Engineering Differential Equations: Theory and Applications, Springer 2010

This page contains supplementary material for the book, Engineering Differential Equations: Theory and Applications, by Bill Goodwine. 2010, Springer.

Movies

Chapter 11 considers solutions to partial differential equations.

The One-Dimensional Wave Equation

Section 11.1 considers the one-dimensional wave equation.

Some movies:

• String of length 3 plucked at L=1 (modeling a guitar):
  • The first (fundamental) mode of the plucked string
  • The first five modes of the plucked string plotted separately
  • The sum of the first five modes of the plucked string
  • The sum of the first fifty modes of the plucked string
• String of length 3 that is impacted near L=1 (modeling a piano):
  • The sum of the first ten modes of the impacted string
  • The sum of the first fifty modes of the impacted string

The One-Dimensional Heat Conduction Equation

Section 11.3 considers the one-dimensional heat conduction equation.

Some movies:

• The solution to the heat equation with inhomogeneous boundary conditions
• The solution to the heat equation with an insulated end

The Two-Dimensional Heat Equation

Movie:

• Solution to the heat conduction equation in two dimensions

Vibrating Membranes

Partial differential equations describing vibrating membranes in rectangular and polar coordinates are considered in Section 11.5.

Section 11.5.1 presents the two-dimensional wave equation in rectangular coordinates.

Some movies:

• 1-1 mode for rectangular drum
• 2-1 mode for rectangular drum
• 2-2 mode for rectangular drum
• Rectangular drum impacted near a corner

Sections 11.5.2-11.5.4 presents the wave equation in polar coordinates. The prototypical example would be a vibrating drum head.

Some movies:

• 1-0 mode for a circular drum
• 1-1 mode for a circular drum
• 1-2 mode for a circular drum
• 1-3 mode for a circular drum
• 2-0 mode for a circular drum
• 2-1 mode for a circular drum
• 4-3 mode for a circular drum
• Circular drum that is impacted by a drum stick
• Another drum impact movie

Errata

Chapter 9

• p. 244, line -11: the first component of the vector on the far right should contain t, not tau.
• p. 348, line 4: the numerator for both expressions of G1 should be 5.
• p. 348, line 6: the numerator for both expressions of G1 should be 9.
• p. 349, caption to Figure 9.16: the numerator for G1 should be 4 in the numerator for G2 should be 9.