# Acoustic Horn

## Introduction

An acoustic horn is a tapered sound guide that aims to maximize the efficiency of sound transfer, and can be used both at the sound transmitting and sound receiving ends such as musical instruments and hearing aid equipments.

A way to quantify the performance of an acoustic horn is to evaluate the amount of sound reflection in the horn. The following model simulates the sound propagation within an acoustic horn varying from the frequency to . The index of reflection intensity (IRI), which is defined as the average amplitude of the reflected wave, is then calculated across the frequency spectrum to show the acoustic performance of the horn.

The symbols and corresponding units used here are summarized in the Nomenclature section.

Please refer to the information provided in "Acoustics in the Frequency Domain" for more general theoretical background for acoustics.

## Pressure Acoustics Model

The propagation of harmonic sound waves can be described by the source-free Helmholtz partial differential equation (PDE) (1):

Air is used as the sound medium.
Define the variables and parameters for a frequency domain acoustics model.

## Domain

The acoustic horn geometric model consists of two parts: a semi-infinite sound channel with width , and a conical termination with width at the free end. The geometry is assumed to be infinite in the direction normal to the plane, which reduces the model into a two dimensional simulation.

Due to the symmetry along the axis it is efficient to construct the simulation domain with only the upper half of the horn, where and denote the inlet and outlet boundary. The wall boundary and the symmetric boundary are symbolized as and , respectively.

Specify the parameters of the geometry.
Define the 2D domain .

In acoustics simulations the wavelength of a sound wave needs to be resolved by a sufficiently fine mesh in order to get an accurate numerical solution. Here we set the max edge length to 12 nodes per , which means that there will be at least twelve elements per wavelength in each direction of the wave propagation.

Set the mesh size according to the maximum frequency of interest.
Discretize the domain .

## Boundary Conditions

There are three types of the boundary conditions involved in this example. At the sound inlet a radiation boundary condition is used to model the incoming sound wave.

Specify a radiation boundary condition at the sound inlet with a incoming sound amplitude ,

On the outer boundary an absorbing boundary condition is specified to model the outgoing cylindrical wave.

Set an absorbing boundary condition for a cylindical wave on the outer boundary.

On the wall boundary and the symmetric boundary a default sound hard boundary conditions is implicitly used.

## Solve the PDE Model

To analyze the performance of the acoustic horn over the frequency range , the PDE model is solved repetitively with ParametricNDSolveValue.

Create the PDE.
Solve the PDE varying from to with an increment of .

## Post-processing and Visualization

### Sound Pressure Distribution

To visualize the sound propagation within the acoustic horn, the solution is transformed into the time domain with the harmonic wave relation (2):

More information on the relation between time domain and frequency domain can be found here.

Find the maximum pressure amplitude at , and set up a legend bar and ContourPlot options.
Visualize the sound pressure distribution at frequency .

See this note about improving the visual quality of the animation.

As explained in the acoustics tutorial, if the acoustic impedance of two media are inconsistent then part of the sound wave will be reflected at the interface. The acoustic horn serves as a transformer that matches the impedance between the waveguide and the surrounding air , so that the reflected sound wave is reduced [3]. The remaining reflected wave results in a flickering sound pressure pattern within the wave channel.

Next, a curved acoustic horn is analyzed as a comparison. By smoothening the shape of the horn the wave reflection can be further minimize.

Define the 2D domain of a curved-shape horn with the curvature .
Discretize the domain .
Solve the PDE model of the curved-shape acoustic horn.
Visualize the sound pressure distribution at frequency .

See this note about improving the visual quality of the animation.

In contrast to the rectangular acoustic horn, the curved-shape boundary has smoothed the gradient of the sound impedance.

The following section demonstrates the procedure to quantify the sound reflection of an acoustic horn.

### Index of Reflection Intensity (IRI)

One quantity to measure the performance of an acoustic horn is the index of reflection intensity (IRI), which is defined as the absolute value of the average reflected wave at the sound inlet . The formula of the reflection intensity are given by:

Calculate the length of the sound inlet.
Calculate the reflection intensity for both rectangular and curved-shape horn over the frequency spectrum.
Visualize the spectrum of the reflection intensity.

As expected, the curved horn has a better performance over most frequencies. Note that the reflection spectrum appears in a decaying pattern with few deep dips, which implies that the acoustic horn is more efficient at higher frequencies. The dips correspond to the resonance frequencies of the acoustic horn, where the sound transmission is further enhanced. The specific frequencies at which the sound resonates are known as the "Eigenfrequencies". Advanced designs such as exponential horns and multicell horns [4] can be utilized to improve the horn efficiency at lower frequencies.

## Nomenclature

 Symbol Description Unit ρ density of a medium [kg/m3] c speed of sound in a medium [m/s] p sound pressure [Pa] ω sound wave angular frequency [rad/s] f sound wave frequency [Hz] X position vector [m] a, b, l, d geometry parameters [m] Γin inlet boundary N/A Γsym symmetric boundary N/A Γwall wall boundary N/A Γout far-field boundary N/A Ω computational domain N/A θ flare angle of acoustic horn [rad] J reflection intensity [m]

## References

1.  E. Bängtsson et al. Shape optimization of an acoustic horn. Computational Methods in Apply Mechanics and Engineering. 192 (2003) 15331571.

2.  F. Negri et al. Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. Journal of Computational Physics. 303 (2015) 431454.

3.  K. Bjørn. Horn Theory: An Introduction. audioXpress. (2008).