COMSOL 6.3 - Organ Pipe Design

In this model, an organ flue pipe is designed and analyzed using a pipe acoustics model. Because it is a 1D model, it is fast to solve but still retains most of the relevant physical parameters when designing an organ pipe. The model includes the elastic properties of the organ pipe walls, the end impedance properties at the open pipe end, and the possibility to add a small background airflow inside the pipe. In this model, the pipe is driven at 440 Hz which is the A4 note (or a’).

A sketch of an organ pipe is shown in Figure 1. An airflow is pushed in at the bottom of the organ pipe and out via the mouth. At the mouth, an air jet strikes the sharp upper lip and this sets the air into vibration. The vibrations resonate with the organ pipe body to create the note of the pipe. In an open pipe, like the one sketched here, the fundamental tone corresponds to a half wave resonance in the pipe. The harmonics are then multiples of this frequency.

Figure 1: Sketch of an organ pipe including the mouth and the pipe body.

This model requires the Acoustics Module or the Pipe Flow Module.

The timbre of the organ pipe depends on the combination of the fundamental tone and all the harmonics. This depends on the shape of the pipe (the length and diameter) as well as on the elastic properties of the pipe walls and their thickness. Moreover, a small residual airflow in the pipe u0, may alter the natural frequencies slightly (this effect is not modeled here). Changing any of these parameters influences not only the natural frequencies of the organ pipe, but also the damping and Q value of the corresponding frequency response resonance peaks. This in turn yields a different timbre.

Model Definition

The organ pipe geometry is defined in terms of its length L, inner pipe diameter d, wall thickness dw (see Figure 1), and cross-section shape (here circular). Only the length is used when drawing the pipe geometry as a straight line segment. The inner radius, wall thickness, and pipe shape are parameters entering the governing equations. The elastic properties of the pipe wall are Young’s modulus Ew and Poisson’s ratio νw. The model parameters are given in the table below.

Table 1: Model parameters.
Name	Expression	Description
f	440 Hz	Frequency of an A4 note
Lguess	c0/(2 f)	Half wavelength at f
L	0.3805 m	Pipe length giving a resonance at 440 Hz
d	3 cm	Inner pipe diameter
dw	2 mm	Wall thickness
Ew	109 Pa	Pipe wall Young's modulus
νw	0.4	Pipe wall Poisson's ratio
c0	343 m/s	Speed of sound
ka	0.12	Wave number times tube radius
hmin	c0/3000 Hz/20	Mesh size at 3000 Hz
T0	20°C	Ambient temperature
p0	1 atm	Ambient pressure
dL	0.6a	End correction
fest	c0/(2(L+dL)	Estimated resonance frequency

The open end of the pipe is modeled by adding an end impedance property. This is an engineering relation for the case of a pipe of circular cross section ending in free space (an unflanged pipe).

Results and Discussion

The frequency response of the pipe is obtained by plotting the sound pressure level Lp at the open pipe end,

where pref is the reference pressure for air, 20 μPa, and * is the complex conjugate.

The frequency response around the first resonance frequency is plotted in Figure 2 for several values of the pipe diameter. Changing the pipe diameter clearly shifts the resonance frequency but also changes the damping and Q value, that is, the width of the peak. Hence this is an important factor when designing organ pipes.

Figure 2: Resonance peak of the fundamental frequency at 440 Hz for different inner pipe diameters.

The response for different values of the pipe wall width is plotted in Figure 3. Here it is also seen that changing the pipe wall width (in general any of the pipe wall properties) changes the resonance slightly. This is because the elastic properties of the pipe wall have influence on the effective compressibility of the system in a given cross section. This in turn changes the effective speed of sound in the pipe and thus the resonance. The value of Young’s modulus in this example model was chosen to demonstrate this effect. A typical circular organ pipe is however made of a tin-lead alloy, which has a much higher value of Young’s modulus.

In the final plot in Figure 4, the parameter values giving a fundamental resonance at 440 Hz are selected (see the parameters list) and the response is plotted for frequencies from 100 Hz to 3000 Hz. The plot shows the fundamental resonance at 440 Hz as well as the following five resonance frequencies of the organ pipe. The shape of this curve is related to the timbre of the pipe.

Figure 3: Resonance peak of the fundamental frequency at 440 Hz for different pipe wall thicknesses.

Figure 4: Resonance peaks of the six first natural frequencies of the pipe from 100 to 3000 Hz.

Pipe_Flow_Module/Pipe_Acoustics/organ_pipe_design

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file organ_pipe_design_parameters.txt.

Geometry 1

Polygon 1 (pol1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


x (m)	y (m)	z (m)
0	0	0
0	0	L

Click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Pipe Acoustics, Frequency Domain (pafd)

Set the cross-section shape of the organ pipe and the elastic properties of the pipe walls.

Fluid Properties 1

In the window for , locate the section.

In the T0 text field, type T0.

In the p0 text field, type p0.

Pipe Properties 1

In the window, click .

In the window for , locate the section.

From the list, choose .

In the di text field, type d.

Locate the section. From the list, choose .

From the E list, choose . In the associated text field, type Ew.

From the ν list, choose . In the associated text field, type nuw.

In the Δw text field, type dw.

End Impedance 1

At the open end, the organ pipe is sitting in free air. Use the end impedance to get the correct acoustic behavior here. Note that a low ka limit version, which is valid for k·a << 1, is also available. However, it is not applicable for this model with k·a ≥ 0.12, as seen in the parameters list.

In the toolbar, click

and choose .

Select Point 2 only.

In the window for , locate the section.

From the list, choose .

Pressure 1

In the toolbar, click

and choose .

Select Point 1 only.

In the window for , locate the section.

In the pin text field, type 1.

Mesh 1

In this model, the mesh is set up manually. Proceed by directly adding the desired mesh component. In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. Here, use 20 elements per wavelength at the maximum frequency.

Edge 1

In the toolbar, click

and choose .

Select Edge 1 only.

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type h_min.

In the text field, type h_min/2.

Click

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(f_est-30,0.5,f_est+30).

Here the variable f_est corresponds to the estimated resonance frequency of the pipe taking the end correction into account. For narrow pipes the calculated resonance f_est is close to the ideal A4 (or a’) note of f = 440 Hz.

Parametric Sweep

The sweep over the pipe diameter is done using a Parametric Sweep.

In the toolbar, click