[1][2] The theory, in particular, provides an explicit formula for the wave drag, which converts the kinetic energy of the moving body into outgoing sound waves behind the body.
[3] Consider a slender body with pointed edges at the front and back.
The supersonic flow past this body will be nearly parallel to the
-axis everywhere since the shock waves formed (one at the leading edge and one at the trailing edge) will be weak; as a consequence, the flow will be potential everywhere, which can be described using the velocity potential
characterising the small deviation from the uniform flow.
is the sound speed in the incoming flow and
is the Mach number of the incoming flow.
is a disturbance propagated with an apparent time
be located at the leading end of the pointed body.
) can be propagated only into the region behind the Mach cone.
The weak Mach cone for the leading-edge is given by
, whereas the weak Mach cone for the trailing edge is given by
The disturbance far away from the body is just like a cylindrical wave propagation.
when the slender body is a solid of revolution.
If this is not the case, the solution is valid at large distances will have correction associated with the non-linear distortion of the shock profile, whose strength is proportional to
and a factor depending on the shape function
To calculate this, consider a cylindrical surface with a large radius and with an axis along the
The momentum flux density crossing through this surface is simply given by
over the cylindrical surface gives the drag force.
upon integration gives zero since the net mass flux
The second term gives the non-zero contribution, At large distances, the values
(the wave region) are the most important in the solution for
This means that we can approximate the expression in the denominator as
appearing in front of the integral need not to be differentiated since this gives rise to the small correction proportional to
Effecting the differentiation and returning to the original variables, we find Substituting this in the drag force formula gives us This can be simplified by carrying out the integration over
When the integration order is changed, the limit for
The final formula for the wave drag force may be written as or
, indicating that the drag coefficient is proportional to the square of the cross-sectional area and inversely proportional to the fourth power of the body length.
The shape with smallest wave drag for a given volume
can be obtained from the wave drag force formula.