Tractrix Horn Calculator

Tractrix Horn Calculator

(Default is 343 m/s for dry air at 20°C)

Calculation Results:

Calculated Horn Length: —

Flare Constant (m): —

Min. Recommended Mouth Diameter: —

Throat Area: —

Min. Mouth Area: —

Expansion Ratio (Am/At): —

Wavelength at Cutoff (λc): —

Understanding the Tractrix Horn

The Tractrix horn is a type of exponential horn profile widely used in high-fidelity audio loudspeaker designs. It's renowned for its smooth flare rate, which helps in achieving a more uniform acoustic impedance transformation from the driver to the air, resulting in excellent sound dispersion and a natural, uncolored sound.

Why Choose a Tractrix Horn?

  • Smooth Acoustic Loading: The Tractrix curve provides a very smooth transition, minimizing reflections and standing waves within the horn.
  • Wide Dispersion: Compared to other horn types, Tractrix horns often offer a wider and more consistent dispersion pattern, leading to a larger "sweet spot" for listeners.
  • Natural Sound: Many audiophiles prefer Tractrix horns for their perceived natural and uncolored sound reproduction, especially in the mid-range and treble frequencies.

Key Parameters Explained:

This calculator helps you determine the physical dimensions of a Tractrix horn based on your desired acoustic performance.

  • Desired Cutoff Frequency (Fc): This is the lowest frequency the horn is designed to reproduce efficiently. Below this frequency, the horn's acoustic loading becomes less effective, and its output rolls off. A lower Fc generally means a larger and longer horn.
  • Throat Diameter (Dt): This is the diameter of the horn at its narrowest point, where the compression driver or loudspeaker unit attaches. It's typically determined by the driver's exit diameter.
  • Speed of Sound (c): The speed at which sound waves travel through the medium (usually air). It varies with temperature and humidity, but 343 m/s is a common value for dry air at 20°C.
  • Calculated Horn Length (L): This is the physical length of the horn from its throat to the point where its radius equals the flare constant (m). This represents the effective length of the pure Tractrix profile.
  • Flare Constant (m): A fundamental parameter of the Tractrix curve, directly related to the cutoff frequency. It defines the rate at which the horn expands. For a Tractrix horn, m = c / (2 * π * Fc). It also represents the maximum radius a pure Tractrix curve can reach from its origin.
  • Minimum Recommended Mouth Diameter (Dm_min): For a horn to effectively load the air at its cutoff frequency, its mouth diameter should be at least a quarter of the wavelength at that frequency (λc / 4). This is a general guideline to prevent acoustic short-circuiting.
  • Throat Area (At): The cross-sectional area at the throat of the horn.
  • Minimum Mouth Area (Am_min): The cross-sectional area corresponding to the minimum recommended mouth diameter.
  • Expansion Ratio (Am/At): The ratio of the mouth area to the throat area. This indicates how much the horn expands from its throat to its mouth.
  • Wavelength at Cutoff (λc): The physical length of one complete sound wave cycle at the cutoff frequency (c / Fc).

How to Use the Calculator:

  1. Enter Desired Cutoff Frequency (Fc): Choose the lowest frequency you want your horn to reproduce. For a tweeter horn, this might be 800-1500 Hz. For a mid-range horn, it could be 300-500 Hz.
  2. Enter Throat Diameter (Dt): Input the diameter of your compression driver's exit or the throat of your chosen loudspeaker.
  3. Adjust Speed of Sound (Optional): The default of 343 m/s is suitable for most indoor conditions.
  4. Click "Calculate": The calculator will instantly provide the horn's length, flare constant, recommended mouth diameter, and other critical dimensions.

Important Considerations:

The calculated horn length represents the length of the pure Tractrix profile up to its flare constant radius. In practice, the actual mouth diameter of a built horn might be larger than the calculated minimum recommended diameter for better loading or aesthetic reasons. However, a pure Tractrix profile is mathematically defined such that its radius cannot exceed the flare constant (m). If you design a horn with a mouth radius larger than 'm', it means the horn's profile deviates from a pure Tractrix beyond that point, often blending into a straight flare or another curve type.

Always consider the physical size constraints, material properties, and the specific driver characteristics when designing and building a Tractrix horn.

function calculateTractrixHorn() { var cutoffFrequency = parseFloat(document.getElementById('cutoffFrequency').value); var throatDiameter = parseFloat(document.getElementById('throatDiameter').value); var speedOfSound = parseFloat(document.getElementById('speedOfSound').value); var errorMessageDiv = document.getElementById('errorMessage'); errorMessageDiv.textContent = "; // Clear previous errors if (isNaN(cutoffFrequency) || cutoffFrequency <= 0) { errorMessageDiv.textContent = 'Please enter a valid positive Cutoff Frequency.'; return; } if (isNaN(throatDiameter) || throatDiameter <= 0) { errorMessageDiv.textContent = 'Please enter a valid positive Throat Diameter.'; return; } if (isNaN(speedOfSound) || speedOfSound Rt_cm if (m_cm Rt.'; document.getElementById('resultHornLength').innerHTML = 'Calculated Horn Length: –'; document.getElementById('resultFlareConstant').innerHTML = 'Flare Constant (m): ' + m_cm.toFixed(2) + ' cm'; document.getElementById('resultMinMouthDiameter').innerHTML = 'Min. Recommended Mouth Diameter: –'; document.getElementById('resultThroatArea').innerHTML = 'Throat Area: ' + (Math.PI * Rt_cm * Rt_cm).toFixed(2) + ' cm²'; document.getElementById('resultMinMouthArea').innerHTML = 'Min. Mouth Area: –'; document.getElementById('resultExpansionRatio').innerHTML = 'Expansion Ratio (Am/At): –'; document.getElementById('resultWavelengthCutoff').innerHTML = 'Wavelength at Cutoff (λc): ' + (c_cm_s / cutoffFrequency).toFixed(2) + ' cm'; return; } // Calculate Horn Length (L) in cm // Using the formula x = m * arccosh(m/y) – sqrt(m^2 – y^2) // where y is the radius, and x is the distance from the point where y=m. // The length L is from the throat (y=Rt) to the point where y=m (x=0). var term_acosh = m_cm / Rt_cm; var acosh_val = Math.log(term_acosh + Math.sqrt(term_acosh * term_acosh – 1)); // Polyfill for Math.acosh var L_cm = m_cm * acosh_val – Math.sqrt(m_cm * m_cm – Rt_cm * Rt_cm); // Calculate Wavelength at Cutoff (λc) in cm var lambda_c_cm = c_cm_s / cutoffFrequency; // Calculate Minimum Recommended Mouth Diameter (Dm_min) in cm (1/4 wavelength rule) var Dm_min_cm = lambda_c_cm / 4; var Rm_min_cm = Dm_min_cm / 2; // Calculate Throat Area (At) in cm² var At_cm2 = Math.PI * Rt_cm * Rt_cm; // Calculate Minimum Mouth Area (Am_min) in cm² var Am_min_cm2 = Math.PI * Rm_min_cm * Rm_min_cm; // Calculate Expansion Ratio (Am_min/At) var expansionRatio = Am_min_cm2 / At_cm2; // Display results document.getElementById('resultHornLength').innerHTML = 'Calculated Horn Length: ' + L_cm.toFixed(2) + ' cm'; document.getElementById('resultFlareConstant').innerHTML = 'Flare Constant (m): ' + m_cm.toFixed(2) + ' cm'; document.getElementById('resultMinMouthDiameter').innerHTML = 'Min. Recommended Mouth Diameter: ' + Dm_min_cm.toFixed(2) + ' cm'; document.getElementById('resultThroatArea').innerHTML = 'Throat Area: ' + At_cm2.toFixed(2) + ' cm²'; document.getElementById('resultMinMouthArea').innerHTML = 'Min. Mouth Area: ' + Am_min_cm2.toFixed(2) + ' cm²'; document.getElementById('resultExpansionRatio').innerHTML = 'Expansion Ratio (Am/At): ' + expansionRatio.toFixed(2); document.getElementById('resultWavelengthCutoff').innerHTML = 'Wavelength at Cutoff (λc): ' + lambda_c_cm.toFixed(2) + ' cm'; }

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