7.6.4 Predicting sound radiation from “virtual setup”

It is possible to do a complete “virtual setup”, combining theoretical models for the bridge and soundpost with measurements on a violin body without bridge or soundpost. In terms of the mechanical response of the violin body, this simply requires a further application of the coupling methodology presented in section 7.6.3. But to extend the method to give a prediction of sound radiation requires a further theoretical ingredient. I am indebted to Robin Langley for the derivation to be presented below.

We will first derive a mathematical result in algebraic form, then give a physical explanation of what the result means and how it works. It is useful to describe the result in very general terms, then give the particular application to the violin problem. In general terms, then, we have two groups of degrees of freedom: there are the ones involved in the coupling calculation (in our case, the velocities at four points, at the bridge feet and post ends) then there are “extra” variables, which could for example be sound pressure at chosen microphone positions or velocities at other points on the structure. Denote these by vectors $\underline{u}_1$ and $\underline{u}_2$ respectively.

Corresponding to these are two vectors of generalised forces, which we denote $\underline{f}_1$ and $\underline{f}_2$. We are dealing entirely with linear systems, and we restrict attention to harmonic quantities at frequency $\omega$, so these degrees of freedom and forces are linked via a matrix of frequency response functions. We can partition this matrix into four blocks, to match the two groups of degrees of freedom:

$$\begin{bmatrix} Y_{11} & Y_{12} \\Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} \underline{f}_1 \\ \underline{f}_2 \end{bmatrix} = \begin{bmatrix} \underline{u}_1 \\ \underline{u}_2 \end{bmatrix} \tag{1}$$

where the block $Y_{11}$ is the admittance matrix linking the first group of velocities to their forces (so it will be a $4 \times 4$ matrix for our particular problem).

In order to do the coupling calculation (to add the soundpost, in our case) we need to follow the procedure used in section 7.6.3: we replace $Y_{11}$ by $\hat{Y}_{11}$ according to

$$\hat{Y}_{11}^{-1} = Y_{11}^{-1} + X^{-1} \tag{2}$$

where $X$ is the corresponding admittance matrix for the system to be coupled. In our problem, this will be a $4 \times 4$ matrix of which only one $2 \times 2$ block will be non-zero, corresponding to the degrees of freedom at the ends of the post. $\hat{Y}_{11}$ will be one block of a larger matrix describing the system after the constraint has been applied (i.e. after the soundpost has been inserted):

$$\begin{bmatrix} \hat{Y}_{11} & \hat{Y}_{12} \\\hat{Y}_{21} & \hat{Y}_{22} \end{bmatrix} \begin{bmatrix} \underline{f}_1 \\ \underline{f}_2 \end{bmatrix} = \begin{bmatrix} \underline{u}_1 \\ \underline{u}_2 \end{bmatrix} . \tag{3}$$

The challenge now is to find an expression for $\hat{Y}_{21}$ which does not involve $Y_{12}$ or $Y_{22}$, because we don’t want to have to measure these transfer functions: they would involve applying forces $\underline{f}_2$, which for the case of radiated sound would involve acoustic sources. However, we are happy to measure the terms of $Y_{21}$, which only involve observing the second group of variables while driving the first group — for example, tapping on the structure while measuring the sound pressure at the microphone.

We can denote the inverse of the admittance matrix by a similarly partitioned impedance matrix:

$$\begin{bmatrix} Y_{11} & Y_{12} \\Y_{21} & Y_{22} \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\Z_{21} & Z_{22} \end{bmatrix}^{-1} \tag{4}$$

and a corresponding expression involving hats on all the terms for the constrained system. Now we can make use of some standard results for the inverse of a partitioned matrix, which can be derived by straightforward matrix algebra:

$$Y_{11} = \left[ Z_{11} – Z_{12} Z_{22}^{-1} Z_{21} \right]^{-1} \tag{5}$$

and

$$Y_{21} = -Z_{22}^{-1} Z_{21} \left[ Z_{11} – Z_{12} Z_{22}^{-1} Z_{21} \right]^{-1} = -Z_{22}^{-1} Z_{21} Y_{11} , \tag{6}$$

and the corresponding result for the constrained system

$$\hat{Y}_{21} = -Z_{22}^{-1} Z_{21} \left[ (Z_{11}+X^{-1}) – Z_{12} Z_{22}^{-1} Z_{21} \right]^{-1}$$

$$= -Z_{22}^{-1} Z_{21} \left[Y_{11}^{-1} + X^{-1} \right]^{-1}$$

$$= Y_{21} Y_{11}^{-1} \left[Y_{11}^{-1} + X^{-1} \right]^{-1} , \tag{7}$$

where the last line has made use of equation (6). This is the result we want: this expression for $\hat{Y}_{21}$ allows us to calculate the response of the sound pressure (or other “extra” variables) in the constrained system from readily measurable quantities relating to the unconstrained system, plus the admittance matrix $X$ of the soundpost. For the case we are interested in there no forces $\underline{f}_2$, so

$$\underline{u}_2 = \hat{Y}_{21} \underline{f}_1 = Y_{21} Y_{11}^{-1} \left[Y_{11}^{-1} + X^{-1} \right]^{-1} \underline{f}_1 . \tag{8}$$

This equation has a simple physical interpretation. The final expression $\left[Y_{11}^{-1} + X^{-1} \right]^{-1} \underline{f}_1$ gives the vector $\underline{u}_1$ of velocities for the constrained system. Pre-multiplying by the inverse matrix $Y_{11}^{-1}$ converts this to a vector of forces that, when acting on the unconstrained system, would give exactly the same set of velocities at the measured points. But no forces act on either system at any other points, so the motion of the entire system is the same as that of the constrained system. In particular, this means that the sound radiation is the same, allowing us to calculate the sound pressure at the chosen microphone positions using the matrix $Y_{21}$ which describes sound radiation by forces applied at these points to the unconstrained system, something we can measure readily.

It is worth describing explicitly how this result can be applied to the soundpost problem. We need to consider four points in the coupling calculation, so $\underline{u}_1$ has four components: we will assign them to the velocities at the post top, the post bottom, the treble bridge foot and the bass bridge foot, in that order. It is enough to consider a single “extra” variable, the sound pressure at a single microphone position, so $\underline{u}_2$ has only a single component. We can measure the $4 \times 4$ matrix $Y_{11}$ by tapping on each of the four points in turn and measuring the velocity response at all four of them. We can also measure the terms of the $4 \times 1$ matrix $Y_{21}$ by tapping on each point and measuring sound pressure at the microphone.

To add the soundpost constraint, we follow the procedure from section 7.6.3. The matrix $X$ is given by equation (16) from that section (called $Y_p$ there). So to implement equation (2) from this section we have to invert the measured matrix $Y_{11}$ at each frequency, add the inverse $X^{-1}$ to the $2 \times 2$ block in the top left corner of $Y_{11}^{-1}$, then re-invert to give $\hat{Y}_{11}$.

The $2 \times 2$ block in the bottom right-hand corner of $\hat{Y}_{11}$ gives us the admittance matrix at the bridge feet, with the soundpost in place. This can be used in the analysis from section 7.5.1, in conjunction with the parameter values for the bridge model, to evaluate the forces at the bridge feet in response to an applied force at the bridge top — equation (51) of section 7.5.1.

Finally, we calculate the combination $Y_{11}^{-1} \left[Y_{11}^{-1} + X^{-1} \right]^{-1}$ needed in equation (7), and combine this with the measured $Y_{21}$ and the deduced values of the forces at the bridge feet so that the sound pressure can be calculated from equation (8).