Laser Reflection Pattern from a Vibrating Circular Membrane

Physical System

Consider a circular membrane of radius $$R$$ vibrating under tension. A laser reflects off a point $$(r_0, \theta_0)$$ on the membrane and projects onto a screen at distance $$L$$. We derive the pattern traced by the laser during vibration.

Governance Equation

The transverse displacement $$z(r,\theta,t)$$ satisfies the wave equation in polar coordinates: $$ \nabla^2 z = \frac{1}{v^2} \frac{\partial^2 z}{\partial t^2} $$ For time-harmonic solutions, this reduces to the Helmholtz equation: $$ \left(\nabla^2 + k^2\right) z = 0 $$

General Solution

The complete solution is a linear combination of normal modes: $$ z(r,\theta,t) = \sum_{m=0}^\infty \sum_{n=1}^\infty A_{mn} J_m\left(\frac{\alpha_{mn}r}{R}\right) \cos(m\theta + \phi_{mn}) \cos(\omega_{mn}t + \psi_{mn}) $$

Where:

• $$J_m$$: Bessel function of first kind, order $$m$$
• $$\alpha_{mn}$$: $$n$$-th positive root of $$J_m$$
• $$\omega_{mn} = c\frac{\alpha_{mn}}{R}$$: Mode frequency
• $$A_{mn}$$: Mode amplitude
• $$\phi_{mn}, \psi_{mn}$$: Phase constants

Reflection Geometry

The surface normal vector $$\mathbf{n}$$ determines reflection direction. For small displacements ($$|\nabla z| \ll 1$$): $$ \mathbf{n} \approx \left(-\frac{\partial z}{\partial x}, -\frac{\partial z}{\partial y}, 1\right) $$

Incident Laser

Assume vertical incidence (along $$\hat{z}$$): $$ \mathbf{v}_i = (0,0,1) $$

Reflected Direction

Using the law of reflection: $$ \mathbf{v}_r = \mathbf{v}_i - 2\frac{\mathbf{v}_i \cdot \mathbf{n}}{|\mathbf{n}|^2}\mathbf{n} \approx (-2\frac{\partial z}{\partial x}, -2\frac{\partial z}{\partial y}, 1) $$

Screen Projection

At screen distance $$L$$, coordinates become: $$ x_{\text{screen}} = -2L \frac{\partial z}{\partial x}\bigg|{\substack{r=r_0\ \theta=\theta_0}},\quad y{\text{screen}} = -2L \frac{\partial z}{\partial y}\bigg|_{\substack{r=r_0\ \theta=\theta_0}} $$

Gradient Calculation

We need $$\partial z/\partial x$$ and $$\partial z/\partial y$$ at $$(r_0, \theta_0)$$.

Polar Gradient Components

First compute polar derivatives:

Radial derivative: $$ \begin{aligned} \frac{\partial z}{\partial r} &= \sum_{m,n} A_{mn} \frac{\alpha_{mn}}{R} J_m'\left(\frac{\alpha_{mn}r_0}{R}\right) \cos(m\theta_0 + \phi_{mn}) \cos(\omega_{mn}t + \psi_{mn}) \ J_m'(x) &= \frac{1}{2}[J_{m-1}(x) - J_{m+1}(x)] \quad \text{(Bessel derivative identity)} \end{aligned} $$

Angular derivative: $$ \frac{\partial z}{\partial \theta} = -\sum_{m,n} A_{mn} m J_m\left(\frac{\alpha_{mn}r_0}{R}\right) \sin(m\theta_0 + \phi_{mn}) \cos(\omega_{mn}t + \psi_{mn}) $$

Cartesian Conversion

Transform to Cartesian coordinates: $$ \begin{aligned} \frac{\partial z}{\partial x} &= \frac{\partial z}{\partial r}\cos\theta_0 - \frac{1}{r_0}\frac{\partial z}{\partial \theta}\sin\theta_0 \ \frac{\partial z}{\partial y} &= \frac{\partial z}{\partial r}\sin\theta_0 + \frac{1}{r_0}\frac{\partial z}{\partial \theta}\cos\theta_0 \end{aligned} $$

Final Parametric Equations

Substitute into screen coordinates:

X-coordinate: $$ \begin{aligned} x_{\text{screen}}(t) &= -2L \sum_{m,n} A_{mn} \cos(\omega_{mn}t + \psi_{mn}) \Biggl[ \frac{\alpha_{mn}}{R} J_m'\left(\frac{\alpha_{mn}r_0}{R}\right) \cos\theta_0 \cos(m\theta_0 + \phi_{mn}) \ &\quad + \frac{m}{r_0} J_m\left(\frac{\alpha_{mn}r_0}{R}\right) \sin\theta_0 \sin(m\theta_0 + \phi_{mn}) \Biggr] \end{aligned} $$

Y-coordinate: $$ \begin{aligned} y_{\text{screen}}(t) &= -2L \sum_{m,n} A_{mn} \cos(\omega_{mn}t + \psi_{mn}) \Biggl[ \frac{\alpha_{mn}}{R} J_m'\left(\frac{\alpha_{mn}r_0}{R}\right) \sin\theta_0 \cos(m\theta_0 + \phi_{mn}) \ &\quad - \frac{m}{r_0} J_m\left(\frac{\alpha_{mn}r_0}{R}\right) \cos\theta_0 \sin(m\theta_0 + \phi_{mn}) \Biggr] \end{aligned} $$

We introduce the following constants to group the constant terms:

$$ B_{mn}^x = \frac{\alpha_{mn}}{R} J_m'!\left(\frac{\alpha_{mn}r_0}{R}\right) \cos\theta_0,\quad C_{mn}^x = \frac{m}{r_0} J_m!\left(\frac{\alpha_{mn}r_0}{R}\right) \sin\theta_0, $$ $$ B_{mn}^y = \frac{\alpha_{mn}}{R} J_m'!\left(\frac{\alpha_{mn}r_0}{R}\right) \sin\theta_0,\quad C_{mn}^y = \frac{m}{r_0} J_m!\left(\frac{\alpha_{mn}r_0}{R}\right) \cos\theta_0. $$

With these definitions, the equations can be rewritten as:

$$ \begin{aligned} x_{\text{screen}}(t) &= -2L \sum_{m,n} A_{mn} \cos(\omega_{mn}t + \psi_{mn}) \Bigl[ B_{mn}^x \cos(m\theta_0 + \phi_{mn}) + C_{mn}^x \sin(m\theta_0 + \phi_{mn}) \Bigr], \\ y_{\text{screen}}(t) &= -2L \sum_{m,n} A_{mn} \cos(\omega_{mn}t + \psi_{mn}) \Bigl[ B_{mn}^y \cos(m\theta_0 + \phi_{mn}) - C_{mn}^y \sin(m\theta_0 + \phi_{mn}) \Bigr]. \end{aligned} $$

We define the following constants that group all time-independent terms:

$$ D_{mn}^x = A_{mn} \bigl[ B_{mn}^x \cos(m\theta_0 + \phi_{mn}) + C_{mn}^x \sin(m\theta_0 + \phi_{mn}) \bigr], $$ $$ D_{mn}^y = A_{mn} \bigl[ B_{mn}^y \cos(m\theta_0 + \phi_{mn}) - C_{mn}^y \sin(m\theta_0 + \phi_{mn}) \bigr]. $$

Substituting these constants into the original equations:

$$ x_{\text{screen}}(t) = -2L \sum_{m,n} D_{mn}^x \cos(\omega_{mn}t + \psi_{mn}), $$

$$ y_{\text{screen}}(t) = -2L \sum_{m,n} D_{mn}^y \cos(\omega_{mn}t + \psi_{mn}). $$

These equations show that the trajectory of the laser point on the screen is a combination of oscillations in $$x$$ and $$y$$, where the coefficients $$D_{mn}^x$$ and $$D_{mn}^y$$ encapsulate all spatial and modal dependencies, leaving the temporal evolution governed solely by the terms $$\cos(\omega_{mn}t + \psi_{mn})$$.

Key Observations

1. Mode Coupling: Each mode contributes terms proportional to $$\omega_{mn}t + \psi_{mn}$$
2. Time Dependence: $$\cos(\omega_{mn}t + \psi_{mn})$$ creates time modulation
3. Frequency Mixing: For multiple modes, cross terms generate: $$ \cos(\omega_{pq}t)\cos(\omega_{rs}t) \propto \cos[(\omega_{pq} \pm \omega_{rs})t] $$
4. Lissajous Figures: When $$\omega_{pq}/\omega_{rs}$$ is rational, curves close; irrational ratios produce non-repeating patterns

Special Case: Single Mode

For a single $$(m,n)$$ mode: $$ \begin{cases} x_{\text{screen}}(t) = C_x \cos(\omega t + \psi) \ y_{\text{screen}}(t) = C_y \cos(\omega t + \psi) \end{cases} $$ This traces an ellipse. Adding modes breaks this symmetry.