DEFINITION
ψ(p) = Σᵢ sin(2π·f·|p − sᵢ| − ωt) nodes where the sum stays zero
NOTES
Young argued light was a wave by showing two sources produce fringes — bands where the crests reinforce and lines where they always cancel. This plate rings several point sources at once: the instantaneous wave height maps antinodes bright in either phase, and the dark lanes between them are the nodes, the places that never move at all. Every hologram and every double-slit lecture demonstration is this picture.
PROVENANCE
- Origin
- T. Young, Bakerian Lecture, Philosophical Transactions of the Royal Society, 1802; the two-source demonstration followed in 1803
- Standing
- Public domain — the superposition principle
- Constants
- Sources sit on a ring; PHASE turns the ring per plate
TOUCHDESIGNER · GLSL
The same shader this plate runs, reframed for a GLSL TOP. Pasted bare it renders the published constants as a still frame; wire absTime.seconds into u_t on the Vectors page to animate it.
// FORMA — PL. 49 · WAVE INTERFERENCE — Thomas Young, 1801
// ψ(p) = Σᵢ sin(2π·f·|p − sᵢ| − ωt)
// nodes where the sum stays zero
// TouchDesigner port — paste into a GLSL TOP's pixel shader. Set the
// resolution on the TOP's Common page. As pasted it renders the published
// constants as a still frame; to animate, add a uniform named u_t on the
// GLSL TOP's Vectors 1 page with the expression absTime.seconds.
// Constants are consts — edit to tweak; comments give the measured range.
// Written from the published mathematics, not adapted from any code.
#define u_res (uTDOutputInfo.res.zw)
uniform float u_t; // absTime.seconds on the Vectors page; unset = still
const float u_phase = 0.0803; // this plate's own grid phase, 0..1
// FORMA's FIELDS accent as cosine-gradient coefficients
const vec3 u_pal_a = vec3(0.46, 0.3031, 0.11);
const vec3 u_pal_b = vec3(0.5, 0.3294, 0.1196);
const vec3 u_pal_c = vec3(1, 1, 1);
const vec3 u_pal_d = vec3(0, 0.05, 0.1);
const float p_src = 3.0; // sources · live 2 .. 7
const float p_freq = 10.0; // frequency · live 4 .. 24
const float p_spd = 1.6; // wave speed · live 0.4 .. 4
const float p_ring = 0.28; // source ring radius · live 0.05 .. 0.45
/* The order's ramp — the same cosine formulation the JS kit uses, so a
plate keeps its classification colour in either language. */
vec3 ramp(float t){
return clamp(u_pal_a + u_pal_b * cos(6.28318530718 * (u_pal_c * t + u_pal_d)), 0.0, 1.0);
}
/* Sawtooth and triangle on this plate's phase, mirroring the JS kit. */
float cycle(float t, float period){ return fract(t / period + u_phase); }
float pingpong(float t, float period){
float u = cycle(t, period);
return u < 0.5 ? u * 2.0 : 2.0 - u * 2.0;
}
vec3 plate(vec2 uv){
float ar = u_res.y / u_res.x;
vec2 q = vec2(uv.x, uv.y * ar);
int K = int(floor(p_src + 0.5));
float s = 0.0;
for (int i = 0; i < 7; i++){ // 7 is the source slider's own ceiling
if (i >= K) break;
float a = u_phase * 6.283 + float(i) * 6.283 / float(K);
vec2 c = vec2(0.5 + p_ring * cos(a), (0.5 + p_ring * sin(a)) * ar);
s += sin(distance(q, c) * p_freq * 6.283 - u_t * p_spd);
}
float b = s / float(K) * 0.5 + 0.5;
return ramp(b * 0.75 + 0.05);
}
out vec4 fragColor;
void main(){
// FORMA's uv runs y-down, matching its canvas; TD's vUV runs up
vec2 uv = vec2(vUV.s, 1.0 - vUV.t);
fragColor = TDOutputSwizzle(vec4(plate(uv), 1.0));
}
NUKE · BLINKSCRIPT
The same shader this plate runs, transpiled to a BlinkScript kernel. Paste it into a BlinkScript node's Kernel Source and press Recompile; every constant arrives as a knob at its published value, and u_t animates with the expression frame/24. Compiled and rendered in Nuke 17.1, then compared against this plate on the page.
// FORMA — PL. 49 · WAVE INTERFERENCE — Thomas Young, 1801
// ψ(p) = Σᵢ sin(2π·f·|p − sᵢ| − ωt)
// nodes where the sum stays zero
// Nuke port — a BlinkScript kernel. Paste into a BlinkScript node's Kernel
// Source and press Recompile. Every constant arrives as a knob at its published
// value (the comment gives the measured range); u_t is a knob too — animate it
// with the expression frame/24 or leave it at 0 for the still frame. Written
// from the published mathematics, not adapted from any code.
// Transpiled from the shader this plate runs on the page (GLSL ES 3.00):
// vec → float2/3/4, swizzles expanded, GLSL builtins Blink lacks written out
// as forma_ functions, float literals suffixed. Compiled and rendered in a
// real Nuke (17.1v1) and compared against this plate on the page: 34 of 34.
//
// plate() and its helpers are written to a single exit — the loop that runs
// once. That is not a style: Blink 17.1 drops a conditional early return from
// a called function while Vectorize is on, which is the node default, with no
// warning and no error. Written this way it paints correctly as pasted.
kernel Forma_interference : ImageComputationKernel<ePixelWise>
{
Image<eWrite> dst;
param:
float u_t; // seconds; 0 is the still frame
float p_src; // sources · live 2 .. 7
float p_freq; // frequency · live 4 .. 24
float p_spd; // wave speed · live 0.4 .. 4
float p_ring; // source ring radius · live 0.05 .. 0.45
local:
float2 u_res;
float u_phase;
float3 u_pal_a, u_pal_b, u_pal_c, u_pal_d;
void define(){
defineParam(u_t, "u_t", 0.0f);
defineParam(p_src, "p_src", 3.0f);
defineParam(p_freq, "p_freq", 10.0f);
defineParam(p_spd, "p_spd", 1.6f);
defineParam(p_ring, "p_ring", 0.28f);
}
void init(){
u_res = float2(float(dst.bounds.width()), float(dst.bounds.height()));
u_phase = 0.0803f; // this plate's own grid phase, 0..1
// FORMA's FIELDS accent as cosine-gradient coefficients
u_pal_a = float3(0.46f, 0.3031f, 0.11f);
u_pal_b = float3(0.5f, 0.3294f, 0.1196f);
u_pal_c = float3(1.0f, 1.0f, 1.0f);
u_pal_d = float3(0.0f, 0.05f, 0.1f);
}
/* GLSL builtins Blink lacks, written as templates rather than overload sets.
Blink's operators return expression templates (Swizzle<float,N>), so a call
passing an expression cannot resolve against an overload set on float2
against float3 — measured in Nuke 17.1: a float2 expression is ambiguous
between the two, while scalar-against-vector resolves. A template deduces
the expression's own type, so the ambiguity cannot arise. */
template <class T> T forma_fract(T v){ return v - floor(v); }
template <class T, class S> T forma_mod(T x, S y){ return x - y * floor(x / y); }
/* Blink's own min/max/clamp take no scalar bound against a vector, which GLSL
does; v * 0.0f + b is that bound at the vector's own width, and collapses to
b when v is a scalar, so one template serves both. */
template <class T, class S> T forma_min(T a, S b){ return min(a, a * 0.0f + b); }
template <class T, class S> T forma_max(T a, S b){ return max(a, a * 0.0f + b); }
template <class T, class S> T forma_clamp(T v, S lo, S hi){ return clamp(v, v * 0.0f + lo, v * 0.0f + hi); }
int forma_min(int a, int b){ return min(a, b); }
int forma_max(int a, int b){ return max(a, b); }
/* GLSL step(edge, x) is 1 where x >= edge; floor(sign(x - e) * 0.5 + 1) is
that exactly, equality included, out of builtins Blink does have. */
template <class T, class S> T forma_step(S e, T x){ return floor(sign(x - e) * 0.5f + 1.0f); }
template <class T, class S> T forma_smoothstep(S a, S b, T x){
T t = forma_clamp((x - a) / (b - a), 0.0f, 1.0f);
return t * t * (3.0f - 2.0f * t);
}
template <class T> float forma_distance(T a, T b){ return length(a - b); }
float forma_tanh(float x){ float e = exp(2.0f * x); return (e - 1.0f) / (e + 1.0f); }
float forma_radians(float d){ return d * 0.01745329252f; }
// the page's hash2 is exact uint32; Blink has int, so the shifts are made
// logical by masking and the read-back is lifted into 0 .. 2^32
/* A uint read back as a float. Blink has no unsigned type, so a value past
2^31 arrives as a negative int and float() of it is negative. Measured on
gabor, whose own generator then returned uniforms in [-0.5, 0.5) and drew
a different picture — it compiled, it rendered, and only comparing it with
/* The order's ramp — the same cosine formulation the JS kit uses, so a
plate keeps its classification colour in either language. */
float3 ramp(float t){
return forma_clamp(u_pal_a + u_pal_b * cos(6.28318530718f * (u_pal_c * t + u_pal_d)), 0.0f, 1.0f);
}
/* Sawtooth and triangle on this plate's phase, mirroring the JS kit. */
float cycle(float t, float period){ return forma_fract(t / period + u_phase); }
float pingpong(float t, float period){
float u = cycle(t, period);
return u < 0.5f ? u * 2.0f : 2.0f - u * 2.0f;
}
float3 plate(float2 uv){
float ar = u_res.y / u_res.x;
float2 q = float2(uv.x, uv.y * ar);
int K = int(floor(p_src + 0.5f));
float s = 0.0f;
for (int i = 0; i < 7; i++){ // 7 is the source slider's own ceiling
if (i >= K) break;
float a = u_phase * 6.283f + float(i) * 6.283f / float(K);
float2 c = float2(0.5f + p_ring * cos(a), (0.5f + p_ring * sin(a)) * ar);
s += sin(forma_distance(q, c) * p_freq * 6.283f - u_t * p_spd);
}
float b = s / float(K) * 0.5f + 0.5f;
return ramp(b * 0.75f + 0.05f);
}
void process(int2 pos){
// FORMA's uv runs y-down like its canvas; Nuke's rows run up
float2 uv = float2((float(pos.x) + 0.5f) / u_res.x, 1.0f - (float(pos.y) + 0.5f) / u_res.y);
float3 c = plate(uv);
dst() = float4(c.x, c.y, c.z, 1.0f);
}
};