diff --git a/README.md b/README.md
index 9298d44..5755c3b 100644
--- a/README.md
+++ b/README.md
@@ -32,7 +32,7 @@ static server works too.
 
 | Control | Effect |
 | --- | --- |
-| Scene | Load one of ten built-in set-ups (see below) |
+| Scene | Load one of twelve built-in set-ups (see below) |
 | Frequency | Oscillation rate of every source, in thousandths of a cycle per step |
 | Damping | Uniform energy loss; useful to settle a busy tank |
 | Speed | Simulation steps per animation frame |
@@ -65,6 +65,10 @@ pick a tool.
 - **Shallow-water lens** – a round patch of slow water focuses plane waves.
 - **Refraction at a boundary** – waves slow down and shorten as they cross
   into shallower water.
+- **Doppler shift** – a source gliding at less than half the wave speed;
+  wavefronts bunch up ahead of it and stretch out behind.
+- **Mach cone** – a source outrunning its own waves, leaving a V-shaped shock
+  front behind it.
 - **Empty tank** – start from nothing.
 
 ## How it works
@@ -88,7 +92,10 @@ u_next = (2u − (1 − σ) u_prev + (c·Δt/Δx)² ∇²u) / (1 + σ)
 - Walls pin the displacement to zero, which reflects waves with a phase
   inversion.
 - Sources add `A·sin(2πft + φ)` to their cell every step (a "soft" source), so
-  waves pass through them instead of scattering off a clamped cell.
+  waves pass through them instead of scattering off a clamped cell. A source
+  can also drift with a velocity in cells per step; it is then injected into
+  the four surrounding cells with bilinear weights so it glides smoothly, and
+  it bounces off the tank border and any wall it meets.
 
 Colours: amber for crests, blue for troughs, dark slate for still water. The
 intensity view keeps an exponential running average of `u²`, normalises it by
diff --git a/src/render.js b/src/render.js
index 72347b1..88fb0b3 100644
--- a/src/render.js
+++ b/src/render.js
@@ -170,7 +170,10 @@ export function paintField(field, rgba, { mode = 'displacement', scale = 1, inte
   }
   for (const src of field.sources) {
     if (!src.enabled) continue;
-    const i = field.index(src.x, src.y) * 4;
+    const x = Math.round(src.x);
+    const y = Math.round(src.y);
+    if (!field.inBounds(x, y)) continue;
+    const i = field.index(x, y) * 4;
     rgba[i] = SOURCE_COLOR[0];
     rgba[i + 1] = SOURCE_COLOR[1];
     rgba[i + 2] = SOURCE_COLOR[2];
diff --git a/src/scenes.js b/src/scenes.js
index 9f7a47b..8cd0ebb 100644
--- a/src/scenes.js
+++ b/src/scenes.js
@@ -109,6 +109,22 @@ export const SCENES = [
       field.fillSpeed(px(field, 0.5), 0, field.width - 1, field.height - 1, 0.5);
     },
   },
+  {
+    id: 'doppler',
+    name: 'Doppler shift',
+    description: 'A source gliding across the tank at less than half the wave speed. Wavefronts bunch up ahead of it and stretch out behind, so an observer in front hears a higher pitch.',
+    apply(field, { frequency }) {
+      field.addSource(px(field, 0.2), py(field, 0.5), { frequency, vx: 0.3 });
+    },
+  },
+  {
+    id: 'mach',
+    name: 'Mach cone',
+    description: 'A source outrunning its own waves. No wavefront can get ahead of it, and the crests it leaves behind pile up along a V-shaped shock front.',
+    apply(field, { frequency }) {
+      field.addSource(px(field, 0.1), py(field, 0.5), { frequency, amplitude: 1.5, vx: 0.95 });
+    },
+  },
   {
     id: 'empty',
     name: 'Empty tank',
diff --git a/src/wave.js b/src/wave.js
index 5cdb71d..f2fc58a 100644
--- a/src/wave.js
+++ b/src/wave.js
@@ -118,15 +118,69 @@ export class WaveField {
 
   /**
    * Add a continuously oscillating point source.
-   * frequency is in cycles per step, phase in radians. Returns the source.
+   * frequency is in cycles per step, phase in radians. vx/vy give a drift in
+   * cells per step; a moving source bounces off the tank's inner border and
+   * off walls. Returns the source.
    */
-  addSource(x, y, { frequency = 0.02, amplitude = 1, phase = 0 } = {}) {
+  addSource(x, y, { frequency = 0.02, amplitude = 1, phase = 0, vx = 0, vy = 0 } = {}) {
     if (!this.inBounds(x, y)) throw new RangeError('source outside the grid');
-    const source = { x, y, frequency, amplitude, phase, enabled: true };
+    const source = { x, y, frequency, amplitude, phase, vx, vy, enabled: true };
     this.sources.push(source);
     return source;
   }
 
+  /** Move a drifting source one step, reflecting off the border and walls. */
+  _moveSource(src) {
+    if (!src.vx && !src.vy) return;
+    const minX = 1;
+    const minY = 1;
+    const maxX = this.width - 2;
+    const maxY = this.height - 2;
+    let nx = src.x + src.vx;
+    let ny = src.y + src.vy;
+    if (nx < minX || nx > maxX) {
+      src.vx = -src.vx;
+      nx = Math.min(maxX, Math.max(minX, nx));
+    }
+    if (ny < minY || ny > maxY) {
+      src.vy = -src.vy;
+      ny = Math.min(maxY, Math.max(minY, ny));
+    }
+    // Reflect off walls along each axis independently so a source sliding
+    // past a barrier keeps its tangential motion.
+    if (this.wall[this.index(Math.round(nx), Math.round(src.y))]) {
+      src.vx = -src.vx;
+      nx = src.x;
+    }
+    if (this.wall[this.index(Math.round(src.x), Math.round(ny))]) {
+      src.vy = -src.vy;
+      ny = src.y;
+    }
+    src.x = nx;
+    src.y = ny;
+  }
+
+  /**
+   * Add value into the four cells around a fractional position with bilinear
+   * weights, so a moving source glides instead of hopping cell to cell.
+   */
+  _inject(buffer, x, y, value) {
+    const x0 = Math.floor(x);
+    const y0 = Math.floor(y);
+    const fx = x - x0;
+    const fy = y - y0;
+    const { width, wall } = this;
+    const put = (cx, cy, w) => {
+      if (w <= 0 || !this.inBounds(cx, cy)) return;
+      const i = cy * width + cx;
+      if (!wall[i]) buffer[i] += value * w;
+    };
+    put(x0, y0, (1 - fx) * (1 - fy));
+    put(x0 + 1, y0, fx * (1 - fy));
+    put(x0, y0 + 1, (1 - fx) * fy);
+    put(x0 + 1, y0 + 1, fx * fy);
+  }
+
   removeSource(source) {
     const i = this.sources.indexOf(source);
     if (i >= 0) this.sources.splice(i, 1);
@@ -197,9 +251,14 @@ export class WaveField {
     this.time += 1;
     for (const src of this.sources) {
       if (!src.enabled) continue;
-      const i = this.index(src.x, src.y);
-      if (wall[i]) continue;
-      next[i] += src.amplitude * Math.sin(2 * Math.PI * src.frequency * this.time + src.phase);
+      this._moveSource(src);
+      const value = src.amplitude * Math.sin(2 * Math.PI * src.frequency * this.time + src.phase);
+      if (Number.isInteger(src.x) && Number.isInteger(src.y)) {
+        const i = this.index(src.x, src.y);
+        if (!wall[i]) next[i] += value;
+      } else {
+        this._inject(next, src.x, src.y, value);
+      }
     }
 
     this.prev = cur;
diff --git a/test/render.test.js b/test/render.test.js
index f13b9f2..d029fbf 100644
--- a/test/render.test.js
+++ b/test/render.test.js
@@ -96,6 +96,11 @@ test('paintField writes opaque pixels, marks walls and sources, and honours the
   const px = (x, y) => Array.from(rgba.subarray(f.index(x, y) * 4, f.index(x, y) * 4 + 3));
   assert.deepEqual(px(1, 1), WALL_COLOR);
   assert.deepEqual(px(6, 4), SOURCE_COLOR);
+  // A drifting source between cells is marked on the nearest one.
+  f.sources[0].x = 5.4;
+  paintField(f, rgba, { scale: 0.5 });
+  assert.deepEqual(px(5, 4), SOURCE_COLOR);
+  assert.notDeepEqual(px(6, 4), SOURCE_COLOR);
   assert.deepEqual(px(3, 3), displacementColor(1), 'scale 0.5 maps u=0.5 to full colour');
   assert.deepEqual(px(0, 0), displacementColor(0));
   f.cur[f.index(3, 3)] = -0.3;
diff --git a/test/scenes.test.js b/test/scenes.test.js
index 85cebb5..d98e6ba 100644
--- a/test/scenes.test.js
+++ b/test/scenes.test.js
@@ -24,6 +24,7 @@ test('every scene loads on small and large grids without leaving the field', ()
       assert.equal(applied, scene);
       for (const s of f.sources) {
         assert.ok(f.inBounds(s.x, s.y), `${scene.id}: source at ${s.x},${s.y} outside ${w}x${h}`);
+        assert.ok(Number.isInteger(s.x) && Number.isInteger(s.y), `${scene.id}: sources start on a cell`);
         assert.equal(f.wall[f.index(s.x, s.y)], 0, `${scene.id}: source inside a wall`);
         assert.equal(s.frequency, 0.03);
       }
@@ -88,6 +89,19 @@ test('the antiphase pair produces a node on the midline where the in-phase pair
   assert.ok(antiPhase < inPhase * 0.1, `antiphase ${antiPhase} vs in-phase ${inPhase}`);
 });
 
+test('the Doppler source is slower than the waves and the Mach source faster', () => {
+  const f = new WaveField(120, 80);
+  loadScene(f, 'doppler');
+  assert.ok(Math.hypot(f.sources[0].vx, f.sources[0].vy) < f.courant);
+  loadScene(f, 'mach');
+  assert.ok(Math.hypot(f.sources[0].vx, f.sources[0].vy) > f.courant);
+  // Both keep running for a long time without leaving the tank or blowing up.
+  for (let t = 0; t < 1500; t++) f.step();
+  const s = f.sources[0];
+  assert.ok(f.inBounds(Math.round(s.x), Math.round(s.y)));
+  assert.ok(Number.isFinite(f.energy()));
+});
+
 test('the lens slows waves inside the disc only', () => {
   const f = new WaveField(200, 100);
   loadScene(f, 'lens');
diff --git a/test/wave.test.js b/test/wave.test.js
index 24f33e1..04bbf0a 100644
--- a/test/wave.test.js
+++ b/test/wave.test.js
@@ -125,6 +125,72 @@ test('an oscillating source keeps injecting energy', () => {
   assert.throws(() => f.addSource(100, 100), RangeError);
 });
 
+test('a source at a fractional position is split between neighbouring cells', () => {
+  const f = new WaveField(21, 21);
+  f.addSource(10.5, 10, { frequency: 0.25, amplitude: 1 }); // sin(pi/2) = 1 on step 1
+  f.step();
+  const a = f.cur[f.index(10, 10)];
+  const b = f.cur[f.index(11, 10)];
+  assert.ok(Math.abs(a - 0.5) < 1e-6 && Math.abs(b - 0.5) < 1e-6, `${a} ${b}`);
+  assert.equal(f.cur[f.index(10, 11)], 0);
+});
+
+test('a drifting source moves each step and bounces off the border', () => {
+  const f = new WaveField(41, 21);
+  const s = f.addSource(35, 10, { vx: 1, vy: 0 });
+  f.step();
+  assert.equal(s.x, 36);
+  for (let i = 0; i < 100; i++) {
+    f.step();
+    assert.ok(s.x >= 1 && s.x <= 39, `source left the tank at x=${s.x}`);
+    assert.equal(s.y, 10);
+  }
+  assert.ok(s.vx < 0 || s.x < 35, 'the source turned around');
+  const t = f.addSource(20, 2, { vx: 0, vy: -0.5 });
+  for (let i = 0; i < 10; i++) f.step();
+  assert.ok(t.y >= 1);
+  assert.ok(t.vy > 0);
+});
+
+test('a drifting source reflects off walls', () => {
+  const f = new WaveField(41, 21);
+  f.fillWall(30, 0, 31, 20);
+  const s = f.addSource(20, 10, { vx: 1 });
+  for (let i = 0; i < 15; i++) {
+    f.step();
+    assert.ok(s.x < 30, `source entered the wall at x=${s.x}`);
+  }
+  assert.ok(s.vx < 0 && s.x < 29, `source did not turn back: x=${s.x} vx=${s.vx}`);
+  // A still source is left alone.
+  const r = f.addSource(5, 5);
+  f.step();
+  assert.equal(r.x, 5);
+  assert.equal(r.y, 5);
+});
+
+// Count sign changes of the displacement along a row segment.
+function zeroCrossings(field, y, x0, x1) {
+  let n = 0;
+  let last = 0;
+  for (let x = x0; x <= x1; x++) {
+    const v = field.cur[field.index(x, y)];
+    if (v !== 0 && last !== 0 && Math.sign(v) !== Math.sign(last)) n++;
+    if (v !== 0) last = v;
+  }
+  return n;
+}
+
+test('a moving source shows a Doppler shift: shorter waves ahead, longer behind', () => {
+  const f = new WaveField(321, 61, { spongeWidth: 16 });
+  const s = f.addSource(60, 30, { frequency: 0.05, vx: 0.3 });
+  for (let t = 0; t < 200; t++) f.step();
+  const x = Math.round(s.x);
+  assert.ok(Math.abs(x - 120) <= 1);
+  const ahead = zeroCrossings(f, 30, x + 10, x + 70);
+  const behind = zeroCrossings(f, 30, x - 70, x - 10);
+  assert.ok(ahead > behind * 1.8, `ahead ${ahead} vs behind ${behind}`);
+});
+
 test('sources can be removed and cleared', () => {
   const f = new WaveField(11, 11);
   const a = f.addSource(3, 3);