21. Waves on Realistic Strings

The lecture begins with an introduction to the topic of waves, specifically focusing on partial differential equations related to waves on a string. The speaker outlines the plan to discuss realistic waves, including the effects of gravity and friction, and mentions the numerical solution of wave equations. The concept of a wave as a signal traveling through space is explained, and the standard wave equation is introduced. The lecture also hints at exploring quantum wave packets and electromagnetic waves, as well as computational fluid dynamics involving water waves and tides. A visual example of a wave on a chain under the influence of gravity is presented, illustrating the damping effect of friction over time.

This paragraph delves into the specifics of setting up a model for waves on a string. It describes the physical setup, including a string of length 'l' with fixed ends, and the variables involved such as the density (rho), tension, and the displacement 'y' from the equilibrium position. The assumptions made for simplicity, such as constant density and tension, and small displacements, are outlined. The paragraph also explains the concept of strain and how it relates to the slope of the string, and discusses the restoring force due to the tension in the string that acts on a small element of the string.

The paragraph focuses on the derivation of the wave equation for one-dimensional waves. It starts with the premise that the restoring force is the difference in the y-component of the tension along the string. The wave speed 'c' is derived as the square root of the tension divided by the density. The paragraph also clarifies the difference between the velocity of the wave and the velocity of the string. The derivation of the wave equation using the formula f=ma is explained, leading to a second-order partial differential equation that relates the second spatial derivative of the wave function to the second temporal derivative.

This section discusses the conditions necessary to solve the wave equation. It emphasizes that two pieces of information are required: initial conditions and boundary conditions. The initial condition described is a triangular pluck of the string, and the second condition is that the string is released from rest, implying an initial velocity of zero. The importance of these conditions in solving the second-order wave equation is highlighted, and the paragraph also touches on the implications of the boundary conditions set by the fixed ends of the string.

The paragraph introduces the concept of normal modes as the analytic solution to the wave equation. It describes how normal modes appear as standing waves and can be visualized using an applet. The explanation includes how to derive normal modes by assuming a separable solution and applying boundary conditions to determine the allowable wave numbers 'k'. The paragraph also mentions the relationship between the wave vector 'k', angular frequency 'omega', and the wave speed 'c', and how the initial and boundary conditions lead to a series solution representing the sum of normal modes.

This section outlines the numerical approach to solving the wave equation. It describes the process of discretizing the derivatives in the wave equation using a central difference approximation, which transforms the continuous equation into a form suitable for a discrete grid in space and time. The paragraph explains the leapfrog method for time-stepping, which uses known values at the present and past to predict the future state of the wave. The stability of the numerical solution and the importance of choosing appropriate time and space step sizes are also discussed.

The paragraph discusses the stability conditions for numerical solutions of wave equations, emphasizing the importance of selecting appropriate values for delta x and delta t to avoid numerical instability. It introduces the concept of algorithmic velocity (c prime) and its relation to the physical wave velocity (c). The stability condition derived is that c prime must be less than c for the solution to remain stable. The paragraph also explains the leapfrog algorithm in more detail, highlighting its efficiency due to the minimal storage requirements and the need for small time steps for high precision.

This section provides a deeper analysis of the stability conditions for numerical solutions of the wave equation. It introduces the concept of the von Neumann stability condition, which is crucial for ensuring that the numerical solution does not diverge. The paragraph explains the process of substituting a discrete plane wave solution into the wave equation to derive the stability criterion. It also presents exercises for the audience, encouraging them to explore the stability condition computationally and to understand the limitations of the numerical method, such as the inability to use relaxation techniques due to the lack of boundary conditions.

The final paragraph focuses on the visualization and analysis of numerical solutions for wave equations. It suggests using techniques such as animation and surface plots to observe the behavior of waves over time and space. The paragraph encourages experimenting with different space and time steps to verify the stability conditions and to ensure the accuracy of the numerical method. It also prompts the audience to compare their numerical results with the analytic solution for normal modes and to estimate the wave speed visually, providing a comprehensive understanding of wave behavior.