Role of the Richardson extrapolation in the evaluation of acoustic data: Applications

Abstract

Three acoustical applications of the Richardson extrapolation procedure are described, where in each case it is not possible to make a measurement at the zero value of the independent variable: (1) the vibrational relaxation time of nitrogen in the limit to zero humidity; (2) the speed of sound of methane in the limit to zero pressure (and, subsequently, its second and third acoustical virial coefficients); (3) the acoustic impedance of a grass covered field in the limit to zero frequency. In example (1), a linear regression extrapolated to zero humidity yields a physically untenable value of the reciprocal relaxation time. In example (2), in which the number of terms in the governing physical law is infinite, the results of a least-squares best-fit depends on the selected number of terms in the representative polynomial (truncation error). In example (3) the governing physical law is unknown, but semi-empirical models have a low-frequency limit that precludes the prediction of the impedance of the ground in the limit to zero frequency. The Richardson extrapolation procedure is shown to resolve these difficulties.