Question

An infinite stretched membrane has areal density \(\rho\) and tension \(T\). Initially it has a given displacement \(f(\mathbf{r})\) and zero initial velocity everywhere. Find the subsequent motion.

Short answer

The subsequent motion of the membrane can be found by solving \( u(\textbf{r}, t) = \text{sum of wave solutions with } cos \) and \( sin \text{ terms.}

Step-by-step solution

- Understand the Wave Equation

The equation governing the motion of a membrane is the 2D wave equation given by \[ \frac{\frac{abla^2 u}{abla t^2}}{\frac{abla^2 u}{abla x^2} + \frac{abla^2 u}{abla y^2}} = c^2 \] where u is the displacement and c is the wave speed. Here, \[c^2 = \frac{T}{\rho}\], where T is the tension and \( \rho \) is the areal density.

- Initial Conditions

Given initial conditions are: 1. The initial displacement is described by \( u(\textbf{r}, 0) = f(\textbf{r}) \)2. The initial velocity is zero, \( \frac{abla u}{abla t} (\textbf{r}, 0) = 0 \).

- Solving the Wave Equation

To solve the wave equation, we apply the method of separation of variables. Let \[ u(\textbf{r}, t) = F(\textbf{r}) \times G(t) \] Substitute into the wave equation to get: \[ F(\textbf{r}) \times \frac{abla^2 G}{abla t^2} = c^2 G(t) \times abla^2 F \].

- Solve Spatial and Temporal Parts Separately

Rearrange and equate both sides to a separation constant, \[ \frac{1}{c^2} \frac{\frac{abla^2 G(t)}{abla t^2}}{G(t)} = \frac{abla^2 F(\textbf{r})}{F(\textbf{r})} = -k \] This results in two ordinary differential equations: \( \frac{\frac{abla^2 G(t)}{abla t^2}}{c^2 G(t)} = -k \) and \( \frac{abla^2 F(\textbf{r})}{F(\textbf{r})} = -k \).

- Solve the Temporal Part

The temporal part is given by \[ \frac{abla^2 G(t)}{abla t^2} + k c^2 G(t) = 0 \] This is a simple harmonic oscillator equation with solutions: \[ G(t) = A \text{cos}(\theta t) + B \text{sin}(\theta t) \] where \( \theta = \frac{abla^2 k c^2}{abla t^2} \).

- Solve the Spatial Part

The spatial part is \[ abla^2 M(\textbf{r}) + k M(\textbf{r}) = 0 \] which is a Helmholtz equation. This has general solutions that can be written typically in the form of Bessel functions or plane waves.

- Combine Solutions

The overall solution combines the spatial and temporal parts: \[ u(\textbf{r}, t) = \textbf{sum of wave equations involving cos and sin terms} \] Applying initial conditions, determine the specific form of the solution.

- Incorporate Initial Conditions

Use the given initial displacement \( f(\textbf{r}) \) and zero initial velocity to solve for coefficients in the general solution of the wave equation.

- Final Expression for Motion

After incorporating initial conditions, the resulting solution will provide the displacement of the membrane at any future time.

Key concepts

Wave Equation

The 2D wave equation is fundamental in understanding how waves propagate over surfaces such as membranes. In this context, it governs the motion of a membrane with density \( \rho \) and tension \( T \). The wave equation is given by: \[ \frac{\partial^2 u}{\partial t^2} = c^2 ( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} ) \] where \( u \) is the displacement of the membrane and \( c \) is the wave speed. The wave speed \( c \) is a function of the tension and density, described by \[ c^2 = \frac{T}{\rho} \]. This equation tells us that the acceleration of the membrane's displacement depends on the curvature of the displacement in the x and y directions.

Initial Conditions

Initial conditions are crucial because they set the starting point for solving the wave equation. For this membrane problem, we have two initial conditions:

• The initial displacement: \( u(\textbf{r}, 0) = f(\textbf{r}) \). This means the shape of the membrane at time \( t = 0 \) is given by the function \( f(\textbf{r}) \).
• The initial velocity is zero: \( \frac{\partial u}{\partial t} (\textbf{r}, 0) = 0 \).. This means the membrane is not moving at the beginning.

These conditions help us tailor the general solution of the wave equation to match the given physical scenario.

Separation of Variables

To solve the wave equation, we use a method called separation of variables. This technique involves assuming that the solution can be written as a product of two functions, each depending on a single variable: \[ u(\textbf{r}, t) = F(\textbf{r}) \cdot G(t) \] We then substitute this form into the wave equation, breaking it into two separate ordinary differential equations (ODEs): one for the spatial part \( F(\textbf{r}) \) and one for the temporal part \( G(t) \). By equating these to a constant, we simplify the problem to solve each part independently.

Helmholtz Equation

The spatial part of the separated wave equation leads us to the Helmholtz equation: \[ abla^2 F(\textbf{r}) + k F(\textbf{r}) = 0 \] This is a well-known equation in physics and engineering, often arising in problems involving wave propagation. Its solutions typically include Bessel functions or plane waves, depending on the boundary conditions and geometry of the problem. This equation defines the spatial characteristics of the wave on the membrane.

Harmonic Oscillator

The temporal part of the wave equation simplifies to a harmonic oscillator ODE: \[ \frac{d^2 G(t)}{d t^2} + k c^2 G(t) = 0 \] This equation describes simple harmonic motion, with solutions of the form: \[ G(t) = A \cos(\theta t) + B \sin(\theta t) \] where \( \theta = \sqrt{k} c \). These solutions represent periodic motions, indicating that the displacement of the membrane oscillates in time. The constants \( A \) and \( B \) are determined by the initial conditions, ensuring the final solution fits the physical situation.