In this section, a crude model of a violin body is developed. A rectangular box is assumed, as sketched in Fig. 1. The top and the back are flat rectangular plates with constant thickness and plan dimensions $a \times b$, hinged at the edges to a rigid rectangular frame. Bending vibration of these two plates is allowed. The displacement in the $z$ direction of the top plate is expressed as
$$w_t(x,y)=\sum_{n,m=1}^{N}{\alpha_{nm} \sin \dfrac{n \pi x}{a} \sin \dfrac{m \pi y}{b}} + A +B\dfrac{x}{a} +C\dfrac{y}{b} \tag{1}$$
where $\alpha_{nm}$ are a set of constant coefficients, and the terms involving $A$, $B$ and $C$ describe the rigid-body motion of the entire box. The displacement of the back plate is similarly described by
$$w_b(x,y)=\sum_{n,m=1}^N{\beta_{nm} \sin \dfrac{n \pi x}{a} \sin \dfrac{m \pi y}{b}} + A +B\dfrac{x}{a} +C\dfrac{y}{b} \tag{2}$$
with coefficients $\beta_{nm}$, and the same rigid-body motion as the top.
Inside the box is a model of the soundpost. The ends of the post contact the two plates at the position with coordinates $(x_p,y_p)$. The post has length $L_p$ and radius $a_p$. It can execute axial vibration with displacement
$$w_p(z) = \sum_{n=0}^{(N_p-1)}{\gamma_n \cos \dfrac{n \pi z}{L_p}} \tag{3}$$
and it can also execute bending motion with lateral displacement
$$u_p(z) = \sum_{n=1}^{N_p}{\delta_n \sin \dfrac{n \pi z}{L_p}} \tag{4}$$
in the $x-z$ plane and
$$v_p(z) = \sum_{n=1}^{N_p}{\epsilon_n \sin \dfrac{n \pi z}{L_p}} \tag{5}$$
in the $y-z$ plane. Notice that equation (3) includes a term with $n=0$ to describe rigid axial motion of the post, but no such terms are needed for the bending motion. Each end of the post is coupled to the plate motion by a linear spring of stiffness $k_p$ in the $z$ direction, and by a pair of torsional springs of stiffness $K_p$ aligned in the $x-z$ plane and the $y-z$ plane.
The next step is to follow the procedure outlined in section 2.2.5: the coefficients $A, B, C, \alpha_{nm}, \beta_{nm},$ $\gamma_n, \delta_n$ and $\epsilon_n$ are all stacked together into a vector, then by evaluating a number of integrals derived from expressions for the potential and kinetic energies of the system, we can populate a stiffness matrix and a mass matrix. The eigenvalues and eigenvectors of this matrix pair then give us the natural frequencies and mode shapes.
For the mass matrix, we need to evaluate integrals derived from the kinetic energy, except that displacement is used in place of velocity. For the bending motion of the top plate, we thus need
$$T_1=\dfrac{\rho_t h_t}{2} \int_0^a{\int_0^b{w_t^2 dx dy}} \tag{6}$$
where $\rho_t$ is the density of the top plate and $h_t$ is its thickness. For the back plate we need the similar expression
$$T_2=\dfrac{\rho_b h_b}{2} \int_0^a{\int_0^b{w_b^2 dx dy}} \tag{7}$$
where $\rho_b$ is the density of the back plate and $h_b$ is its thickness. For the rigid frame representing the rib garland of the violin, the corresponding expression involves only the coefficients $A, B$ and $C$: the result is
$$T_3=\dfrac{m_r}{2}\left[ 2(a+b) A^2 + (2a/3+b) B^2) +(a+2b/3) C^2\right.$$
$$\left.+ 2(a+b) AB + 2(a+b) AC + (a+b)BC \right] \tag{8}$$
where $m_r$ is the mass per unit length of the “ribs”. For the axial motion of the post, we need
$$T_4 = \dfrac{\pi a_p^2 \rho_p}{2} \int_0^{L_p}{w_p^2 dz} \tag{9}$$
where $\rho_p$ is the density of the post. Finally, for the two components of bending motion of the post we need
$$T_5 = \dfrac{\pi a_p^2 \rho_p}{2} \int_0^{L_p}{u_p^2 dz} \tag{10}$$
and
$$T_6 = \dfrac{\pi a_p^2 \rho_p}{2} \int_0^{L_p}{v_p^2 dz} \tag{11}$$
Using information from equations (1–5), these integrals can all be evaluated straightforwardly (if a little laboriously).
For the stiffness matrix, we need corresponding expressions for potential energy. For the bending motion of the top plate this is
$$V_1=\dfrac{h_t^3}{2} \int_0^a{ \int_0^b{\left[ D_1^t \left(\dfrac{\partial^2 w_t}{\partial x^2}\right)^2 + D_2^t \left(\dfrac{\partial^2 w_t}{\partial x^2}\right)\left(\dfrac{\partial^2 w_t}{\partial y^2}\right) \right.}}$$
$$\left. + D_3^t \left(\dfrac{\partial^2 w_t}{\partial y^2}\right)^2 + D_4^t \left(\dfrac{\partial^2 w_t}{\partial x \partial y}\right)^2 \right] dx dy \tag{12}$$
where $D_1^t$, $D_2^t$, $D_3^t$ and $D_4^t$ are four elastic constants for the top plate (see section 10.3.2 and reference [1] for details). For the bending motion of the back plate we need the similar expression
$$V_2=\dfrac{h_b^3}{2} \int_0^a{ \int_0^b{\left[ D_1^b \left(\dfrac{\partial^2 w_b}{\partial x^2}\right)^2 + D_2^b \left(\dfrac{\partial^2 w_b}{\partial x^2}\right)\left(\dfrac{\partial^2 w_b}{\partial y^2}\right) \right.}}$$
$$\left. + D_3^b \left(\dfrac{\partial^2 w_b}{\partial y^2}\right)^2 + D_4^b \left(\dfrac{\partial^2 w_b}{\partial x \partial y}\right)^2 \right] dx dy \tag{13}$$
where $D_1^b$, $D_2^b$, $D_3^b$ and $D_4^b$ are the corresponding elastic constants for the back plate. For the axial vibration of the post we obtain
$$V_3 = \dfrac{\pi a_p^2 E_p}{2} \int_0^{L_p}{\left(\dfrac{\partial w_p}{\partial z} \right)^2 dz} \tag{14}$$
where $E_p$ is the Young’s modulus of the post. For post bending in the $x-z$ plane we obtain
$$V_4 = \dfrac{\pi a_p^4 E_p}{8} \int_0^{L_p}{\left(\dfrac{\partial^2 u_p}{\partial z^2} \right)^2 dz} \tag{15}$$
with a corresponding result for the $y-z$ plane
$$V_5 = \dfrac{\pi a_p^4 E_p}{8} \int_0^{L_p}{\left(\dfrac{\partial^2 v_p}{\partial z^2} \right)^2 dz} . \tag{16}$$
Again, all the integrals can be evaluated straightforwardly: see reference [2] for some of the gory details.
Finally, we need to include contributions from the contact springs at the ends of the post. For the axial spring at the top of the post this is
$$V_6 = \dfrac{k_p}{2}\left[w_t(x_p,y_p)-w_p(L_p) \right]^2 , \tag{17}$$
while for the bottom it is
$$V_7 = \dfrac{k_p}{2}\left[w_b(x_p,y_p)-w_p(0) \right]^2 . \tag{18}$$
For the torsional springs in the $x-z$ plane we have
$$V_8 = \dfrac{K_p}{2}\left[\dfrac{\partial w_t}{\partial x}(x_p,y_p)-\dfrac{\partial u_p}{\partial z}(L_p) \right]^2 \tag{19}$$
for the top and
$$V_9 = \dfrac{K_p}{2}\left[\dfrac{\partial w_b}{\partial x}(x_p,y_p)-\dfrac{\partial u_p}{\partial z}(0) \right]^2 \tag{20}$$
for the bottom. There are similar expressions for the $y-z$ plane:
$$V_{10} = \dfrac{K_p}{2}\left[\dfrac{\partial w_t}{\partial y}(x_p,y_p)-\dfrac{\partial v_p}{\partial z}(L_p) \right]^2 \tag{21}$$
for the top and
$$V_{11} = \dfrac{K_p}{2}\left[\dfrac{\partial w_b}{\partial y}(x_p,y_p)-\dfrac{\partial v_p}{\partial z}(0) \right]^2 \tag{22}$$
for the bottom.
The totals $\sum_j{T_j}$ and $\sum_j{V_j}$ are both quadratic combinations of the parameters $A, B, C,$ $\alpha_{nm},$ $\beta_{nm},$ $\gamma_n,$ $\delta_n$ and $\epsilon_n$, whose coefficients give the mass and stiffness matrices directly.
Using these results, together with the parameter values listed in Table 1, we can obtain a model which is converged for all frequencies up to 20 kHz using a total of well under 1000 degrees of freedom, a far smaller number than would be possible with anything like a Finite-Element model of a violin body able to cover the same frequency range. For the parameter values used here, the model predicts about 400 modes in the frequency range 0–20 kHz.
| $a$ | 320 mm | $b$ | 170 mm |
| $h_t$ | 2.9 mm | $h_b$ | 4.0 mm |
| $D_1^t$ | 1100 MPa | $D_2^t$ | 67 MPa |
| $D_3^t$ | 84 MPa | $D_4^t$ | 230 MPa |
| $D_1^b$ | 860 MPa | $D_2^b$ | 140 MPa |
| $D_3^b$ | 170 MPa | $D_4^b$ | 230 MPa |
| $\rho_t$ | 420 kg/m$^3$ | $\rho_b$ | 650 kg/m$^3$ |
| $L_p$ | 60 mm | $a_p$ | 3.0 mm |
| $E_p$ | 4.4 GPa | $\rho_p$ | 420 kg/m$^3$ |
| $x_p$ | 150 mm | $y_p$ | 74 mm |
| $m_r$ | 31.2 g/m | ||
| $N$ | 20 | $N_p$ | 10 |
Once we have the mode shapes, we can calculate transfer functions between any points on the structure: the only thing we need to add is an estimate of the modal damping, because all the calculations so far have been undamped. For the purposes of the results shown here, all modes above 500 Hz have been assigned the same Q-factor, with the value 30. Modes with lower frequency are assigned the value 15: without this increased damping, it was found that the strong peak around 300 Hz contributed an undesirable “booming” quality to the sound examples. This pragmatic approach was motivated by the fact that low modes like this are not in any case captured realistically by the cigar box model, and it was important to have a datum case that sounded reasonably violin-like so that the effect of small changes could be heard clearly.
One useful thing to do is to calculate the $2 \times 2$ matrix of admittances between the positions of the bridge feet (typical positions were indicated by the green rectangles in Fig. 1). That matrix can be used in exactly the same way that we used a measured matrix from a real violin in section 7.5: our cigar-box violin can be coupled to the bridge model developed earlier, and this can be used to estimate the bridge admittance and the force spectra at the bridge feet, in response to excitation at any of the string notches.
However, there is still an important missing ingredient to our model. It would be of great interest to be able to predict the sound radiated by our vibrating cigar box. This is a harder problem then predicting the vibration of the box, though. Within the spirit of this crude model, we can only hope to obtain an approximate prediction.
There are two limiting cases which give simple formulae, both based on material from Chapter 4 and both relating to the far-field sound pressure, at a distance which is very large compared both to the size of the structure and to the wavelength of sound. At very low frequency, when the entire cigar box is small compared to the wavelength of sound, the dominant sound radiation is driven by net volume change of the vibrating structure. The sound field is then a spherically symmetric monopole pattern. In section 4.3.1 we derived the formula for the sound radiated by a pulsating sphere. In section 11.7.1 it will be shown that this same formula applies to any compact source region. As applied to our cigar box model, the sound pressure from a sinusoidal vibration at frequency $\omega$ at a distance $R$ is
$$p_1 = -\omega^2 \dfrac{\rho_0}{4 \pi R} (q_t-q_b) e^{i \omega t – i k R} \tag{23}$$
where $\rho_0$ is the density of air, and $q_t$ and $q_b$ are the net volume displacements, in the positive $z$ direction, of the top plate and the back plate respectively. From equations (1) and (2), it is easy to show that these are given by
$$q_t=ab(A+B/2+C/2) $$
$$+ \sum_{n,m}{\dfrac{ab \alpha_{nm}}{nm \pi^2} \left[1-(-1)^n)\right] \left[1-(-1)^m)\right]} \tag{24}$$
and
$$q_b=ab(A+B/2+C/2) $$
$$+ \sum_{n,m}{\dfrac{ab \beta_{nm}}{nm \pi^2} \left[1-(-1)^n)\right] \left[1-(-1)^m)\right]} . \tag{25}$$
To obtain an approximation for the radiated sound at higher frequencies, we can make use of the Rayleigh integral derived in section 4.3.2. This approach applies strictly to a single plate vibrating within an infinite rigid baffle. At high frequency, it could be imagined that the sound radiation in the direction normal to the top plate will be dominated by vibration of the top plate, because the back plate will be “shadowed” by the box. Furthermore, the radiated sound might not be very different to what it would be if a baffle was added around the top plate. Given these assumptions, we can obtain an approximation to the sound pressure that would be measured by a single microphone at a large distance $R$ in the direction normal to the top plate. For that special case, every point of the vibrating plate is more or less the same distance away from the microphone, so that the Rayleigh integral reduces to the very simple result
$$p_2 = -\omega^2 \dfrac{\rho_0}{2 \pi R} q_t e^{i \omega t – i k R} \tag{26}$$
in terms of the same displaced volume given by equation (24). This formula is remarkably similar to equation (23): it differs only by a factor of 2, and the fact that $q_t$ replaces the net volume change $q_t-q_r$.
Finally, we would like to obtain a single formula that we can use to give an indication of the change in radiated sound when a model parameter is changed. We can do this in an ad hoc way by interpolating between $p_1$ and $p_2$: for frequencies less that 500 Hz, we will use $p_1$; for frequencies above 1000 Hz we will use $p_2$; between these frequencies, we will use a smooth interpolation between the two. Figure 2 shows a plot, based on the parameter values listed above and with all contact springs stiff enough that the post behaves as if it were glued rigidly to both plates. The blue and red curves show $p_1$ and $p_2$ respectively, and the dashed magenta curve shows the interpolated combination.
The argument based on the Rayleigh integral can be extended to predict the far-field sound pressure in other microphone directions. It was shown by Wallace [3], and conveniently summarised in the book by Fahy [4], that for an observation at declination angle $\theta$ to the normal and azimuth angle $\phi$ relative to the $x$-axis, the sound pressure due to a single sinusoidal mode of the pinned-pinned plate can be expressed in terms of the quantity
$$q_{nm}(\theta,\phi) \approx \dfrac{ab \alpha_{nm}}{nm \pi^2} \left[\dfrac{1-(-1)^n e^{-i \alpha}}{1 – (\alpha/n \pi)^2}\right] \left[\dfrac{1-(-1)^m e^{-i \beta}}{1 – (\beta/m \pi)^2}\right] \tag{27}$$
where $\alpha_{nm}$ is the modal amplitude as before, and
$$\alpha = k a \sin \theta \cos \phi,~~~\beta=kb \sin \theta \sin \phi \tag{28}$$
where $k=\omega/c$ is the usual wavenumber for sound waves. This is a natural extension of the expression appearing in the summation in equation (24). The corresponding contributions to the Rayleigh integral from the rigid-body degrees of freedom can be calculated using the same approximation: they are
$$\lambda_A \approx -Aab \left[\dfrac{e^{i \alpha} – 1}{\alpha}\right]~\left[\dfrac{e^{i \beta} – 1}{\beta} \right] , \tag{29}$$
$$\lambda_B \approx Bab \left[\dfrac{e^{i \alpha}}{i\alpha} +\dfrac{e^{i \alpha} – 1}{\alpha^2} \right]~\left[\dfrac{e^{i \beta} – 1}{\beta}\right] \tag{30}$$
and
$$\lambda_C \approx Cab \left[\dfrac{e^{i \alpha} – 1}{\alpha}\right] ~ \left[\dfrac{e^{i \beta}}{i\beta} +\dfrac{e^{i \beta} – 1}{\beta^2} \right] . \tag{31}$$
These terms can be assembled into a combined expression equivalent to equation (24):
$$q(\theta,\phi) = \lambda_A+\lambda_B + \lambda_C + \sum_{n,m}{q_{nm}} \tag{32}$$
and this can be substituted into equation (26) in place of $q_t$ to give the sound pressure.
[1] M. E. McIntyre and J. Woodhouse, “On measuring the elastic and damping constants of orthotropic sheet materials”, Acta Metallurgica 36, 1397—1416 (1988).
[2] Myles Cameron Nadarajah, “The mechanics of the soundpost in the violin”, PhD dissertation, University of Cambridge (2018).
[3] C. E. Wallace, “Radiation resistance of a rectangular panel”, Journal of the Acoustical Society of America 51, 946–952 (1972).
[4] Frank Fahy, “Sound and structural vibration”, Academic Press (1989) (see section 2.4).