Traveling Waves on a String: From Normal Modes to Progressive Wave Solutions

Overview

In this MIT OpenCourseWare lecture, the instructor analyzes traveling wave solutions of the wave equation on a string. It is shown that a traveling wave can be written as a function of X minus V T, propagating without distortion, with the shape carried along at the wave speed. The talk connects this progressing wave form to the traditional normal modes and Fourier series decomposition, and discusses how energy and boundary conditions constrain wave behavior.

Key concepts

The discussion covers progressive waves, the equivalence of F(X minus V T) and F(X plus V T) forms, how a stationary pulse decomposes into two traveling waves moving in opposite directions, and how two traveling waves can pass through or interact without losing their shape. The speed of propagation is tied to the string tension and mass per unit length, and the role of boundary conditions in coupling different media is introduced.

Introduction to traveling wave solutions

The lecture begins with a reminder of normal modes as standing waves and introduces a second class of solutions called progressing or traveling waves. A traveling wave is written in the form F of X minus V P times T, where F is a well behaved function describing the shape of the wave. By defining a new variable tau equal to X minus V P T, the speaker shows how partial derivatives of F with respect to X and T relate to the function F tau. This leads to the key result: any function of X minus V P T is a solution to the wave equation, propagating at the speed V P without changing shape. The complementary form F of X plus V P T is also a traveling wave solution, moving in the opposite direction. The physical interpretation is that a triangular or Gaussian pulse on the string appears to move along the string while the points of the string move up and down, not along the string itself.

Physical picture and sampling

Using a Gaussian example, the instructor demonstrates how the waveform is sampled at fixed X as time increases. The parameter tau decreases if time increases, which makes the waveform appear to move to the right. The intuitive message is that a traveling wave is a propagating distortion of the string’s shape, carried along by the tension and inertia of the string rather than a particle moving along the string.

Energy and small amplitude approximations

The energy content of a segment of string is analyzed by considering kinetic energy, derived from the velocity of each string element, and potential energy stored in the stretched string. With small amplitude assumptions, the length change is approximated, and the energy expressions reduce to standard forms involving the tension and the slope of the string. These preparations set the stage for calculating the energy carried by traveling waves and understanding superposition principles for multiple wave shapes.

Superposition and stationary shapes

A stationary shape on the string can be understood as a superposition of two traveling waves moving in opposite directions with equal and opposite velocities. The speaker emphasizes that any stationary profile can be decomposed into two traveling waves, enabling an intuitive shortcut versus performing a full Fourier decomposition into normal modes. The boundary between left and right media is used to illustrate how different wave speeds in each medium affect transmission and refraction of traveling waves.

Interface between two strings

The lecture then discusses connecting two strings with different mass per unit length and the same tension. The wave speeds differ on each side, leading to reflection and transmission when a traveling wave reaches the boundary. Boundary conditions are established: the displacement must be continuous across the boundary, and the slope must be continuous as there is no net force at the boundary. By imposing these conditions on incident, reflected, and transmitted waves, the reflection and transmission coefficients are derived, showing how the incident wave can partially reflect with a phase change and partially transmit with altered amplitude on the other side.

Experimental demonstrations

Real-time demonstrations with a Bell Labs apparatus illustrate the predicted behavior: an incident pulse from the left side produces a transmitted pulse on the right side and a reflected pulse on the left, with the relative amplitudes depending on the media on each side. The results align with the theoretical predictions, confirming the boundary conditions and energy partitioning across the interface.

Conclusion and outlook

The lecture closes by connecting traveling waves to broader topics in wave physics such as dispersion relations and more complex media, paving the way for future discussion on additional systems described by the wave equation and their dispersion properties.