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If a guitarist plays a low note on an outdoor festival stage on a freezing winter afternoon, does that sound wave reach the crowd at the exact same speed as a screaming high-frequency guitar solo on a sweltering summer night?

To master audio engineering and acoustic design, we need to uncover the mathematical relationship connecting pitch, physical wave size, and the medium carrying them.

Sound is a mechanical pressure wave consisting of alternating compressions and rarefactions traveling through an elastic medium like air.

**Frequency** ($f$), measured in **Hertz** ($	ext{Hz}$), is the number of complete pressure cycles passing a single point each second.

**Wavelength** ($\lambda$), measured in meters ($	ext{m}$), is the physical distance between two consecutive identical pressure peaks in space.

Since speed is simply distance divided by time, a wave covering distance $\lambda$ over one cycle period $T$ moves at $v = rac{\lambda}{T}$, which simplifies directly to $v = f\lambda$ because $f = rac{1}{T}$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      user-select: none;
    }
    .formula-card {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      align-items: center;
      justify-content: space-around;
      font-weight: 700;
      font-size: 15px;
      flex-wrap: wrap;
      gap: 8px;
    }
    .var-box {
      display: inline-flex;
      flex-direction: column;
      align-items: center;
      gap: 2px;
    }
    .var-val {
      font-size: 14px;
      font-weight: 600;
    }
    .c-v { color: #0284c7; }
    .c-f { color: #2563eb; }
    .c-l { color: #059669; }
    .op { color: #94a3b8; font-size: 16px; }

    .canvas-container {
      position: relative;
      width: 100%;
      height: 180px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      overflow: hidden;
    }
    canvas {
      width: 100%;
      height: 100%;
      display: block;
    }
    .speed-badge {
      position: absolute;
      top: 8px;
      right: 8px;
      background: rgba(255,255,255,0.92);
      border: 1px solid #e2e8f0;
      border-radius: 6px;
      padding: 3px 8px;
      font-size: 12px;
      font-weight: 600;
      color: #0284c7;
    }

    .controls {
      display: grid;
      grid-template-columns: repeat(3, 1fr) auto;
      gap: 8px;
      align-items: center;
    }
    .preset-btn {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 4px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }
    .preset-btn.active {
      border-color: #2563eb;
      background: #eff6ff;
      color: #2563eb;
      box-shadow: 0 0 0 1px #2563eb;
    }
    .audio-btn {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 12px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      color: #1e2945;
      display: flex;
      align-items: center;
      gap: 4px;
      white-space: nowrap;
    }
    .audio-btn.playing {
      background: #eff6ff;
      border-color: #2563eb;
      color: #2563eb;
    }
    @media (max-width: 440px) {
      .controls {
        grid-template-columns: repeat(3, 1fr);
      }
      .audio-btn {
        grid-column: span 3;
        justify-content: center;
      }
    }
  </style>
</head>
<body>

  <div class="formula-card">
    <div class="var-box">
      <span class="c-v">Velocity (v)</span>
      <span class="var-val c-v" id="valV">340 m/s</span>
    </div>
    <span class="op">=</span>
    <div class="var-box">
      <span class="c-f">Frequency (f)</span>
      <span class="var-val c-f" id="valF">340 Hz</span>
    </div>
    <span class="op">×</span>
    <div class="var-box">
      <span class="c-l">Wavelength (λ)</span>
      <span class="var-val c-l" id="valL">1.00 m</span>
    </div>
  </div>

  <div class="canvas-container">
    <div class="speed-badge">Speed v = 340 m/s (Constant) →</div>
    <canvas id="waveCanvas"></canvas>
  </div>

  <div class="controls">
    <button class="preset-btn" data-freq="170">Low Pitch<br><span style="font-size:11px;font-weight:500;color:#64748b;">170 Hz</span></button>
    <button class="preset-btn active" data-freq="340">Mid Pitch<br><span style="font-size:11px;font-weight:500;color:#64748b;">340 Hz</span></button>
    <button class="preset-btn" data-freq="680">High Pitch<br><span style="font-size:11px;font-weight:500;color:#64748b;">680 Hz</span></button>
    <button class="audio-btn" id="soundBtn">Play 🔊</button>
  </div>

  <script>
    const canvas = document.getElementById('waveCanvas');
    const ctx = canvas.getContext('2d');
    const valF = document.getElementById('valF');
    const valL = document.getElementById('valL');
    const soundBtn = document.getElementById('soundBtn');
    const presetBtns = document.querySelectorAll('.preset-btn');

    let currentFreq = 340;
    const soundSpeed = 340; // m/s
    const speedPxPerSec = 180; // visual wave travel speed in px/s

    let audioCtx = null;
    let osc = null;
    let gainNode = null;
    let isPlayingSound = false;

    function resize() {
      const rect = canvas.getBoundingClientRect();
      canvas.width = rect.width * window.devicePixelRatio;
      canvas.height = rect.height * window.devicePixelRatio;
      ctx.scale(window.devicePixelRatio, window.devicePixelRatio);
    }
    window.addEventListener('resize', resize);
    resize();

    function setFrequency(f) {
      currentFreq = f;
      const wavelength = (soundSpeed / f).toFixed(2);
      valF.textContent = f + ' Hz';
      valL.textContent = wavelength + ' m';

      if (osc && isPlayingSound) {
        osc.frequency.setValueAtTime(currentFreq, audioCtx.currentTime);
      }
    }

    presetBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        presetBtns.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        setFrequency(Number(btn.dataset.freq));
      });
    });

    soundBtn.addEventListener('click', () => {
      if (!audioCtx) {
        audioCtx = new (window.AudioContext || window.webkitAudioContext)();
      }
      if (isPlayingSound) {
        gainNode.gain.setValueAtTime(gainNode.gain.value, audioCtx.currentTime);
        gainNode.gain.exponentialRampToValueAtTime(0.0001, audioCtx.currentTime + 0.08);
        setTimeout(() => {
          if (osc) { osc.stop(); osc.disconnect(); osc = null; }
        }, 100);
        isPlayingSound = false;
        soundBtn.textContent = 'Play 🔊';
        soundBtn.classList.remove('playing');
      } else {
        if (audioCtx.state === 'suspended') {
          audioCtx.resume();
        }
        osc = audioCtx.createOscillator();
        gainNode = audioCtx.createGain();
        osc.type = 'sine';
        osc.frequency.setValueAtTime(currentFreq, audioCtx.currentTime);
        gainNode.gain.setValueAtTime(0.0001, audioCtx.currentTime);
        gainNode.gain.exponentialRampToValueAtTime(0.15, audioCtx.currentTime + 0.05);
        osc.connect(gainNode);
        gainNode.connect(audioCtx.destination);
        osc.start();
        isPlayingSound = true;
        soundBtn.textContent = 'Stop ⏹';
        soundBtn.classList.add('playing');
      }
    });

    let startTime = performance.now();

    function render(now) {
      const t = (now - startTime) / 1000;
      const w = canvas.width / window.devicePixelRatio;
      const h = canvas.height / window.devicePixelRatio;
      const centerY = h * 0.54;
      const amp = 34;

      ctx.clearRect(0, 0, w, h);

      // Baseline
      ctx.beginPath();
      ctx.setLineDash([3, 3]);
      ctx.strokeStyle = '#cbd5e1';
      ctx.lineWidth = 1;
      ctx.moveTo(0, centerY);
      ctx.lineTo(w, centerY);
      ctx.stroke();
      ctx.setLineDash([]);

      // Visual wavelength in px inversely proportional to frequency
      const lambdaPx = (340 / currentFreq) * 160;
      const k = (2 * Math.PI) / lambdaPx;
      const omega = k * speedPxPerSec;

      // Draw traveling sine wave
      ctx.beginPath();
      ctx.lineWidth = 3;
      ctx.strokeStyle = '#2563eb';
      for (let x = 0; x <= w; x += 2) {
        const y = centerY + amp * Math.sin(k * x - omega * t);
        if (x === 0) ctx.moveTo(x, y);
        else ctx.lineTo(x, y);
      }
      ctx.stroke();

      // Find crest positions for wavelength marker
      const phase = omega * t;
      const firstPeakX = ((Math.PI / 2 + phase) % (2 * Math.PI) + 2 * Math.PI) % (2 * Math.PI) / k;
      let p1 = firstPeakX;
      while (p1 < 30) p1 += lambdaPx;
      let p2 = p1 + lambdaPx;

      if (p2 <= w - 10) {
        const markerY = centerY - amp - 14;
        ctx.strokeStyle = '#059669';
        ctx.fillStyle = '#059669';
        ctx.lineWidth = 1.5;

        // Wavelength bracket
        ctx.beginPath();
        ctx.moveTo(p1, centerY - amp);
        ctx.lineTo(p1, markerY);
        ctx.lineTo(p2, markerY);
        ctx.lineTo(p2, centerY - amp);
        ctx.stroke();

        // Label
        ctx.font = '600 12px system-ui, sans-serif';
        ctx.textAlign = 'center';
        ctx.fillText(`λ = ${(soundSpeed / currentFreq).toFixed(2)} m`, (p1 + p2) / 2, markerY - 4);
      }

      requestAnimationFrame(render);
    }
    requestAnimationFrame(render);
  </script>
</body>
</html>

Does cranking up the pitch to a higher frequency make the wave travel across the room any faster?

## How Temperature Dictates Sound Velocity

In air, wave velocity ($v$) does not depend on frequency; instead, it depends almost entirely on the **kinetic energy** of the air molecules transmitting the collisions.

In 1816, French physicist Pierre-Simon Laplace corrected Isaac Newton's early acoustic model by proving that sound compressions happen so rapidly that heat cannot escape, creating **adiabatic compression**.

For practical audio calculations in dry air, we calculate the speed of sound ($v$) in meters per second ($	ext{m/s}$) using the linear approximation $v \approx 331.3 + 0.606T$, where $T$ is the temperature in Celsius ($^\circ	ext{C}$).

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif; }
    body { background: #f8fafc; color: #1e2945; padding: 12px; display: flex; flex-direction: column; gap: 10px; width: 100%; min-height: 100%; }
    
    .panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 14px;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 12px;
      flex-wrap: wrap;
    }

    .ctrl-group {
      flex: 1 1 140px;
      display: flex;
      flex-direction: column;
      gap: 4px;
    }

    .ctrl-label {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      display: flex;
      justify-content: space-between;
    }

    .val-badge {
      color: #2563eb;
      font-weight: 700;
    }

    input[type="range"] {
      width: 100%;
      accent-color: #2563eb;
      cursor: pointer;
    }

    .calc-row {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
      text-align: center;
    }

    .calc-card {
      background: #f1f5f9;
      border-radius: 8px;
      padding: 6px 4px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    .calc-title {
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
      text-transform: uppercase;
    }

    .calc-val {
      font-size: 16px;
      font-weight: 700;
      color: #0f172a;
      margin-top: 2px;
    }

    .calc-card.highlight {
      background: #eff6ff;
      border: 1px solid #bfdbfe;
    }
    .calc-card.highlight .calc-val {
      color: #2563eb;
    }

    .presets {
      display: flex;
      gap: 6px;
      overflow-x: auto;
      padding-bottom: 2px;
    }

    .btn {
      border: 1px solid #cbd5e1;
      background: #ffffff;
      padding: 4px 10px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      white-space: nowrap;
      transition: all 0.2s;
    }

    .btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }

    .wave-container {
      position: relative;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      height: 140px;
      overflow: hidden;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    canvas {
      width: 100%;
      height: 100%;
      display: block;
    }
  </style>
</head>
<body>

  
  <div class="panel">
    <div class="row">
      <div class="ctrl-group">
        <div class="ctrl-label">
          <span>Temperature (T)</span>
          <span class="val-badge" id="tempVal">20°C</span>
        </div>
        <input type="range" id="tempSlider" min="-10" max="40" step="1" value="20">
      </div>

      <div class="ctrl-group">
        <div class="ctrl-label">
          <span>Frequency (f)</span>
          <span class="val-badge" id="freqVal">100 Hz</span>
        </div>
        <input type="range" id="freqSlider" min="20" max="1000" step="5" value="100">
      </div>
    </div>

    
    <div class="presets">
      <button class="btn" onclick="setFreq(41.2, this)">41.2 Hz (Low E)</button>
      <button class="btn active" onclick="setFreq(100, this)">100 Hz (Kick)</button>
      <button class="btn" onclick="setFreq(440, this)">440 Hz (A4)</button>
      <button class="btn" onclick="setFreq(1000, this)">1 kHz</button>
    </div>
  </div>

  
  <div class="calc-row">
    <div class="calc-card">
      <span class="calc-title">Speed (v)</span>
      <span class="calc-val" id="dispSpeed">343.4 m/s</span>
    </div>
    <div class="calc-card">
      <span class="calc-title">Frequency (f)</span>
      <span class="calc-val" id="dispFreq">100 Hz</span>
    </div>
    <div class="calc-card highlight">
      <span class="calc-title">Wavelength (λ)</span>
      <span class="calc-val" id="dispLambda">3.43 m</span>
    </div>
  </div>

  
  <div class="wave-container">
    <canvas id="waveCanvas"></canvas>
  </div>

  <script>
    const canvas = document.getElementById('waveCanvas');
    const ctx = canvas.getContext('2d');

    const tempSlider = document.getElementById('tempSlider');
    const freqSlider = document.getElementById('freqSlider');
    const tempVal = document.getElementById('tempVal');
    const freqVal = document.getElementById('freqVal');
    const dispSpeed = document.getElementById('dispSpeed');
    const dispFreq = document.getElementById('dispFreq');
    const dispLambda = document.getElementById('dispLambda');
    const presetBtns = document.querySelectorAll('.presets .btn');

    let phase = 0;

    function resizeCanvas() {
      const rect = canvas.getBoundingClientRect();
      canvas.width = rect.width * window.devicePixelRatio;
      canvas.height = rect.height * window.devicePixelRatio;
      ctx.scale(window.devicePixelRatio, window.devicePixelRatio);
    }
    window.addEventListener('resize', resizeCanvas);
    resizeCanvas();

    function updateCalculations() {
      const T = parseFloat(tempSlider.value);
      const f = parseFloat(freqSlider.value);

      tempVal.textContent = `${T}°C`;
      freqVal.textContent = f >= 1000 ? `${(f/1000).toFixed(1)} kHz` : `${f} Hz`;

      // Formula: v = 331.3 + 0.606 * T
      const v = 331.3 + (0.606 * T);
      const lambda = v / f;

      dispSpeed.textContent = `${v.toFixed(1)} m/s`;
      dispFreq.textContent = f >= 1000 ? `${(f/1000).toFixed(1)} kHz` : `${f.toFixed(1)} Hz`;
      dispLambda.textContent = `${lambda.toFixed(2)} m`;
    }

    function setFreq(val, btn) {
      freqSlider.value = val;
      presetBtns.forEach(b => b.classList.remove('active'));
      if (btn) btn.classList.add('active');
      updateCalculations();
    }

    tempSlider.addEventListener('input', () => {
      updateCalculations();
    });

    freqSlider.addEventListener('input', () => {
      presetBtns.forEach(b => b.classList.remove('active'));
      updateCalculations();
    });

    function draw() {
      const rect = canvas.getBoundingClientRect();
      const w = rect.width;
      const h = rect.height;

      ctx.clearRect(0, 0, w, h);

      const T = parseFloat(tempSlider.value);
      const f = parseFloat(freqSlider.value);
      const v = 331.3 + (0.606 * T);
      const lambda = v / f;

      // Visual scaling: map 10 meters across the canvas width
      const maxMeters = 8.0;
      const pxPerMeter = w / maxMeters;
      const wavePx = Math.max(8, lambda * pxPerMeter);

      const centerY = h * 0.52;
      const amplitude = Math.min(32, h * 0.28);

      // Baseline
      ctx.strokeStyle = '#e2e8f0';
      ctx.lineWidth = 1;
      ctx.beginPath();
      ctx.moveTo(0, centerY);
      ctx.lineTo(w, centerY);
      ctx.stroke();

      // Traveling Sound Wave
      ctx.strokeStyle = '#2563eb';
      ctx.lineWidth = 2.5;
      ctx.beginPath();
      for (let x = 0; x <= w; x += 2) {
        const y = centerY + Math.sin((x / wavePx) * Math.PI * 2 - phase) * amplitude;
        if (x === 0) ctx.moveTo(x, y);
        else ctx.lineTo(x, y);
      }
      ctx.stroke();

      // Dimension Arrow for 1 Wavelength (λ)
      const startX = 24;
      const endX = Math.min(w - 24, startX + wavePx);
      const dimY = 22;

      ctx.strokeStyle = '#0f172a';
      ctx.fillStyle = '#0f172a';
      ctx.lineWidth = 1.5;

      // Span line
      ctx.beginPath();
      ctx.moveTo(startX, dimY);
      ctx.lineTo(endX, dimY);
      ctx.stroke();

      // Boundary ticks
      ctx.beginPath();
      ctx.moveTo(startX, dimY - 6);
      ctx.lineTo(startX, centerY + amplitude);
      ctx.moveTo(endX, dimY - 6);
      ctx.lineTo(endX, centerY + amplitude);
      ctx.setLineDash([3, 3]);
      ctx.strokeStyle = '#94a3b8';
      ctx.stroke();
      ctx.setLineDash([]);

      // Arrow heads
      const arrowSize = 4;
      ctx.fillStyle = '#0f172a';
      ctx.beginPath();
      ctx.moveTo(startX, dimY);
      ctx.lineTo(startX + arrowSize, dimY - arrowSize);
      ctx.lineTo(startX + arrowSize, dimY + arrowSize);
      ctx.fill();

      ctx.beginPath();
      ctx.moveTo(endX, dimY);
      ctx.lineTo(endX - arrowSize, dimY - arrowSize);
      ctx.lineTo(endX - arrowSize, dimY + arrowSize);
      ctx.fill();

      // Label
      ctx.font = '600 12px system-ui, sans-serif';
      ctx.fillStyle = '#0f172a';
      ctx.textAlign = 'center';
      const midX = (startX + endX) / 2;
      ctx.fillText(`λ = ${lambda.toFixed(2)} m`, midX, dimY - 6);

      // Phase increment proportional to frequency
      phase += 0.04;

      requestAnimationFrame(draw);
    }

    updateCalculations();
    draw();
  </script>
</body>
</html>

How do we apply this temperature-dependent velocity to find the physical footprint of a bass note in a recording studio?

## Worked Example: Calculating Sub-Bass Wavelength

Suppose a mixing engineer calibrates a subwoofer producing a fundamental bass tone of $f = 68.7	ext{ Hz}$ in a studio kept at $T = 20^\circ	ext{C}$.

First, calculate the local speed of sound: $$v = 331.3 + 0.606(20) = 331.3 + 12.12 = 343.42	ext{ m/s}$$

Next, rearrange the wave equation $v = f\lambda$ to solve for wavelength $\lambda = rac{v}{f}$: $$\lambda = rac{343.42	ext{ m/s}}{68.7	ext{ Hz}} \approx 5.00	ext{ m}$$

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8fafc;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    .math-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      font-size: 16px;
      font-weight: 600;
      flex-wrap: wrap;
    }
    .math-val {
      padding: 2px 8px;
      border-radius: 6px;
      font-variant-numeric: tabular-nums;
    }
    .val-lambda { background: #eff6ff; color: #2563eb; border: 1px solid #bfdbfe; }
    .val-v { background: #f0fdf4; color: #16a34a; border: 1px solid #bbf7d0; }
    .val-f { background: #fef3c7; color: #d97706; border: 1px solid #fde68a; }

    .canvas-container {
      position: relative;
      width: 100%;
      height: 180px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      overflow: hidden;
    }
    canvas {
      width: 100%;
      height: 100%;
      display: block;
    }

    .controls {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
    }
    @media (max-width: 480px) {
      .controls { grid-template-columns: 1fr; }
    }

    .control-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      display: flex;
      flex-direction: column;
      gap: 4px;
    }
    .control-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
    }
    .control-header span:last-child {
      color: #1e2945;
      font-weight: 700;
    }

    input[type="range"] {
      width: 100%;
      accent-color: #2563eb;
      cursor: pointer;
      margin: 4px 0;
    }

    .btn-row {
      display: flex;
      gap: 8px;
    }
    .btn {
      flex: 1;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: background 0.15s, border-color 0.15s;
    }
    .btn:hover {
      background: #f1f5f9;
      border-color: #cbd5e1;
    }
    .btn.active {
      background: #eff6ff;
      border-color: #2563eb;
      color: #2563eb;
    }
  </style>
</head>
<body>

  <div class="math-bar">
    <span>λ =</span>
    <span class="math-val val-v" id="lblV">343 m/s</span>
    <span>÷</span>
    <span class="math-val val-f" id="lblF">68.7 Hz</span>
    <span>=</span>
    <span class="math-val val-lambda" id="lblLambda">5.00 m</span>
  </div>

  <div class="canvas-container">
    <canvas id="waveCanvas"></canvas>
  </div>

  <div class="controls">
    <div class="control-card">
      <div class="control-header">
        <span>Frequency (f)</span>
        <span id="freqText">68.7 Hz</span>
      </div>
      <input type="range" id="freqSlider" min="40" max="350" step="0.1" value="68.7">
      <div class="btn-row">
        <button class="btn" id="btn68">68.7 Hz</button>
        <button class="btn" id="btn171">171.5 Hz</button>
        <button class="btn" id="btn343">343 Hz</button>
      </div>
    </div>

    <div class="control-card">
      <div class="control-header">
        <span>Speed (v) / Medium</span>
        <span id="speedText">343 m/s (20°C)</span>
      </div>
      <div class="btn-row" style="margin-top: 4px;">
        <button class="btn" id="btnCold">0°C (331 m/s)</button>
        <button class="btn active" id="btnWarm">20°C (343 m/s)</button>
      </div>
      <button class="btn" id="audioBtn" style="margin-top: 4px;">🔊 Play Tone</button>
    </div>
  </div>

  <script>
    const canvas = document.getElementById('waveCanvas');
    const ctx = canvas.getContext('2d');

    const freqSlider = document.getElementById('freqSlider');
    const freqText = document.getElementById('freqText');
    const speedText = document.getElementById('speedText');

    const lblV = document.getElementById('lblV');
    const lblF = document.getElementById('lblF');
    const lblLambda = document.getElementById('lblLambda');

    const btn68 = document.getElementById('btn68');
    const btn171 = document.getElementById('btn171');
    const btn343 = document.getElementById('btn343');
    const btnCold = document.getElementById('btnCold');
    const btnWarm = document.getElementById('btnWarm');
    const audioBtn = document.getElementById('audioBtn');

    let v = 343.42;
    let f = 68.7;
    let time = 0;
    const maxMeters = 6.0;

    function resize() {
      const dpr = window.devicePixelRatio || 1;
      const rect = canvas.getBoundingClientRect();
      canvas.width = rect.width * dpr;
      canvas.height = rect.height * dpr;
      ctx.setTransform(dpr, 0, 0, dpr, 0, 0);
    }
    window.addEventListener('resize', resize);
    resize();

    function updateMath() {
      const lambda = v / f;
      lblV.textContent = `${v.toFixed(1)} m/s`;
      lblF.textContent = `${f.toFixed(1)} Hz`;
      lblLambda.textContent = `${lambda.toFixed(2)} m`;
      freqText.textContent = `${f.toFixed(1)} Hz`;
    }

    freqSlider.addEventListener('input', (e) => {
      f = parseFloat(e.target.value);
      updateMath();
      clearActivePreset();
    });

    function setFreq(val) {
      f = val;
      freqSlider.value = f;
      updateMath();
    }

    btn68.addEventListener('click', () => { setFreq(68.7); highlightBtn(btn68); });
    btn171.addEventListener('click', () => { setFreq(171.5); highlightBtn(btn171); });
    btn343.addEventListener('click', () => { setFreq(343.0); highlightBtn(btn343); });

    function clearActivePreset() {
      [btn68, btn171, btn343].forEach(b => b.classList.remove('active'));
    }
    function highlightBtn(btn) {
      clearActivePreset();
      btn.classList.add('active');
    }

    btnCold.addEventListener('click', () => {
      v = 331.3;
      btnCold.classList.add('active');
      btnWarm.classList.remove('active');
      speedText.textContent = '331 m/s (0°C)';
      updateMath();
    });

    btnWarm.addEventListener('click', () => {
      v = 343.42;
      btnWarm.classList.add('active');
      btnCold.classList.remove('active');
      speedText.textContent = '343 m/s (20°C)';
      updateMath();
    });

    // Tone Player
    let audioCtx = null;
    let osc = null;
    let gainNode = null;
    let isPlaying = false;

    audioBtn.addEventListener('click', () => {
      if (!audioCtx) {
        audioCtx = new (window.AudioContext || window.webkitAudioContext)();
      }
      if (isPlaying) {
        stopTone();
      } else {
        startTone();
      }
    });

    function startTone() {
      if (audioCtx.state === 'suspended') {
        audioCtx.resume();
      }
      osc = audioCtx.createOscillator();
      gainNode = audioCtx.createGain();
      osc.frequency.setValueAtTime(f, audioCtx.currentTime);
      gainNode.gain.setValueAtTime(0.12, audioCtx.currentTime);
      osc.connect(gainNode);
      gainNode.connect(audioCtx.destination);
      osc.start();
      isPlaying = true;
      audioBtn.textContent = '⏹ Stop Tone';
      audioBtn.classList.add('active');
    }

    function stopTone() {
      if (osc) {
        osc.stop();
        osc.disconnect();
        osc = null;
      }
      isPlaying = false;
      audioBtn.textContent = '🔊 Play Tone';
      audioBtn.classList.remove('active');
    }

    // Animation Loop
    function draw() {
      const rect = canvas.getBoundingClientRect();
      const w = rect.width;
      const h = rect.height;

      ctx.clearRect(0, 0, w, h);

      const lambda = v / f;
      const pxPerMeter = w / maxMeters;
      const centerY = h * 0.45;
      const amplitude = h * 0.28;

      // Draw Grid / Meter markers
      ctx.strokeStyle = '#e2e8f0';
      ctx.lineWidth = 1;
      ctx.fillStyle = '#94a3b8';
      ctx.font = '11px system-ui, sans-serif';
      ctx.textAlign = 'center';

      for (let m = 0; m <= maxMeters; m += 1) {
        const x = m * pxPerMeter;
        ctx.beginPath();
        ctx.moveTo(x, 0);
        ctx.lineTo(x, h - 22);
        ctx.stroke();
        ctx.fillText(`${m}m`, x + (m === 0 ? 10 : m === maxMeters ? -10 : 0), h - 8);
      }

      // Draw baseline
      ctx.strokeStyle = '#cbd5e1';
      ctx.setLineDash([3, 3]);
      ctx.beginPath();
      ctx.moveTo(0, centerY);
      ctx.lineTo(w, centerY);
      ctx.stroke();
      ctx.setLineDash([]);

      // Draw Travelling Wave
      ctx.strokeStyle = '#2563eb';
      ctx.lineWidth = 3;
      ctx.beginPath();

      const k = (2 * Math.PI) / lambda; // spatial frequency
      const omega = 2 * Math.PI * (f > 200 ? 2 : f > 100 ? 1.5 : 1); // visually comfortable speed
      const phase = time * omega;

      for (let px = 0; px <= w; px += 2) {
        const xMeters = px / pxPerMeter;
        const y = centerY - Math.sin(k * xMeters - phase) * amplitude;
        if (px === 0) ctx.moveTo(px, y);
        else ctx.lineTo(px, y);
      }
      ctx.stroke();

      // Find first visible peak to anchor wavelength bracket
      // sin(k * x - phase) = 1 => k * x - phase = PI/2 + 2*PI*n
      // x = (PI/2 + phase + 2*PI*n) / k
      let peak1M = ((Math.PI / 2 + phase) % (2 * Math.PI)) / k;
      while (peak1M < 0.2) peak1M += lambda;
      const peak2M = peak1M + lambda;

      const p1x = peak1M * pxPerMeter;
      const p2x = peak2M * pxPerMeter;

      if (p1x < w) {
        // Highlight crest dots
        ctx.fillStyle = '#2563eb';
        ctx.beginPath();
        ctx.arc(p1x, centerY - amplitude, 5, 0, Math.PI * 2);
        ctx.fill();

        if (p2x < w) {
          ctx.beginPath();
          ctx.arc(p2x, centerY - amplitude, 5, 0, Math.PI * 2);
          ctx.fill();

          // Dimension Line (Wavelength Bracket)
          const dimY = 22;
          ctx.strokeStyle = '#d97706';
          ctx.lineWidth = 2;
          ctx.fillStyle = '#d97706';

          ctx.beginPath();
          // vertical guides
          ctx.moveTo(p1x, centerY - amplitude);
          ctx.lineTo(p1x, dimY);
          ctx.moveTo(p2x, centerY - amplitude);
          ctx.lineTo(p2x, dimY);
          // horizontal span
          ctx.moveTo(p1x, dimY);
          ctx.lineTo(p2x, dimY);
          ctx.stroke();

          // Bracket Label
          ctx.font = 'bold 12px system-ui, sans-serif';
          ctx.textAlign = 'center';
          const midX = (p1x + p2x) / 2;
          ctx.fillStyle = '#ffffff';
          ctx.fillRect(midX - 34, dimY - 9, 68, 18);
          ctx.fillStyle = '#d97706';
          ctx.fillText(`λ = ${lambda.toFixed(2)}m`, midX, dimY + 4);
        }
      }

      time += 0.016;
      if (isPlaying && osc) {
        osc.frequency.setValueAtTime(f, audioCtx.currentTime);
      }

      requestAnimationFrame(draw);
    }

    highlightBtn(btn68);
    updateMath();
    requestAnimationFrame(draw);
  </script>
</body>
</html>

What happens when the physical length is fixed, but the environmental temperature changes dramatically?

## Worked Example: Pitch Drift in Wind Instruments

An open pipe acoustic instrument fixes a half-wavelength resonance of $\lambda = 0.78	ext{ m}$. Let's calculate its sounding frequency at $T = 35^\circ	ext{C}$ during a hot outdoor concert compared to $T = 5^\circ	ext{C}$ in a cold hall.

At $35^\circ	ext{C}$, the velocity is $v_{	ext{hot}} = 331.3 + 0.606(35) = 352.51	ext{ m/s}$. The frequency is $$f_{	ext{hot}} = rac{v}{\lambda} = rac{352.51}{0.78} \approx 451.9	ext{ Hz}$$

At $5^\circ	ext{C}$, the velocity drops to $v_{	ext{cold}} = 331.3 + 0.606(5) = 334.33	ext{ m/s}$. The frequency becomes $$f_{	ext{cold}} = rac{334.33}{0.78} \approx 428.6	ext{ Hz}$$

The instrument drifts sharp in hot weather and flat in cold weather because faster sound velocity forces more cycles per second through a fixed pipe length.

A common mistake is assuming that high-frequency sounds travel faster than low-frequency sounds because they have more energy.

In audible air acoustics, all frequencies travel at the **same velocity** governed entirely by the temperature and medium; as frequency increases, wavelength contracts proportionally to maintain a constant speed.

music-technology

Calculating Sound Velocity, Wavelength, and Frequency. Apply the wave equation v = fλ to calculate frequency, wavelength, and the speed of sound at varying temperatures.

Speed of Sound in Gases, Temperature Dependence, and Wave Speed Formula

Sound Waves and Music: Lesson 2 - The Speed of Sound and Temperature Calculations

Waves Intro & Sound: Interactive Modeling of Frequency, Wavelength, and Wave Speed

Pedagogical strategy for Year 13 (A2 Level / MUSTECH-22.4): Emphasize dimensional rigor and physical causality. Students often conflate mathematical manipulation with physical cause-and-effect; teachers should firmly establish that frequency is locked by the mechanical/electronic source oscillator, speed is determined by thermodynamic properties of the propagation medium, and wavelength is the resulting dependent spatial variable ($\lambda = v(T)/f$). Incorporate real-world music technology and acoustic engineering contexts (such as outdoor stage delay-tower time alignment, mic phasing, and temperature-induced tuning drifts in pipe organs and brass instruments) to solidify theoretical understanding.

Calculating Sound Velocity, Wavelength, and Frequency

The Anatomy of the Wave Equation

How Temperature Dictates Sound Velocity