Ultrasound Physics Calculator: Beam, Resolution & Doppler Aliasing

Pulse and beam

Wavelength is $\lambda=c/f_0$. A pulse of $n$ cycles has spatial length $L_p=n\lambda$, duration $\tau_p=n/f_0$, and ideal axial resolution $R_a=L_p/2$. These relations, the lateral-resolution interpretation, Doppler projection and Nyquist limit are summarized by Kremkau, 10th ed., Appendix B, printed p. 289 (sample PDF p. 17). Jensen's DTU lecture, attached PDF pp. 11–13, supplies the independent two-cycle example.

For an unfocused circular disk, the large-aperture near-field estimate is $N=D^2/(4\lambda)$. The teaching beam envelope uses $W=D-2\lambda z/D$ for $z\le N$ and $W=2\lambda z/D$ beyond it. It joins at $W(N)=D/2$ and gives $W(2N)=D$. This is the elementary near/far-zone approximation described in Hedrick's single-element and pulse-transmission treatment, rather than a calculated point-spread function. Lateral resolution is estimated as this width. It is not a calibrated −6 dB, Rayleigh or pulse-echo PSF width.

The separate far-field half-angle $\beta=\arcsin(1.22\lambda/D)$ refers to the first null of a uniform circular aperture. It uses a different width convention and does not set the teaching envelope. Hoskins et al., printed pp. 24–25, provide the near-field and first-null example.

Propagation loss

For the conventional coefficient $\alpha$ in dB/(cm·MHz), one-way loss is $A_1=\alpha f_{MHz}z_{cm}$; echo loss is $A_2=2A_1$. Budget-limited penetration is $z_{pen}=A_{budget}/(2\alpha f_{MHz})$. Hoskins et al., printed p. 21, give an independent two-way-loss example. Decibels are logarithmic labels; the attenuation law uses explicit 1 MHz and 1 cm references to keep its numeric convention unambiguous.

Sampling and Doppler

The leading edge returns after $T_e=2z/c$. The ideal ceiling is $c/(2z)$; receiving the whole pulse gives $PRF_{cap}=1/(T_e+\tau_p)$. A shared transmit/receive probe also requires $T_e\ge\tau_p$, equivalently $z\ge L_p/2$, so the echo starts after transmission. Shallower depths are excluded from the speed-boundary curve. This ideal transmit dead zone follows pulse geometry; Fabiszewska et al. (2017), printed p. 775, describe the loss of shallow echoes that overlap excitation. The model uses the smaller of this cap and the requested PRF, with no extra hardware guard time. UCSD's lecture, attached PDF p. 6, gives the travel-time check.

For signed flow speed $v$ and acute angle $\theta$, $f_D=2f_0v\cos\theta/c$. With a centred baseline, $f_N=PRF/2$, and the unambiguous range requires $|f_D|<f_N$. Its radial speed boundary is $cf_N/(2f_0)$; the angle-corrected boundary divides by $|\cos\theta|$. The principal alias wraps $f_D$ into $[-f_N,f_N)$. At either Nyquist boundary direction is ambiguous. The two animated phases differ by an integer number of turns at every sampling instant.

The pulse animation follows the leading edge until it returns at $T_e$. Reception continues while the echo tail arrives, through $T_{clear}=T_e+\tau_p$; the receiver is highlighted during that interval. Waiting begins only at $T_{clear}$.

• Educational, linear, homogeneous soft-tissue acoustics with a stationary probe and a single moving scatterer; $|v|\ll c$.
• A uniformly driven, flat circular aperture with diameter at least ten wavelengths. This educational guard makes the omitted last-axial-maximum correction, λ/4, at most 1% of the approximate near-field length. The continuous-wave beam envelope is used as a pulse-width teaching estimate; near-field interference is not resolved.
• Constant sound speed, pulse cycle count and frequency-normalized attenuation throughout both travel paths.
• The loss budget covers propagation alone: it excludes reflection/scattering strength, geometric spreading, receiver noise, gain, sensitivity and required signal-to-noise ratio.
• One pulse-echo line at a time, with full reception possible only when the echo starts at or after transmission ends. The PRF cap includes pulse duration but excludes scanner recovery time, multiplexing, interleaved B-mode, ensemble length and frame-rate constraints.
• Flow sign determines toward/away direction; the angle is acute. A centred signed Doppler display is assumed; no baseline shift, wall filtering, spectral broadening or phase noise is simulated.
• Diagram coordinates share one length scale. The visible leading-edge marker is enlarged and is not a drawing of carrier cycles; the highlighted receiver denotes reception of the echo tail. The sampling animation shows phase evolution at a much slower playback speed than the physical process.
• The attenuation coefficient keeps its customary dB/(cm·MHz) convention in both unit systems; ordinary length and velocity fields use the viewer's metric/US conversions.

Use for sonography physics study, teaching and preliminary engineering intuition. The simple beam-width and lateral-resolution estimates are not predictions for a clinical scanner, focused probe or array. Measured resolution depends on aperture shape, transmit/receive focusing, bandwidth, apodization, electronics and the chosen resolution criterion.

The linear attenuation law is a soft-tissue approximation; it does not describe bone, gas, heterogeneous paths or nonlinear frequency dependence. Budget-limited penetration is a propagation-loss threshold, not guaranteed echo detection.

The ideal depth ceiling omits pulse duration; the reported PRF used includes it and may be lower than the requested rate. Real equipment usually has further timing constraints. At 90° the Doppler shift is zero and angle-corrected speed outputs and the speed-depth curve are deliberately unavailable. Angles above 60° are increasingly sensitive to angle error. A zero aliased shift can also represent an integer multiple of PRF, not necessarily stationary flow.

This tool does not diagnose disease, recommend patient scanning settings, or assess thermal index, mechanical index or acoustic exposure.

The minimum complete-echo depth $L_p/2$ accounts only for the finite transmit pulse. Transducer ring-down, switching and receiver recovery can make an actual scanner's dead zone larger.