In the Home Studio - Part 3 episode we simulated the steady state pressure field for a realistically shaped room governed by the linear wave equation. The boundary condition for the room was that of ideally rigid walls. We seen that the effect of such a boundary condition is an extremely uneven frequency response. This due to how sharp the resonances of the room are. In this episode we will introduce slightly more realistic boundary conditions and see what the effect on the solution is.

In the Home Studio - Part 1 episode we gave a first look at the response of a realistically shaped room with rigid walls. We understood how to develop and run a model with an uniform velocity sphere source placed somewhere in the room. We run the model up to $400$ $\text{Hz}$ with the help of the convergence considerations outlined in the Dealing with Convergence Issues episode. In the Home Studio - Part 2 episode instead we used Elmer’s WaveSolver to compute the eigenmodes of the room. Not only that, but we introduced several improvements such as the use of $p$-elements and locally refined meshes. In this episode we will model in high detail the low frequency response of our room, up to $125$ $\text{Hz}$, making use of the improvements we just mentioned.

In the Home Studio - Part 1 episode we computed the steady state field of a realistic, but still rigid walled, room. We put a source in the room, modelled as a flux boundary condition, and run the study at few different frequencies. The solutions that we found were reminiscent of modal patterns, which is expected as the low frequency response of a room is dominated by its resonances. However, that kind of study does not inform us on the actual resonance frequencies of the room, which are properties of great interest, as well as how the associated modal shapes (eigenmodes, or eigenfunctions) look like.

In the Rigid Walled Room Revisited - Part 2 episode we reviewed the solution provided by the Elmer Wave Equation solver. In that study we solved for the eigenfrequencies and eigenmodes of a rectangular room by using a first order mesh. The results we found were very accurate already, with eigenfrequencies within $1$ $\text{Hz}$ from the exact value and eigenmodes accurate within $1.4\text{%}$. However, we seen in the Mesh Order and Accuracy episode that we can significantly increase the accuracy of results by acting on the mesh fineness and, more effectively, the order. In this episode we will solve the rigid walled room eigenproblem with a second order mesh to quantify what effects this has on the accuracy of the solution.

In the Rigid Walled Room Revisited - Part 1 episode we seen how to setup an eigen system problem with Elmer by making use of the Wave Equation solver. In this episode we will review the simulation results and check the agreement with the analytical solutions in the Acoustic Modes of a Rectangular Room episode.

We covered the rigid walled rectangular room previously in the Rigid Walled Room episode. In that episode we solved for the steady state field in a rectangular rigid walled room as driven by a source placed somewhere in the room. This allowed us to see how the steady state field is sustained by a modal superposition, the purer the lower the driving frequency (assuming that this driving frequency matched an eigenfrequency of the room). The study was setup the way it was in order to work around one limitation of the Elmer Helmholtz solver: the lack of an Eigensolver feature. Had an Helmholtz Eigensolver been available we would have been able to compare with the analytical solution presented in the Acoustic Modes of a Rectangular Room episode directly, in a much more straightforward and meaningful way. It turns out that Elmer does have the capability of solving for the normal modes of cavities. This capability is implemented in the Wave Equation solver. We will explore this solver in this and further episodes, and use this solver to build a new benchmark system for the Home Studio episodes, as we mentioned in the conclusion of the Mesh Order and Accuracy episode.

In the Rigid Walled Room episode we seen how to model a rectangular room with rigid walls. We driven the room at the modal frequencies and compared the solution field with the theoretical modal shapes, finding that the results matched single modal shapes real well until, at a frequency high enough, the contribution of multiple modes (in addition to that related to the driving modal frequency) became important. In this episode we will look at making the model more realistic. To do so, we will investigate the low frequency response of a home studio.

In the Acoustic Modes of a Rectangular Room episode we explored the analytical model of a rigid walled room with some Julia code. We focused on finding the resonance frequencies (or eigenfrequencies) of the room and calculating the related modal patterns (eigenfunctions). Now that, thanks to The Pulsating Sphere episode, we know how to setup Helmholtz problems with Elmer we can approach the problem with the FEM method. In this episode we will solve for the modal superposition in a rectangular rigid walled room and use the results from the Acoustic Modes of a Rectangular Room episode to check the accuracy.

In this episode we will build a model of a pulsating sphere source. The pulsating sphere source is an ideal source which forms the base for the development of point sources. In essence, a point source is a pulsating sphere in the limit of $a$, the radius of the sphere, approaching $0$. For this reason, although abstract, the pulsating sphere is a very powerful theoretical tool that enables the study of point sources which in turn, through integration and wave propagation principles, enable to study of any arbitrary acoustic field source.

In the last episode we seen that our model solution for the modes of vibration of a metal bar wasn’t looking particularly good for the highest mode. In fact, when we clipped into the bar, we seen few bubbles of discontinuity in the displacement field that are not expected for linearly elastic homogeneous and isotropic bodies. To understand this we have to step back a little and think about FEM some more.

In the last episode we examined the analytical solution of the acoustic modes of a rectangular room and we are now ready to take steps into moving in the world of FEM modelling. Elmer is a powerful package, but it is not extremely user friendly. So, it is best to have a gentle introduction to it first before dwelling into the intricacies of FEM modelling of acoustic fields. One of the simplest problems to solve with Elmer is that of the elastic vibration modes of solids. This problem will allow us to get started with Elmer in the most gentle way possible, by only using the GUI.