Ever wondered why a teapot dribbles along the spout or why a runner’s ponytail swings side to side? Dr. Joseph B. Keller, father of the geometrical theory of diffraction (GTD), thought about these questions and found the answers using applied mathematics.

For these quirky discoveries, Keller was twice awarded the Ig Nobel prize: first, for his paper Teapot Effect and again for Ponytail Motion.

Engineers looking for fun tests of their simulation skills can model Keller’s dribbling teapots and swinging ponytails in ANSYS Fluent and ANSYS Mechanical, respectively.

However, Keller’s best-known discovery is GTD, which extends geometrical optics (GO) to include diffraction.

Keller’s methods predict how rays are diffracted when electromagnetic (EM) waves strike scattering objects on their faces, edges, corners or tips. Diffraction can also occur as rays graze a surface, such as the curved exterior of a cylinder.

GTD is an analytical technique belonging to a broad class of asymptotic methods. These methods have evolved to include physical optics (PO), the physical theory of diffraction (PTD), the uniform theory of diffraction (UTD) and others.

There’s a direct lineage from Keller’s contributions to modern asymptotic methods used to solve installed antenna and scattering problems.

Shooting-and-bouncing rays (SBR) is another asymptotic method that emerged after Keller’s research. It advances the science of ray diffraction by hybridizing GO and PO. ANSYS HFSS SBR+ combines SBR with PTD, UTD and creeping wave physics to offer a sophisticated solution for antenna design, radar cross section (RCS) and stealth technology.

### Why Does a Teapot Drip?

Let’s go back to 1957, when Keller published Teapot Effect in the Journal of Applied Physics. In it, he mathematically proves that liquid dribbles along the underside of the spout because it’s supported by atmospheric pressure. Keller’s research shows that surface tension does not accurately account for this phenomenon — despite popular opinions otherwise.

Keller reasons that the liquid’s velocity is highest at the teapot’s lip, so its pressure is lowest there — by Bernoulli’s principle. The surrounding air pressure at the tipping point is greater than the liquid’s pressure. This turns the liquid around the corner and pushes it along the spout and the exterior surface of the teapot.

Years later, Keller identified an optimal shape for teapots, which eliminates drips.

### Why Do Ponytails Swing as You Run?

Fast forward to 2010, when Ponytail Motion is released in the Society for Industrial Applied Mathematics’ (SIAM) Journal of Applied Mathematics.

In the paper, Keller explains why a jogger’s ponytail flicks side to side, although the head it’s attached to bobs up and down. Intuition says the ponytail should follow the head’s vertical motion, but it doesn’t.

To demystify this, Keller considers two alternative models for the ponytail. In one, he treats it as a rigid model and in another he treats it as a flexible string.

In both cases, he shows that the vertical motion of the ponytail is unstable due to lateral perturbations causing it to sway left and right.

Keller goes on to develop equations that govern the shape and motion of a ponytail whose natural frequency is approximately half of the support’s (i.e., the head’s) natural frequency.

Keller uses the Hill equation to show the relationship that causes the ponytail’s sideways motion — no matter how small — to increase exponentially.

### Asymptotic Methods Help Design Antennas, Stealth Technology and Radar

In 1962, Keller published the seminal paper, Geometrical Theory of Diffraction, in the Journal of the Optical Society of America.

Here, he shines light on the ray diffractions that are produced when an electromagnetic wave hits a scattering object, its edges and corners.

GTD lays the groundwork used to develop the asymptotic EM field methods that we use today in ANSYS HFSS SBR+.

According to Dr. Robert Kipp, research and development fellow at ANSYS, “one limitation of GTD is that it works best with simple shapes like spheres, cones and cylinders. However, it is impractical to extend it to irregular shapes like aircraft, ships and automobiles. PO — another asymptotic method — proved better on this front, but it ignored multibounce interactions.

“A key innovation was the development of SBR since it synthesizes GO — which is part of GTD — and PO to eliminate their respective limitations. SBR solves diverse EM problems like installed antenna performance, RCS and other radar applications.”

Advancements to asymptotic methods were made through the introduction of PTD and UTD. PTD and UTD accurately capture the ray diffraction from sharp edges on objects (like aircraft, missiles, unmanned aerial vehicles and more) better than SBR simulation.

ANSYS HFSS SBR+ builds on top of the SBR foundation. It synthesizes UTD and PTD to predict radar signatures of electrically large targets.

Asymptotic methods can be used to simulate pivotal applications in many other industries. For instance, with HFSS SBR+ engineers can simulate how a radar detects cars on a road. Designing these automotive radar systems in HFSS SBR+ can help improve the safety and reliability of traditional and autonomous vehicles.>

Joseph Keller’s incredibly curious mind took him into unusual and complex scientific domains.

To learn how modern asymptotic techniques in ANSYS HFSS SBR+ can solve installed antenna and scattering problems, read An Integrated Workflow for Simulating Installed Antenna and RF System Performance on Airborne Platforms.