Worthmann Brian M, Dowling David R
Department of Applied Physics, University of Michigan, Ann Arbor, Michigan 48109, USA.
Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA.
J Acoust Soc Am. 2017 Jun;141(6):4579. doi: 10.1121/1.4985440.
The frequency-difference and frequency-sum autoproducts are quadratic products of solutions of the Helmholtz equation at two different frequencies (ω and ω), and may be constructed from the Fourier transform of any time-domain acoustic field. Interestingly, the autoproducts may carry wave-field information at the difference (ω - ω) and sum (ω + ω) frequencies even though these frequencies may not be present in the original acoustic field. This paper provides analytical and simulation results that justify and illustrate this possibility, and indicate its limitations. The analysis is based on the inhomogeneous Helmholtz equation and its solutions while the simulations are for a point source in a homogeneous half-space bounded by a perfectly reflecting surface. The analysis suggests that the autoproducts have a spatial phase structure similar to that of a true acoustic field at the difference and sum frequencies if the in-band acoustic field is a plane or spherical wave. For multi-ray-path environments, this phase structure similarity persists in portions of the autoproduct fields that are not suppressed by bandwidth averaging. Discrepancies between the bandwidth-averaged autoproducts and true out-of-band acoustic fields (with potentially modified boundary conditions) scale inversely with the product of the bandwidth and ray-path arrival time differences.
频率差自积和频率和自积是亥姆霍兹方程在两个不同频率（ω 和 ω）下解的二次乘积，可由任何时域声场的傅里叶变换构建。有趣的是，即使原始声场中可能不存在这些频率，自积也可能携带频率差（ω - ω）和频率和（ω + ω）处的波场信息。本文提供了分析和模拟结果，证实并说明了这种可能性，并指出了其局限性。分析基于非齐次亥姆霍兹方程及其解，而模拟针对的是由完美反射面界定的均匀半空间中的点源。分析表明，如果带内声场是平面波或球面波，自积在频率差和频率和处具有与真实声场相似的空间相位结构。对于多径环境，这种相位结构相似性在自积场中未被带宽平均抑制的部分仍然存在。带宽平均自积与真实带外声场（可能具有修改后的边界条件）之间的差异与带宽和路径到达时间差的乘积成反比。
