FEATool Multiphysics
v1.10 Finite Element Analysis Toolbox |

Resonance Frequencies of a Room

This example studies the resonance frequencies of an empty room by using the Helmholz equation for the time-harmonic pressure field

\[ \Delta p + k^2 p = 0 \]

The resulting eigenmodes can be compared to and validated against the analytical solution for the resonance frequencies of a boxed enclosure, that is *f = c/2*√( (i/lx) ^{2} + (j/ly)^{2} + (k/lz)^{2} )* where

This model is available as an automated tutorial by selecting **Model Examples and Tutorials...** > **Classic PDE** > **Resonance Frequencies of a Room** from the **File** menu. Or alternatively, follow the step-by-step instructions below.

- To start a new model click the
**New Model**toolbar button, or select*New Model...*from the*File*menu. - Select the
**3D**radio button.

Both the *Poisson Equation* and *Custom Equation* physics modes are suitable for this model example but in the following tutorial the *Custom Equation* will be used.

- Select the
**Custom Equation**physics mode from the*Select Physics*drop-down menu. - Enter
`p`

into the*Dependent Variable Names*edit field. - Press
**OK**to finish the physics mode selection. - Press the
**Create cube/block***Toolbar*button. - Enter
`4`

into the*x*edit field._{max} - Enter
`3`

into the*y*edit field._{max} - Enter
`2.6`

into the*z*edit field._{max} - Press
**OK**to finish and close the dialog box. - Switch to
**Grid**mode by clicking on the corresponding*Mode Toolbar*button. - Enter
`0.5`

into the*Grid Size*edit field. - Press the
**Generate**button to call the grid generation algorithm. - Switch to
**Equation**mode by clicking on the corresponding*Mode Toolbar*button. - Press the
**edit**button.

For this eigenmode analysis the right hand side of the Helmholz equation is not involved and is therefore set to zero.

- Enter
`p' - (px_x + py_y + pz_z) = 0`

into the*Equation for p*edit field. - Press
**OK**to finish and close the dialog box. - Press
**OK**to finish the equation and subdomain settings specification. - Switch to
**Boundary**mode by clicking on the corresponding*Mode Toolbar*button.

Homogenous Neumann boundary conditions are used for all boundaries to represent solid and impermeable walls.

- Select all boundaries (
**1-6**) in the list box. - Select the
**Neumann, g_p**radio button. - Press
**OK**to finish the boundary condition specification. - Switch to
**Solve**mode by clicking on the corresponding*Mode Toolbar*button. - Press the
**Settings***Toolbar*button.

Select the **Eigenvalue** solver in the *Solver Settings* dialog box, and increase the number of eigenmodes to compute to `9`

- Press the
**Solve**button.

Open the *Postprocessing* settings dialog box, plot and visualize some eigenmodes to compare them. One can see how the lower modes resonate in a single direction, and increase in complexity and frequency as the number of resonance modes are added.

- Press the
**Plot Options***Toolbar*button. - Clear the
**Surface Plot**check box. - Select the
**Iso Plot**check box. - Press
**Apply**to plot and visualize the selected postprocessing options. - Select
**0.611429 (0.124449 Hz)**from the*Available solutions/eigenvalues (frequencies)*drop-down menu. - Press
**Apply**to plot and visualize the selected postprocessing options. - Select
**1.07499 (0.165015 Hz)**from the*Available solutions/eigenvalues (frequencies)*drop-down menu. - Press
**OK**to plot and visualize the selected postprocessing options.

Compare the frequency of the computed modes to the analytical results assuming a speed of sound *c = 1* so that, *f = 0, 0.125, 0.1667, 0.1923, 0.2083, 0.2294, 0.25, 0.2545*, and *0.2835*.

The *resonance frequencies of a room* classic pde model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file, or GUI script (.fes) file.