Question

The Sturm-Liouville equation can be extended to two independent variables, \(x\) and \(z\), with little modification. In equation \((22.22) y^{\prime 2}\) is replaced by \((\nabla y)^{2}\) and the integrals of the various functions of \(y(x, z)\) become two-dimensional, i.e. the infinitesimal is \(d x d z\). The vibrations of a trampoline 4 units long and 1 unit wide satisfy the equation $$ \nabla^{2} y+k^{2} y=0 $$ By taking the simplest possible permissible polynomial as a trial function, show that the lowest mode of vibration has \(k^{2} \leq 10.63\) and, by direct solution, that the actual value is \(10.49\)

Short answer

Lowest mode of vibration has \( k^2 \leq 10.63 \) and actual value is 10.49.

Step-by-step solution

Formulation of the Trial Function

Assume the simplest permissible polynomial trial function for the field variable. A suitable choice is a separable function of the form: \[ y(x,z) = X(x) Z(z) \]

Applying the Boundary Conditions

For the given problem, the trampoline's dimensions impose boundary conditions. Let's use: \[ y(0, z) = y(4, z) = 0 \] \[ y(x, 0) = y(x, 1) = 0 \] This suggests: \[ X(0) = X(4) = 0 \] \[ Z(0) = Z(1) = 0 \]

Choosing Polynomial Forms

A simple polynomial form satisfying the boundary conditions would be: \[ X(x) = C \, \sin\left(\frac{n \pi x}{4}\right) \] \[ Z(z) = D \, \sin\left(m \pi z\right) \] Hence, the trial function is: \[ y(x,z) = C D \, \sin\left(\frac{n \pi x}{4}\right) \sin(m \pi z) \]

Substituting into the Helmholtz Equation

Substitute the trial function into the Helmholtz equation \( abla^2 y + k^2 y = 0 \). Compute the Laplacian: \[ abla^2 y = \frac{\partial^2 y}{\partial x^2} + \frac{\partial^2 y}{\partial z^2} \] Given \( y(x,z) = X(x) Z(z) \): \[ \frac{\partial^2 y}{\partial x^2} = - \left(\frac{n \pi}{4}\right)^2 y \] \[ \frac{\partial^2 y}{\partial z^2} = - (m \pi)^2 y \] Therefore: \[ abla^2 y = - \left(\left(\frac{n \pi}{4}\right)^2 + (m \pi)^2\right) y \]

Determining the Eigenvalue Equation

Combine the results from step 4: \[ -\left(\left(\frac{n \pi}{4}\right)^2 + (m \pi)^2 \right) y + k^2 y = 0 \] This leads to: \[ k^2 = \left(\frac{n \pi }{4}\right)^2 + (m \pi)^2 \]

Evaluating the Smallest Eigenvalue

For the lowest mode of vibration, use \( n = 1 \) and \( m = 1 \): \[ k^2 = \left(\frac{1 \pi }{4}\right)^2 + (1 \pi)^2 = \frac{\pi^2}{16} + \pi^2 = \frac{17 \pi^2}{16} \approx 10.63 \]

Verifying the Actual Value

By direct solution of the boundary value problem, the actual value for \( k^2 \) is computed as 10.49 for the lowest mode.

Key concepts

boundary conditions

In solving differential equations, boundary conditions are crucial. They specify the solution behavior at the boundaries of the problem domain.
Boundary conditions can be of different types such as Dirichlet, Neumann, or mixed.
In our exercise, we use Dirichlet boundary conditions which set the value of the function to zero at the boundaries. Specifically, for the trampoline, these conditions are:

• \( y(0, z) = y(4, z) = 0 \)
• \( y(x, 0) = y(x, 1) = 0 \)

This means the vibration at the edges of the trampoline is zero. These conditions help in determining the eigenfunctions that satisfy both the differential equation and the boundary constraints.
They simplify the problem by reducing the possible forms of the solution function.

Helmholtz equation

The Helmholtz equation is a partial differential equation commonly used in various fields including acoustics, electromagnetics, and vibration analysis. It has the form: \[ abla^2 y + k^2 y = 0 \] Here, \( abla^2 \) is the Laplacian operator, which measures the divergence of the gradient of a function.
In our exercise, it describes the vibrations of a trampoline. The parameter \( k^2 \) is related to the frequency of the vibration modes.
By substituting the trial function into the Helmholtz equation, we can solve for the eigenvalue \( k^2 \), which gives us information about the system's natural frequencies.

eigenvalue problem

An eigenvalue problem seeks to find values (eigenvalues) for which a given differential equation has non-trivial solutions (eigenfunctions). In our context: \[ abla^2 y + k^2 y = 0 \] becomes: \[ k^2 = \frac{n^2 \pi^2}{16} + m^2 \pi^2 \] when we apply a permissible trial function. The smallest eigenvalue (the fundamental mode) is of particular interest because it corresponds to the lowest natural frequency of the system.
This eigenvalue tells us how the trampoline will naturally vibrate under no external forces. Solving for these eigenvalues and their corresponding eigenfunctions helps us understand the dynamics of the system.

vibration modes

Vibration modes refer to the characteristic patterns with which a system vibrates. Each mode corresponds to a specific eigenvalue and has a distinct shape. For our trampoline: \[ y(x,z) = C D \sin \left(\frac{n \pi x}{4} \right) \sin (m \pi z) \] describes the vibration modes, where \( n \) and \( m \) are integers determining the number of nodes in each direction.
The lowest mode corresponds to \( n = 1 \) and \( m = 1 \), indicating the simplest form of vibration with only one node along each direction.
Understanding these modes is crucial for various applications, from engineering structures to musical instruments, as each mode can resonate at a specific frequency.

Laplacian operator

The Laplacian operator \( abla^2 \) is a second-order differential operator in the n-dimensional Euclidean space, representing the sum of all unmixed second partial derivatives. In 2D, it is given by: \[ abla^2 y = \frac{\partial^2 y}{\partial x^2} + \frac{\partial^2 y}{\partial z^2} \] In our problem, applying the Laplacian to the trial function:

• \( \frac{\partial^2 y}{\partial x^2} = - \left(\frac{n \pi}{4} \right)^2 y \)
• \( \frac{\partial^2 y}{\partial z^2} = - (m \pi)^2 y \)

produces: \[ abla^2 y = - \left( \left( \frac{n \pi}{4} \right)^2 + (m \pi)^2 \right)y \] By analyzing the Laplacian operator, we can deduce how the differentials impact the overall behavior of the function, leading us to accurately determine the eigenvalues and subsequently the vibration modes.