Linear to Nonlinear Relations in Wave Science (Acoustics)

In the realm of acoustics or wave science, the transition from linear to nonlinear physics marks a significant evolution in the understanding of tones and their generation. The foundation of this understanding dates back to Pythagoras, who established a linear relationship between the length of a plucked string and the resultant musical tone. This principle posited that the pitch produced by a string can be modulated linearly by altering its length.

A shift occurred in the 1580s with Vincenzo Galilei, the father of the renowned Galileo, challenging the prevailing linear thinking. Galilei’s experiments revealed a more complex, nonlinear relationship in the production of musical tones, particularly when it came to varying the tension of a string. Contrary to the linear assumption that increasing tension produced higher pitches in a directly proportional manner, Galilei discovered that the pitch interval was related to the square of the string’s tension, \(T^2\). This finding showed a nonlinear relationship in the generation of acoustic tones, extending beyond strings to wind instruments, where the pitch interval varied as the cube of the vibrating air volume, \(V^3\).

The implications of Galilei’s discovery were profound, demonstrating that an interval of a perfect fifth could be achieved through multiple nonlinear pathways: strings differing in length by a ratio of 3:2, in tension by a factor of 9:4, or wind instrument air volumes by a ratio of 27:8. This nonlinear understanding fundamentally altered the approach to musical acoustics, showing a method for a exploration of the intricate relationships that govern the generation of musical tones. This led to the study of nonlinear systems.

Brian Spalding

One last poem by turbulence / numerics researcher Prof. Brian Spalding

I shall have no regrets when I am dead.

Of deadlines none will matter but my own.

Unwritten papers? Hopelessly misled.

Inheritors? All claimants I’ll disown.

Yet hope, while still alive, there’ll be but few

Who think: I was a fool to trust him.

Now that he’s gone, what am I going to do?

None I would hope; but guess the chance is slim.

Yet, in that soon-to-close window of time,

There’s much I want to do; and think I can.

Always too optimistic is what I’m

Dismissed as. To disprove it is my plan.

‘After such labours,’ I would have it said,

‘It must be truly blissful to be dead.’

Brian Spalding

Returning to Ludwig Prandtl’s One-Equation Model

In my turbulence class this semester, I recently reviewed Prandtl’s one-equation model, which was developed over 20 years since the time of boundary theory in the early 1900s. The major paper by Ludwig Prandtl was published in the early 1940s. He presented the first one-equation turbulence model for the closure of the boundary layer equations, specifically for incompressible flow. He calibrated coefficients via channel measurements, which were carefully conducted. Therefore, the prediction of the model in terms of modeled coefficients corresponded exactly with his experiments. Many later investigators, through the early 1990s and one particular recent paper in the Royal Society published in 2023, reexamined the model and its analysis. Many have programmed this model in computational fluid dynamics, and it is the basis of many one-equation models, especially those based on a k or turbulent kinetic energy equation. However, it is generally shown that the model is not predictive for many flows without modification. Some have misattributed Prandtl’s model to a reduction of Andrei Kolmogorov’s two-equation model from 1945. Prandtl was unlikely to know about the Soviet invention during World War II, and it can be truly attributed to him. We should attribute to Ludwig Prandtl the invention of the world’s first one-equation turbulence model, even though it may be flawed, this is only because of the lack of the contemporary digital computer.

Geometrics and Art

The Renaissance, a period of significant intellectual, artistic, and cultural rebirth, marked the combination of art and science, especially through the application of geometric principles in artistic representation. This era witnessed the pioneering development of linear perspective, a technique that revolutionized the way depth and three-dimensional objects were portrayed on two-dimensional surfaces. The mathematical foundation for perspective, which shows that parallel lines appear to converge at a distant point, was established in Italy during the 15th century and was instrumental in creating more lifelike and spatially coherent artworks​​.

Leonardo da Vinci, emblematic of the Renaissance “universal genius,” exemplified the integration of scientific inquiry with artistic mastery. His work, along with that of other notable figures such as Paulo Uccello and Piero della Francesca, underscores the period’s drive towards a deeper understanding of the natural world and its representation through art. This pursuit was not confined to Italy, but over Europe, influencing a wide range of artistic expressions and leading to a distinct new art form in mid-16th century Germany, characterized by polyhedral-based geometrical designs​​.

The Renaissance was not only a time of artistic flourishing, but also a critical juncture in the history of science, with the synthesis of mathematics, geometry, and art propelling forward the modern scientific worldview. The artist-engineers of the Renaissance, with their detailed studies of nature and commitment to empirical observation, laid the groundwork for the subsequent developments in science and engineering. Their legacy is a testament to the enduring power of interdisciplinary inquiry and the intrinsic relationship between art and mathematics​​​​.

Additional Thoughts on Half-Equation Model of Johnson and King

The Johnson King turbulence model represented a significant advancement in the understanding and modeling of turbulent flows. Introduced amidst the exploration of first and second equation models, the Johnson King model distinguished itself through the innovative concept of a half-equation model, emphasizing the critical role of memory in turbulence phenomena.

The early stages of turbulence model development were characterized by efforts to create predictive models, a task complicated by the limitations of computational power and the scarcity of high-quality experimental data. Unlike its contemporaries, the Johnson King model introduced a new approach by tracking the ratio of non-equilibrium flow through a single ordinary differential equation (ODE). This innovative strategy, which led to the designation of the model as a half-equation model, did not rely on a new closure equation involving a partial differential equation but rather utilized an ODE to describe the evolution of turbulence.

A key strength of the Johnson King model lies in its predictive capabilities, especially when compared to previous models, such as those based on the work of Cebeci and Smith. The model’s inclusion of turbulence history and the consideration of non-equilibrium effects allowed for a more accurate depiction of the amplification or dissipation of turbulent kinetic energy, particularly in the separation in boundary layers due to shock waves. This focus on the history and memory effects in turbulent flows marked a significant departure from equilibrium-based models and contributed to a deeper understanding of turbulence dynamics. Furthermore, it got away from concepts of one-point statistics of algebraic models.

The practical implications of the Johnson King model were underscored by its performance on VAX computer systems (the most popular DEC system of the day), where it demonstrated superior efficiency compared to one or two equation models. The model’s ODE was specifically valid across a line in the axial or streamwise direction through the flow, at points of maximum Reynolds’ stress. This focus on a streamlined computational approach not only enhanced the model’s speed and accuracy but also laid the groundwork for future developments in the field.

The Johnson King turbulence model, with its emphasis on memory and history effects, has played a pivotal role in the evolution of turbulence modeling. By introducing the half-equation concept and highlighting the importance of non-equilibrium effects, this model has contributed to a more nuanced understanding of turbulent flows and their underlying mechanisms in shock separation. The legacy of the Johnson King model continues to influence contemporary turbulence research, underscoring the lasting impact of their innovative approach.

Hindu-Arabic Numerical System

The decimal number system, an integral part of daily life, traces its origins back to 6th-century India. Characterized by the digits zero through nine, revolutionized numerical computation and record-keeping, setting the stage for advancements in mathematics, science, and commerce. Despite its apparent simplicity and utility to the contemporary observer, the widespread adoption of this system across the globe was a gradual process that spanned over a millennium.

The genesis of the modern digits can be linked to the Brahmi numerals from the 3rd century, with the concept of zero emerging concurrently as a placeholder within the Babylonian base-60 counting system. This incorporation of zero into the Hindu-Arabic numerals was a significant leap forward, enabling a positional numeral system akin to the one in use today. Unlike the Roman numerals used in the Western world, which complicated written calculations and hindered efficient computation, the Hindu-Arabic system offered a streamlined approach by introducing a zero and limiting the numerals to nine, with each position signifying an increasing power of ten.

The spread of the Hindu-Arabic numeral system was propelled by the Islamic conquests of the 7th century, eventually being known as the Hindu-Arabic number system. However, its adoption in Europe faced considerable resistance, with Roman numerals predominating until the 12th century. The turning point came with Leonardo of Pisa, better known as Fibonacci, who, through his travels in Arab lands, was exposed to this efficient numeral system. His seminal work, Liber Abaci in 1202, introduced the Hindu-Arabic number system to Europe, but also demonstrated its practical application in commerce, thereby catalyzing a shift in European mathematical practices and commercial operations.

The transition from Roman to Hindu-Arabic numerals in Europe was marked by a protracted debate between the algorists, who advocated for the Hindu-Arabic system, and the abacists, who favored the traditional Roman numerals and counting boards. The dispute eventually subsided in the 16th century, leading to the relegation of Roman numerals to specific contexts and the ascendance of the Hindu-Arabic system as the preeminent numerical standard.

Diophantine Equations

Contributions of Diophantus of Alexandria hold a distinguished place. His seminal work, Arithmetica, unveiled in the 3rd century CE, is a key in the study of number theory, particularly in the realm of integers. This ancient text, encapsulating 130 equations, laid the foundation for what are now known as Diophantine equations—equations constrained to integer solutions.

Diophantine equations are a specialized subset of polynomial equations, which in their essence, are expressions consisting of one or more algebraic terms with at least one unknown variable, typically denoted by $x$. Diophantus’s pioneering approach, which sought integer solutions to these equations, earned him the title “the father of algebra,” a testament to his profound influence on the field, despite the modern algebraic notations and concepts that were to emerge centuries later.

A quintessential Diophantine problem might pose a scenario such as: “A father’s age is one less than twice the age of his son. If the digits of the father’s age are reversed to represent the son’s age, what are their ages?” The resolution to such problems often involves an initial trial-and-error methodology, culminating in a singular solution—in this instance, a father aged 73 and a son aged 37.

Despite being occasionally perceived as mere mathematical puzzles, Diophantine equations have posed significant challenges, with some remaining unsolved for millennia. A notable moment in the history of these equations occurred in 1637 when Pierre de Fermat, while engaging with Diophantine puzzles, conjectured an equation that appeared unsolvable: “If integer $n$ is greater than 2, then there are no three integers where $x^n + y^n = z^n$.” This marginal note in his copy of Arithmetica, devoid of a proof, laid the groundwork for what would become known as Fermat’s Last Theorem—a conundrum that would not be resolved until 1994.

Diophantus’s legacy extends beyond his mathematical works, encapsulated poetically in the riddle inscribed on his tombstone. This riddle, a polynomial equation in itself, challenges the reader to deduce Diophantus’s age based on key life events. The solution, ingeniously woven into the fabric of his life’s timeline, reveals his age to be 84 years.

The Arithmetica of Diophantus and the ensuing Diophantine equations exemplify the enduring nature of mathematical inquiry. These equations serve as a bridge between the ancient and modern mathematical worlds, illustrating the timeless quest for knowledge and the inherent beauty found in the pursuit of solutions to seemingly insurmountable problems. Diophantus’s work, transcending the centuries, continues to inspire mathematicians, affirming the profound impact of ancient scholarship on contemporary mathematical thought and exploration.

An Improbable Life by D.C. Wilcox, and the $k-\omega$ Model

I just finished reading the autobiography of D. C. Wilcox. He wrote a number of books that were published through his own company. One of the most popular is on fluid dynamics. A less known book is on turbulence modeling. He was famous for a particular two-equation turbulence model in the form of $k-\omega$. It is mostly known as the Wilcox $k-\omega$ model today. Dr. Wilcox had an interesting life, and one publication is called “An Improbable Life,” which is an autobiography. In the autobiography, he discusses his childhood, father, mother, growth as a child through an arrest. While incarcerated he studied and was helped by volunteers. He was admitted to MIT through American standardized testing. He then went on to obtain a Ph.D. at Caltech. He was very focused later in life, and tried to give back to the community that helped him. He was an outspoken conservative and a huge fan of Ayn Rand, and in particular Atlas Shrugged. So many people in his generation were influenced by the book. My only dissapointment with the autobiography was that he did not talk at all about his interest and development / motivation for his famous turbulence model. Otherwise, the little hard to find book is a short read, which was motivated to help other young men such as himself. Here are some favorite quotes:

Just over nine years passed from the day I woke up in a six foot by nine foot prison cell determined to reclaim my life to the day Caltech President Harold Brown handed me my Doctorate. I had taken a journey that required a great deal of hard work, some good fortune and the assistance of some wonderful people. Among those wonderful people, the gracious and generous Jean Kane Foulke du Pont stood out not only as a benefactor but, eventually, as a dear friend.

I visited with Mrs. du Pont just after I graduated from MIT in 1966 to thank her for all she had done for me. I offered to pay her back with installments over time so other scholars could get the kind of education she had made possible for me. “Oh, heavens no,” she said, “we wouldn’t know how to handle it with the IRS.”

She went on to tell me that I could repay her by making sure my children went to their college of choice. It wasn’t much of a repayment I thought, because Barbara and I couldn’t imagine doing otherwise.

I sent her a letter in 1986 to tell her that I had just fulfilled part of my promise to make sure my children received the college education they wanted. My daughter Kinley would be graduating from college almost twenty years to the day from when I had graduated from MIT. Sadly, a reply came back telling me that Mrs. du Pont had passed away.

D. C. Wilcox, “An Improbable Life,” Published by DCW Industries, Inc., 2007, ISBN 10: 1928729509

In late June of 1966, Barbara and I moved to California where a job awaited me with Douglas Aircraft in Long Beach. This was yet another one of my childhood dreams come true. I had never forgotten my wonderful days in California with Aunt Isabel and her son Warren. This time I was coming to California to stay.

After working for a year at Douglas Aircraft, I was accepted for graduate study at the California Institute of Technology where I met my third great teacher, Dr. Philip Saffman. In addition to being a wonderful teacher and $\mathrm{PhD}$ thesis adviser, he told me that he felt a truly dedicated student should pursue his studies like a monk in a monastery. For students who did that, he added, six years from high school to $\mathrm{PhD}$ should be the norm.

In June of 1970, after just three years of study under the guidance of this brilliant mathematician/scientist, I graduated with a PhD in Aeronautics. I had accomplished the second of my highschool goals by earning a PhD with just six years of college.

While I was at Caltech, my son Robert Sabatino Wilcox named after his two grandfathers – was born. The year was 1969. Dad would have been tickled to know that his grandson was born in the year that Neil Armstrong became the first man to walk on the moon.

After a brief time working for various Southern California aerospace companies, I founded my own company, which I named DCW Industries. The company came into existence on July 19 , 1973. Since I was twenty-nine years old, I had accomplished my third goal. Initially focused on aerospace research, the company prospered and I have published more than seventy scientific reports and journal articles in some of the aerospace industry’s most prestigious journals. The company now specializes in book publishing, and I have written several college-level textbooks that are used in universities all over the world.

Mom and I wrote each other from time to time until she passed away in 1977. In one letter she told me something that proved to be one of the nicest things she ever did for me. She told me that I had a relative who was a professor at UCLA. His name was Bill Meecham and he was my Aunt Mabel’s son. Since Mabel was Dad’s sister, we were first cousins.

I contacted Bill and discovered that we had more than our bloodline in common – we were both working in the same field! We became friends and he helped me obtain a part-time teaching job at UCLA in 1981. I have been a fixture in the Mechanical and Aerospace Engineering Department ever since.

When Aunt Mabel died in 1987, Bill showed me some genealogical information that was among her belongings. I noticed that Dad had a brother named Arthur who had died in 1947 when I was three years old. At the age of 43 , I had just discovered that I had an uncle I was completely unaware of. Of even greater significance, this information revealed a great deal about my family roots dating back through ten generations in America.

D. C. Wilcox, “An Improbable Life,” Published by DCW Industries, Inc., 2007, ISBN 10: 1928729509

On Algorithm

An algorithm, a concept rooted in 9th-century Arabic scholarship, is a methodical procedure for problem resolution, eliminating the need for trial-and-error. This term, reflecting centuries of intellectual endeavor, denotes the evolution from Euclid’s ancient formulations to Al-Khwarizmi’s systematic methods and Ada Lovelace’s 19th-century innovations, highlighting algorithms’ integral role in computational development.

Salome

It is my yearly tradition to listen to the opera Salome by Richard Strauss on Valentine’s Day. To celebrate, I treated the department staff to donuts. Working on my research while listening to the opera can be a bit distracting, but it makes for a perfect Valentine’s Day tradition. I first saw the opera while working at NASA Langley Research Center in Norfolk, Virginia, as part of the Virginia Opera Guild.