lead-210 in the white lead is once more in radioactive equilibrium with the small amount of  radium present, that is, the disintegration of lead-210 is exactly balanced by the  disintegration of  the radium.

Let us now  use this information to compute the amount of lead-210 present in a sample in terms of the amount originally present at the  time of manufacture. Let y(t) be the amount of  lead-210 per grain of white lead at time t, and r(t), the number of disintegrations of radium-226 per minute per gram of  white lead at time t. If \lambda is the decay constant for lead-210, then

\frac {dy}{dt}=-\lambda y +r(t) with y(t_{0})=y_{0} equation IV

Since we are only interested in a time period of at most 300 years, we may  assume that the radium-226, whose half-life is 1600 years, remains constant, so that r(t) is a constant r. Multiplying both sides of the differential equation by the  integrating factor

\mu (t)=e^{\lambda t} we obtain that

\frac {d}{dt} e^{\lambda t}(y)=re^{\lambda t}


or y(t)= (r/\lambda)(1- e^{-\lambda (t-t_{0})})+y_{0}e^{-\lambda (t-t_{0})}) equation V

Now, y(t) and r can be easily measured. Thus, if we knew y_{0} we could use equation V to compute

(t-t_{0}) and consequently, we could determine the age of the painting. As we pointed out, though, we cannot measure

y_{0} directly. One possible way out of this difficulty is to  use the fact that original quantity of lead-210 was in radioactive equilibrium with the larger amount of radium-226 in the ore from which the metal was extracted. Let us, therefore, take samples of different ores and count the number of disintegrations of  the radium-226 in the ores. This was done for a variety of ores and the results were tabulated. These numbers vary from 0.18 to 140. Consequently, the number of disintegrations of  the lead-210 per minute per gram of white lead at the time of manufacture will vary from 0.18 to 140. This implies that y_{0} will also vary over a very large interval, since the number of disintegrations of lead-210 is  proportional to the amount present. Thus, we cannot use Equation V to obtain an accurate, or even a crude estimate of the age of  a painting.

To be continued in the next part,

— Nalin