Why Waveriders Matter
At hypersonic speeds, drag is the dominant challenge. Conventional aircraft shapes allow the bow shock wave to spill around the vehicle, wasting the high pressure air that could otherwise generate lift. A waverider solves this by design: the vehicle is shaped so that its bow shock attaches along the entire leading edge, trapping compressed air beneath the lower surface and eliminating spillage. The result is a class of vehicles with exceptionally high lift-to-drag ratios at Mach numbers well above 5.
The concept was first proposed by Nonweiler in 1959 [1] and has since been central to programs like the Boeing X-51, which demonstrated scramjet-powered flight at Mach 5+. More recently, waverider forebodies have appeared in programs like HIFiRE-4 and in ongoing hypersonic glide vehicle research worldwide.
But designing a waverider is not like designing a conventional aircraft. You cannot simply sketch a shape and check whether the shock attaches. Instead, you must work backward.
The Inverse Design Approach
Traditional aerodynamic design is a forward problem: define a geometry, then compute the flow around it. Waverider design inverts this. The engineer starts by defining the desired shock wave, then derives the vehicle geometry that would produce it. This three-step process has remained the standard since Nonweiler [1]:
Step 1 — Generate a basic flowfield. Choose a simple, well-understood flow, such as the supersonic flow around a cone or a wedge. Solve for the complete flowfield including the shock wave shape and all flow properties behind it.
Step 2 — Define the leading edge. Draw a curve on the shock surface. This curve will become the leading edge of the waverider. Because it sits on the shock, the shock is guaranteed to be attached along the entire leading edge by construction.
Step 3 — Trace streamlines. From each point on the leading edge, trace a streamline downstream through the known flowfield. These streamlines collectively form the lower (compression) surface of the vehicle. The upper surface is typically a freestream surface, parallel to the incoming flow, ensuring no disturbance is generated on the top of the vehicle.
The elegance of this approach is that the waveriding condition (attached shock along the full leading edge) is guaranteed by construction rather than verified after the fact. The challenge is that you are constrained by the generating flowfield: the vehicle you can build depends entirely on the flow solution you started from.
Two Methods, Two Levels of Freedom
The Kilia Waverider Designer implements two inverse design methods, each offering a different trade-off between simplicity and design flexibility.
Cone Derived (SHADOW Method)
The simpler of the two methods uses the axisymmetric supersonic flow around a right circular cone as the generating flowfield. The flow behind a conical shock is described by the Taylor-Maccoll equations, a second-order nonlinear ODE that relates the radial and angular velocity components in the conical flow region.
Given a freestream Mach number and a shock cone angle, the Taylor-Maccoll solution provides the complete velocity field from the shock surface down to the cone body. The designer then projects a two-dimensional leading edge curve onto the three-dimensional conical shock surface and traces streamlines from each leading edge point back through the known Taylor-Maccoll flow.
The key design parameters are the Mach number, the shock cone angle (which must be compatible with the Mach number for an attached shock to exist), and the polynomial that defines the leading edge curve. By varying the polynomial coefficients, different waverider shapes can be generated from the same conical flow solution.
The name SHADOW comes from the code developed at Utah State University by Weaver [2]: Stability of Hypersonic Aerodynamic Derivatives Of Waveriders. The method is well suited for exploring cone-derived waverider families quickly and understanding how leading edge parameterization affects vehicle shape, volume, and aerodynamic properties.
Strengths: Computationally fast. The Taylor-Maccoll solution is a single ODE integration, and streamline tracing through the resulting conical field is straightforward. This makes it ideal for parametric studies and design space exploration.
Limitation: For a given Mach number and shock angle, the only way to change the vehicle shape is to modify the leading edge curve. The shock itself is always a simple cone, which limits the range of achievable vehicle geometries.
Osculating Cones Method
To overcome the geometric limitations of cone-derived waveriders, the osculating cones method, developed by Sobieczky [3], treats the generating shock not as a single cone but as a continuous assembly of infinitesimal conical shock segments.
At each point along the leading edge, the local shock surface is approximated by a tangent (osculating) cone whose properties, such as axis orientation and local cone angle, can vary spanwise. This means the designer can define a much more complex, three-dimensional shock shape, and the method will still produce a valid waverider as long as the variation in curvature between adjacent osculating planes remains smooth.
In practice, the designer defines a two-dimensional shock shape on the base plane and discretizes it into points. At each point, the radius of curvature of the shock determines the properties of the local osculating cone. The Taylor-Maccoll solution is computed locally for each osculating plane, and streamlines are traced within each plane independently. The resulting streamlines are then assembled spanwise to form the complete lower surface.
This approach, as detailed by Jones et al. [4], Kontogiannis et al. [5], and Son et al. [6], dramatically expands the design freedom. The vehicle is no longer limited to shapes derivable from a single conical flow. Instead, the designer can control volume distribution, upper surface contours, and even features relevant to propulsion integration by tailoring the shock shape.
Strengths: Far greater design flexibility. The shock shape can be tailored to meet volume, stability, and integration requirements simultaneously. The osculating cones method is the basis for most practical waverider forebody designs, including the HIFiRE-4 flight experiment.
Limitation: More complex to implement and more computationally expensive. The design space is also harder to define: not all prescribed shock shapes lead to physically valid geometries, and determining which combinations of parameters produce viable designs requires careful analysis.
From Inverse Design to Real Vehicles
It is important to understand what inverse design gives you and what it does not. The output of these methods is a starting geometry, an idealized, inviscid waverider shape that guarantees shock attachment at the design Mach number. This is not a flight-ready vehicle.
Real vehicles require viscous corrections (boundary layers thicken leading edges and reduce effective L/D), thermal protection, structural considerations, propulsion integration, and stability analysis across a range of flight conditions, not just the single design point. The sharp leading edges produced by inverse design must be blunted for thermal management, which introduces some shock detachment and pressure leakage.
The value of the inverse design tool is in rapidly exploring the design space: understanding how Mach number, shock angle, and leading edge shape influence the vehicle's aerodynamic characteristics. From there, promising candidates can be taken into high-fidelity CFD, structural FEA, and trajectory analysis.
Try It Yourself
The Kilia Waverider Designer implements both the Osculating Cones and Cone Derived (SHADOW) methods in the browser. You can configure flow conditions, adjust leading edge parameters, visualize the resulting geometry in 3D, and export shapes in STEP and STL format for further analysis.
For deeper work, the Waverider Studio desktop application provides additional capabilities for parametric studies and batch generation.
If your project requires high-fidelity analysis of waverider configurations, including viscous CFD, thermal analysis, or stability assessment, get in touch.
References
- Nonweiler, T.R. "Aerodynamic Problems of Manned Space Vehicles." Journal of the Royal Aeronautical Society, 63, 521–528, 1959.
- Weaver, A. "Investigating Stability of Cone-Derived Hypersonic Waverider Vehicles." M.S. Thesis, Utah State University, 2024.
- Sobieczky, H.; Dougherty, F.C.; Jones, K.D. "Hypersonic Waverider Design from Given Shock Waves." In Proceedings of the First International Hypersonic Waverider Symposium, College Park, MD, 1990.
- Jones, J.G.; Moore, K.C.; Pike, J.; Roe, P.L. "A Method for Designing Lifting Configurations for High Supersonic Speeds, Using Axisymmetric Flow Fields." Ingenieur-Archiv, 37, 56–72, 1968.
- Kontogiannis, K.; Sóbester, A.; Taylor, N. "On the Conceptual Design of Waverider Forebody Geometries." AIAA 2015-1009, 2015.
- Son, J.; Son, C.; Yee, K. "A Novel Direct Optimization Framework for Hypersonic Waverider Inverse Design Methods." Aerospace, 9(7), 348, 2022. DOI: 10.3390/aerospace9070348