**Drag at supersonic speeds** include the drag components at subsonic flow
conditions (skin friction drag and induced drag) and a *wave drag*,
characteristic of supersonic flows only. This drag component is in fact due to the
energy radiated by the aerodynamic system through the shock waves.

The wave drag is on turns divided in (wave) drag due to volume and (wave) drag due to
lift. It can be proved that the wave drag is proportional to the volume and inversely
proportional to the fourth power of the body length.

Two bodies are of particular theoretical (but also practical) interest: these are the
Sears-Haack body (Sears, Haack; 1947), and the von Kármán ogive. An elegant
derivation of the drag of these bodies is available in Ashley-Landahl (1965).

#### Sears-Haack Body

The Sears-Haack body is a slender body of revolution pointed at both ends, that
corresponds to the minimum wave drag (for given volume and length) of a linear
distribution of sources at supersonic speeds.

#### von Kármán’s Ogive

The von Kármán ogive is a slender body of revolution pointed at one end
only (ogive). This is also a body of minimum wave drag. Fig. 1 shows the comparative
thickness distribution of the Sears-Haack body and the von Kármán ogive
(the x axis has two different scales).

**Figure 1: Sears-Haack body vs von Karman ogive**

The wave drag of the Sears-Haack body scales with (*V*=volume, *L*=length). Therefore, for given
volume it is convenient to have very long bodies. Doubling the length *L* at
constant *V* will reduce the wave drag by a factor 8.

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