空中超声触觉的声场 Mid-Air Haptics: the Acoustic Field

本页计算的是线性、稳态、连续波条件下的复声压。84 个阵元各被视为自由场点源，按 1/r 衰减；焦距 F 与阵元 pᵢ 的距离给出时延 τᵢ = |F − pᵢ|/c。载波 f = 40 kHz、c = 343 m/s（20 °C），故 λ = c/f = 8.575 mm。截面上任一点 r 的幅值为

This page computes a steady-state, continuous-wave complex pressure field. Each of the 84 elements is a point source in free field with 1/r spreading; the geometric focus F and the element positions pᵢ set the delays τᵢ = |F − pᵢ|/c. At a 40 kHz carrier and c = 343 m/s (20 °C), λ = c/f = 8.575 mm. The amplitude at any point r in the section is

场以焦点幅值 A_F 归一化，取 20·log₁₀ 并截断在 −20 dB。采样网格为 240×150，覆盖 x ∈ [−190, 190] mm、z ∈ [−14, 240] mm；84 元模式的全分辨率样本只在围绕主波束与近场区的包络内计算，包络外按色阶下限处理；16 元模式因旁瓣分布在多个方向而计算全网格。−6 dB 等值线由同一网格的 marching squares 提取，−6 dB 宽度则由 0.4 mm 步长的一维切割得到——它们不是从屏幕像素估计的。

The field is normalised to the focal amplitude A_F, plotted as 20·log₁₀ and floored at −20 dB. The sampling grid is 240×150 over x ∈ [−190, 190] mm and z ∈ [−14, 240] mm. In the 84-element case full-resolution samples are computed only inside an envelope around the main beam and the immediate near field, and cells outside that envelope are set to the colour floor; the sparse 16-element case, whose lobes point in many directions, is computed over the full grid. The −6 dB contour is extracted from the same grid with marching squares, and the −6 dB widths come from 0.4 mm one-dimensional cuts — neither is estimated from screen pixels.

有四类真实效应被刻意省略。第一是非线性传播：在约 150 dB SPL 的估计水平上，空气的 B/A ≈ 0.4，p ≈ 0.8 kPa 时平面波冲击形成距离约 0.2 m，与传播路径同量级；因此线性叠加会高估焦点峰值，并完全略去谐波增长与饱和。第二是换能器指向性：真实 40 kHz 活塞（10 mm 孔径时 ka ≈ 3.7）并非各向同性，点源模型会高估大角度与栅瓣的贡献。第三是空气吸收：40 kHz、20 °C、50% RH 时 α ≈ 1.3 dB/m，200 mm 路径约 0.26 dB，页内忽略。第四是通道校准与互耦合：真实阵列的逐路增益、相位偏差与阵元间耦合均未建模。

Four real effects are deliberately absent. First, nonlinear propagation: at the estimated ~150 dB SPL, air (B/A ≈ 0.4) has a plane-wave shock formation distance of roughly 0.2 m for p ≈ 0.8 kPa — comparable to the propagation path. The linear sum therefore overestimates the focal peak and omits harmonic growth and saturation entirely. Second, transducer directivity: a real 40 kHz piston (ka ≈ 3.7 for a 10 mm aperture) is not isotropic, so the point-source model overweights wide-angle and grating-lobe contributions. Third, air absorption: at 40 kHz, 20 °C and 50% RH, α ≈ 1.3 dB/m, or about 0.26 dB over 200 mm; it is neglected here. Fourth, channel calibration and mutual coupling: per-channel gain, phase offset and element-to-element coupling in the real array are not modelled.

站点内公开的 UMH 原型页记录的是一套 61 单元六角阵列（行数 5-6-7-8-9-8-7-6-5，40 kHz 载波，推荐焦距 80–150 mm）。当前 84 路控制器在连接时通过设备描述符逐路上报实测坐标（x, y, z，µm），这份坐标表并未随仓库发布。因此本页使用一套写明的理想布局：12 列 × 7 行交错六角晶格，最近邻间距 7.83 mm，水平孔径 90.0 mm、垂直孔径 40.7 mm，共 84 个等效点源。16 元模式从同一孔径中抽取 4×4 个子集，用于展示稀疏采样下的栅瓣，并不对应设备的运行模式。实测坐标、单元灵敏度与校准会改变旁瓣和焦斑细节，页面中的绝对声压只应作为量级参考。

The public UMH prototype page on this site documents a 61-element hexagonal array (rows 5-6-7-8-9-8-7-6-5, 40 kHz carrier, recommended focal range 80–150 mm). The current 84-channel controller reports measured coordinates for every element (x, y, z in µm) through its device descriptor; that coordinate table is not published in the repository. This page therefore uses an explicitly stated ideal layout: a 12-column × 7-row staggered hexagonal lattice with a 7.83 mm nearest-neighbour pitch, a 90.0 mm horizontal aperture and a 40.7 mm vertical aperture, giving 84 equivalent point sources. The 16-element mode takes a 4×4 sub-set from the same aperture to demonstrate sparse sampling and grating lobes; it is not an operating mode of the device. Measured coordinates, element sensitivity and calibration will change sidelobes and focal detail, so the absolute pressures shown here are order-of-magnitude estimates only.

读数中的 Rayleigh 极限取 1.22 λR/D（D = 90 mm 水平孔径，圆孔径远场判据）。R = 100 mm 时约 11.6 mm，与模拟的 −6 dB 横向宽度约 11.9 mm 相符。o_x、o_z 分别为过焦点的水平与竖直 −6 dB 切割宽度；o_z 取 x = F_x 的竖直切割，偏转时并非焦斑主轴。焦斑呈细长形（100 mm 处约 12 × 96 mm），这是低 Fresnel 数聚焦的正常结果。声压估计假设每阵元在 1 m 处 1 Pa rms（约 94 dB SPL）并按 1/r 相干叠加，未计驱动电压、灵敏度与饱和。色阶以几何焦点归一化；Fresnel 数较低时，1/r 加权会让轴向最大点早于几何焦点：100 mm、0° 时约 +0.7 dB（z ≈ 84 mm），200 mm、40° 时可达 +7 dB 量级。因此长距离视图中被截白的区域主要对应焦前峰值，几何焦点由十字标出；读数中的“焦点 / 场峰值”给出两者之差。这是模型结果，不是绘图误差。

The Rayleigh limit in the readout is 1.22 λR/D with D = 90 mm, the horizontal aperture; this is the far-field criterion for a circular aperture. At R = 100 mm it is 11.6 mm, consistent with the simulated −6 dB lateral width of about 11.9 mm. The reported widths (o_x, o_z) are horizontal and vertical −6 dB cuts through the focus; o_z is a vertical cut at x = F_x and is not the principal axis of the spot when the beam is steered. The spot is elongated (about 12 × 96 mm at 100 mm), as expected for focusing at a low Fresnel number. The pressure estimate assumes 1 Pa rms at 1 m per element (≈94 dB SPL) with coherent 1/r summation, and ignores drive voltage, sensitivity and saturation. The colour scale is normalised to the geometric focus. At low Fresnel numbers, 1/r weighting places the axial maximum upstream of that focus: about +0.7 dB at z ≈ 84 mm for 100 mm, 0°, and of order +7 dB for 200 mm, 40°. In long-range views the clipped white region therefore corresponds mainly to the pre-focal peak, while the geometric focus is marked by the crosshair; the “Focus / field peak” readout gives the difference. That is a model result, not a plotting error.