A focal law is a list of transmit and receive delays, one per element, that makes every element’s contribution arrive at the target point at the same instant. Geometrically the delay is simply a path difference divided by velocity: element i sits at x_i from the centre of the aperture, the focal point sits at range F and angle θ, and the distance between them is r_i = √(F² + x_i² − 2·F·x_i·sin θ).

Because a delay cannot be negative, the law is referenced to the element with the longest path, which fires first at zero delay. Every other element is delayed by (r_max − r_i)/c. The result is the familiar curved delay profile: a symmetric bowl for a pure focus, a straight ramp for pure steering, and a tilted bowl for both together.

The total delay span matters practically. It must fit inside the instrument’s delay range and its delay resolution – typically 2.5 to 10 ns – because quantising the delays quantises the beam. Coarse delay resolution shows up as raised side lobes and as angular jitter in a sectorial scan.

These are contact or immersion laws, with no wedge. A wedge law also needs the refraction point on the wedge face solved element by element with Snell’s law, which the instrument’s ray tracer does; the wedge contribution is then added to each delay. Use this calculator to sanity check the shape and the span, not to hand-type wedge laws.

Compression or shear

Worked example

Elements in the active group	16
Element pitch	0.6 mm
Element number in the group	12
Steering angle	20 deg
Focal distance	30 mm
Wave mode	compression
Material velocity	5900 m/s
Half aperture (centre to end element)	4.5 mm
Element offset from the aperture centre	2.1 mm
Delay for this element	419.2 ns
Total delay span across the group	516.6 ns

Half aperture (16−1) × 0.6/2 = 4.500 mm; element 12 sits at (12 − 8.5) × 0.6 = 2.100 mm. With F = 30 mm and sin 20° = 0.34202, r_max = √(900 + 20.25 + 92.345) = 31.821 mm and r_12 = √(900 + 4.41 − 43.094) = 29.348 mm, so t = (31.821 − 29.348)/5900 × 10⁶ = 419.2 ns. The shortest path is the far end element at 28.773 mm, giving a 516.6 ns span.

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