= Quadrature ambiguity of pressure-tap data
\b[Quadrature ambiguity of pressure-tap data] means that the same finite set of measured surface <pressure> values can be consistent with different integrated <forces>. A chosen <quadrature> rule fixes an estimate from those readings; it does not, by itself, establish the unsampled distribution. This is an instance of how <finite samples do not certify quadrature accuracy>.
For an original smooth example, use the normalized <airfoil chord> coordinate $\xi\in[0,1]$ and let $f=C_{p,l}-C_{p,u}$ be the lower-minus-upper <pressure coefficient> difference. Put three sample locations at $0$, $1/2$ and $1$, and consider
$$
f_0(\xi)=0.6,\qquad
f_1(\xi)=0.6+84\xi^2(\xi-1/2)^2(\xi-1)^2.
$$
These <polynomials> agree at all three taps. Both the <composite trapezoidal rule> and any other deterministic rule using only these same values produce identical answers for them. Nevertheless,
$$
\int_0^1\xi^2(\xi-1/2)^2(\xi-1)^2\,d\xi=\frac1{840},
\qquad \int_0^1 f_0\,d\xi=0.6,\quad \int_0^1 f_1\,d\xi=0.7.
$$
The discrepancy is not caused by a discontinuity or mislocated tap: both profiles are smooth at every sample. At stipulated <dynamic pressure> $q=400\,\mathrm{Pa}$ and <airfoil chord> $c=0.25\,\mathrm m$, the chord-normal pressure loads are \b[$60$ and $70\,\mathrm{N/m}$]. Their relation to flow-relative <lift> still requires the geometric resolution explained in <surface pressure integration for airfoil lift>.
Additional assumptions can make an error bound possible. If a twice differentiable $f$ has a known bound $|f''|\le M$ over the entire chord, the local linear-interpolation remainder gives an integral error at most $Mh_i^3/12$ on an interval of width $h_i$. Summing gives
$$
\left|\int_0^1 f(\xi)\,d\xi-T\right|\le\frac M{12}\sum_i h_i^3.
$$
For uniform spacing $h$ on this unit interval it becomes $Mh^2/12$. A value of $M$ inferred only from the same sparse readings is not an independently established bound. Conversely, a justified polynomial model of fixed degree would supply different information and could remove the ambiguity.
A proposed experiment compares the pressure reconstruction with an independently corrected force-balance measurement at matched <angle of attack> and <Reynolds number>, then adds taps near suspected sharp features. <Pressure-tap position calibration> and repeated pressure readings address other error sources. Agreement of two sparse-data rules is a check on their chosen interpolants, not proof that an unresolved contribution is absent.
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