## Abstract

This study shows that a dynamic modal characterization of musical instruments with membrane can be carried out using a low-cost device and that the obtained very informative results can be presented as a movie. The proposed device is based on a digital holography technique using the quasi-Fourier configuration and time-average principle. Its practical realization with a commercial digital camera and large plane mirrors allows relatively simple analyzing of big vibration surfaces. The experimental measurements given for a percussion instrument are supported by the mathematical formulation of the problem.

© 2005 Optical Society of America

Full Article |

PDF Article
### Equations (13)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$U({x}_{1},{y}_{1},t)=\delta ({x}_{1}-X,{y}_{1}-Y)+s({x}_{1},{y}_{1},t),$$
(2)
$$U({x}_{2},{y}_{2},t)\propto \mathrm{exp}\left[i(\pi \u2044\lambda d)\left({x}_{2}^{2}+{y}_{2}^{2}\right)\right]F\left\{U({x}_{1},{y}_{1},t)\mathrm{exp}[i(\pi /\lambda d)\left({x}_{1}^{2}+{y}_{1}^{2}\right)\right]\},$$
(3)
$$U({x}_{2},{y}_{2},t)\propto \mathrm{exp}\left\{i(\pi \u2044\lambda d)\left[{\left({x}_{2}-X\right)}^{2}+{\left({y}_{2}-Y\right)}^{2}\right]\right\}+\mathrm{exp}\left[i(\pi \u2044\lambda d)\left({x}_{2}^{2}+{y}_{2}^{2}\right)\right]$$
(4)
$$\times F\left\{s({x}_{1},{y}_{1})\mathrm{exp}\left[i(4\pi \u2044\lambda )h({x}_{1},{y}_{1})\mathrm{sin}2\pi \mathit{ft}+i(\pi \u2044\lambda d)\left({x}_{1}^{2}+{y}_{1}^{2}\right)\right]\right\}.$$
(5)
$${E}_{1}({x}_{2},{y}_{2})\propto \mathrm{exp}[-i(2\pi \u2044\lambda d)\left({x}_{2}X+{y}_{2}Y\right)]$$
(6)
$$\times F\left\{s({x}_{1},{y}_{1}){J}_{0}\left[(4\pi \u2044\lambda )h({x}_{1},{y}_{1})\right]\mathrm{exp}\left[i(\pi \u2044\lambda d)\left({x}_{1}^{2}+{y}_{1}^{2}\right)\right]\right\}$$
(7)
$$\mid U\prime ({x}_{1},{y}_{1})\mid \propto \mid s({x}_{1}-X,{y}_{1}-Y)\mid \mid {J}_{0}\left[(4\pi \u2044\lambda )h({x}_{1}-X,{y}_{1}-Y)\right]\mid .$$
(8)
$$\left[\left(\frac{{\partial}^{2}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial}^{2}}{\partial {\phi}^{2}}\right)-\frac{1}{{v}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}\right]h(r,\phi ,t)=0$$
(9)
$${h}_{\mathit{mn}}(r,\phi ,t)={A}_{\mathit{mn}}{J}_{m}({x}_{\mathit{mn}}r\u2044a)\mathrm{cos}\left(m\phi -{\phi}_{0}\right)\mathrm{cos}\left({\omega}_{\mathit{mn}}t-\delta \right),$$
(10)
$$h(r,\phi ,t)=\sum _{m=0}^{\infty}\sum _{m=1}^{\infty}{h}_{\mathit{mn}}(r,\phi ,t).$$
(11)
$$\frac{{d}^{2}T\left(t\right)}{{\mathit{dt}}^{2}}+\gamma \frac{\mathit{dT}\left(t\right)}{\mathit{dt}}+{\omega}_{\mathit{mn}}^{2}T\left(t\right)=F\left(t\right),$$
(12)
$$\mid {T}_{p}\left(\omega \right)\mid ={P}_{0}\u2044{\left[{\left({\omega}_{\mathit{mn}}^{2}-{\omega}^{2}\right)}^{2}+{\gamma}^{2}{\omega}^{2}\right]}^{1\u20442},$$
(13)
$$h(r,\phi ,\omega )={P}_{0}\sum _{m=0}^{\infty}\sum _{n=1}^{\infty}\{1\u2044{\left[{\left({\omega}_{\mathit{mn}}^{2}-{\omega}^{2}\right)}^{2}+{\gamma}^{2}{\omega}^{2}\right]}^{1\u20442}\}{J}_{m}({x}_{\mathit{mn}}r\u2044a)\mathrm{cos}m\phi ,$$