Question

A square membrane of side \(l\) is distorted into the shape $$f(x, y)=x y(l-x)(l-y)$$ and released. Express its shape at subsequent times as an infinite series. Hint: Use a double Fourier series as in Problem \(5.9 .\)

Short answer

Use a double Fourier series to represent the shape at any time: \[ u(x, y, t) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) \cos(\omega_{mn}t) \].

Step-by-step solution

Understand the Initial Shape Function

Given the initial shape of the membrane as a function $$f(x, y) = xy(l-x)(l-y)$$, the task is to express this shape in terms of a double Fourier series.

Double Fourier Series Representation

The double Fourier series representation for a function over a square region can be written as: \[ f(x, y) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) \].

Calculate Fourier Coefficients

Determine the coefficients \(A_{mn}\) using the formula: \[ A_{mn} = \frac{4}{l^2} \int_0^l \int_0^l f(x, y) \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) dx dy \].

Plug in the Initial Shape Function

Substitute the given initial shape function \(f(x, y) = xy(l-x)(l-y)\) into the coefficient formula and simplify.

Perform the Integration

Compute the integrals to find the values of the coefficients \(A_{mn}\). This involves solving the double integral \[ A_{mn} = \frac{4}{l^2} \int_0^l x(l-x) \sin\left(\frac{m \pi x}{l}\right) dx \int_0^l y(l-y) \sin\left(\frac{n \pi y}{l}\right) dy \].

Construct the Series Solution

Once the coefficients \(A_{mn}\) are determined, the shape of the membrane at any time \(t\) is given by the series \[ u(x, y, t) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) \cos(\omega_{mn}t) \], where \(\omega_{mn} = \sqrt{\frac{m^2 + n^2}{l^2}}\).

Key concepts

square membrane distortion

In mathematical physics, the study of square membrane distortion focuses on how a two-dimensional membrane (like a drumhead) behaves when it is displaced from its equilibrium position. For a square membrane with side length \(l\), the initial distortion or shape can be expressed by a function of the coordinates \(x\) and \(y\). In this specific exercise, the initial shape function is given by \(f(x, y) = xy(l-x)(l-y)\). This function describes how each point on the membrane is displaced.

We must convert this initial shape into a form that allows us to track the future distortions over time. The mathematical tool we use for this is the double Fourier series. This approach breaks down the complex shape into a sum of simpler sine and cosine functions, making it much easier to analyze the behavior of the membrane across time.

Fourier coefficients

Fourier coefficients are critical components of the Fourier series representation. They quantify the contribution of each sine and cosine term to the overall shape of the function. To find these coefficients for our problem, we use the given formula: \[ A_{mn} = \frac{4}{l^2} \int_0^l \int_0^l f(x, y) \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) dx dy \].

By plugging in the initial shape function \(f(x, y) = xy(l-x)(l-y)\) into this formula, we essentially measure how each sine and cosine term fits to the initial shape. This involves performing double integration. The resulting coefficients \(A_{mn}\) are unique to our specific initial shape distribution and serve as the 'weights' in the Fourier series.

• \(A_{mn}\): The weight of each frequency component
• \(m, n\): Indices representing the specific sine/cosine wave components
• \Integral Limits (0 to \(l\)): Cover the entire extent of the initial membrane shape

Fourier series representation

A Fourier series is a way to represent a function as the sum of simple sine and cosine waves. For a function defined over a square region, we use a double Fourier series, which involves two sets of sine functions \( \sin\left(\frac{m \pi x}{l}\right) \) and \( \sin\left(\frac{n \pi y}{l}\right) \). This representation helps to break down the initial complex shape into a series of simpler oscillatory components.

The double Fourier series representation of a function is given by:
\[ f(x, y) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) \].

Here, \(A_{mn}\) are the Fourier coefficients we calculated previously. By summing up these weighted sine terms, we can reconstruct the initial shape function. This makes it a very powerful tool for understanding complex patterns and predicting future behavior.

time-dependent shape function

The shape function of the membrane changes over time due to oscillations. To describe this dynamic behavior, we extend our Fourier series representation to include a time-dependent component. This is done by adding a cosine function that accounts for temporal changes:
\[ u(x, y, t) = \sum_{m=1}^\infty \sum_{n=1}^\infty A_{mn} \sin\left(\frac{m \pi x}{l}\right) \sin\left(\frac{n \pi y}{l}\right) \cos(\omega_{mn} t) \].

Here, \( \omega_{mn} = \sqrt{\frac{m^2 + n^2}{l^2}} \) is the angular frequency of the oscillation. This equation shows how each sine wave component of the initial shape evolves over time. The cosine terms \( \cos(\omega_{mn} t) \) describe the time evolution of each mode of the membrane's vibration.

• \(u(x, y, t)\): Shape of the membrane at time \(t\)
• \(\omega_{mn}\): Angular frequency specific to each mode
• \Cosine term: Represents the temporal evolution

double integral calculation

The calculation of the Fourier coefficients \(A_{mn}\) involves evaluating a double integral. This process may seem complex, but it is crucial for determining the exact contribution of each sine wave component.

To compute the coefficients, we start with the integral formula:
\[ A_{mn} = \frac{4}{l^2} \int_0^l x(l-x) \sin\left(\frac{m \pi x}{l}\right) dx \int_0^l y(l-y) \sin\left(\frac{n \pi y}{l}\right) dy \].

In this formula, the double integral is broken down into two single integrals, one over \(x\) and one over \(y\).

• First Integral: \( \int_0^l x(l-x) \sin\left(\frac{m \pi x}{l}\right) dx \)
• Second Integral: \( \int_0^l y(l-y) \sin\left(\frac{n \pi y}{l}\right) dy \)

Solving these integrals requires integration by parts or using standard integral tables. Once computed, these integrals yield the numerical values of \(A_{mn}\), which we then plug back into the Fourier series formula to get the full representation.