## Final Exam Preparation APMA 0340

APMA 0340

#### Item 1: Guitar string vibration

A guitar string is an example of an incompressible elastic string that is usually stretched between two fixed points. Let u(x, t) denote the vertical displacement experienced by the string at point x at time t. If damping effects, such as air resistance, are neglected, and if the amplitude of the vertical motion is small compared to the string length, then u(x, t) satisfies the partial differential equation, known as the wave equation,

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \qquad \mbox{or for short} \qquad u_{tt} = c^2 u_{xx} .$

Music often involves strings that are fixed at both ends and are elastic and vibrate. The methodsin which these strings are made to vibrate, and therefore emit sound, vary across differentinstruments. In the case of a guitar, specificallyin fingerpicking, a string is plucked by pulling it upward and releasing it from rest; therefore creatinga triangular impulse shape. String vibration in strumming patterns works in a similar way, however, the strings can be forced either upward or downward by the guitar pick. Nonetheless, the initial condition consequently defines some vertical displacement, and zero vertical velocity,at t = 0. The point of which the string is plucked is usually about one third of the way down across the string.

##### Item 1-1

Use the separation of variables method. Its essence consists of considering an auxiliary problem that is obtained by removing its nonhomogeneous part---the initial conditions. Then seek for nontrivial solutions in the form u(x, t) = X(x) T(t), which will lead to the Sturm--Liouville problem for variable X(x). Upon solving this problem, represent the required solutions as an infinite sum over eigenfunctions:

$u(x,t) = \sum_{n\ge 1} X_n (x)\, T_n (t) .$
##### Item 1-2

To plot the solution, follow the code presented in Example 2 of the course website

#### Item 2: Piano string vibration

A piano string is an example of an incompressible elastic string that is stretched between two fixed points. Let u(x, t) denote the vertical displacement experienced by the string at point x at time t. If damping effects, such as air resistance, are neglected, and if the amplitude of the motion is small compared to the string length, then u(x, t) satisfies the partial differential equation, known as the wave equation

$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \qquad \mbox{or for short} \qquad u_{tt} = c^2 u_{xx} .$

Contrary to the guitar string case, a piano involves no initial displacement. Rather, an initial velocity is created by a felt-covered hammer, which strikes the piano string when the pianist presses on a key. This velocity can be small or large,depending on how hard the key is pressed.A harder strike will cause a louder sound, and vice-versa. These piano strings can be arranged in two ways: vertically or horizontally---for upright or grand pianos, respectively. Therefore, a vertical or horizontal velocity at t = 0 is created by the hammer system, and subsequently creates sound.

##### Item 2-1

Use the separation of variables method. Its essence consists of considering an auxiliary problem that is obtained by removing its nonhomogeneous part---the initial conditions. Then seek for nontrivial solutions in the form u(x, t) = X(x) T(t), which will lead to the Sturm--Liouville problem for variable X(x). Upon solving this problem, represent the required solutions as an infinite sum over eigenfunctions:

$u(x,t) = \sum_{n\ge 1} X_n (x)\, T_n (t) .$
##### Item 2-2

To plot the solution, follow the code presented in Example 3 of the course website

#### Item 3: Heat transfer problem

Let us consider a heat conduction problem for a straight bar of uniform cross section and homogeneous material. Let the x-axis be chosen to be along the axis of the bar, and let x = 0 and x = ℓ denote the ends of the bar. Suppose further that the sides of the bar are perfectly insulated so that no heat passes through them. We also assume that the cross-sectional dimensions are so small that the temperature u can be considered constant on any given cross section. Then u is a function of the axial coordinate x and the time t.

The variation of temperature in the bar is governed by the partial differential equation, called the heat equation or diffusion equation:

$\frac{\partial u}{\partial t} = \alpha\, \frac{\partial^2 u}{\partial x^2} \qquad \mbox{or for short} \qquad u_{t} = \alpha\, u_{xx} .$
Here α is a constant known as the thermal diffusivity, and it depends on material in the bar. It is assumed that the initial temperature distribution in the bar is known
$u(x,0) = f(x) , \qquad 0 < x < \ell ,$
where f is a smooth function. Finally, we assume that the ends of the bar are held at fixed temperatures to be zero.
$u(0,t) = 0 , \qquad u(\ell , t) = 0.$
The fundamental problem of heat conduction is to find u(x, t) that satisfies the diffusion differential equation for 0 < x < ℓ and for t > 0, the initial condition u(x, 0) = f(x) when t = 0, and the boundary conditions at x = 0 and x = ℓ.

##### Item 3-1

The initial boundary value problem for the one-dimensional heat equation with the separation of variables method. See section on boundary value problems for heat equation of the course website

##### Item 3-2

To plot the solution, follow the code presented in Example 1 of the course website

#### Item 4: Neumann problem for Laplace's equation

See Example 1B of the course website

#### Item 5: Dirichlet problem

See Example 1B of the course website

#### Item 6: Outer Dirichlet problem for circle

See Examples 3 and 4 of the course website

#### Item 7: Inner Neumann problem for circle

See Example 5 and 6 of the course website

#### Item 8: Fourier series

The Fourier series for a piecewise continuous function is discussed in section of the course website

##### Item 8-1
The Gibbs phenomena is discussed in section of the course website

##### Item 8-2
To eliminate overshoot and undershoot, it is convenient to use the Cesàro summation; see