Kármán–Moore theory is a linearized theory for supersonic flows over a slender body, named after Theodore von Kármán and Norton B. Moore, who developed the theory in 1932. 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.
Mathematical description Consider a slender body with pointed edges at the front and back. The supersonic flow past this body will be nearly parallel to the x {\displaystyle x} -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 φ = x v 1 + ϕ {\displaystyle \varphi =xv_{1}+\phi } , where v 1 {\displaystyle v_{1}} is the incoming uniform velocity and ϕ {\displaystyle \phi } characterising the small deviation from the uniform flow. In the linearized theory, ϕ {\displaystyle \phi } satisfies
∂ 2 ϕ ∂ y 2 + ∂ 2 ϕ ∂ z 2 − β 2 ∂ 2 ϕ ∂ x 2 = 0 β 2 = ( v 1 2 − c 1 2 ) / c 1 2 = M 1 2 − 1 {\displaystyle {\begin{aligned}{\frac {\partial ^{2}\phi }{\partial y^{2}}}+{\frac {\partial ^{2}\phi }{\partial z^{2}}}-\beta ^{2}{\frac {\partial ^{2}\phi }{\partial x^{2}}}=0\\\beta ^{2}=(v_{1}^{2}-c_{1}^{2})/c_{1}^{2}=M_{1}^{2}-1\end{aligned}}}
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