# The wave equation

The wave speed depends on tension We shall give qualitative physical explanations as well as the mathematics. However, before starting this section, you may wish to revise calculus. In the chapter Travelling waves Iwe described the travelling wave and some of its properties. The Speed of a Wave The Wave Equation As was discussed in Lesson 1a wave is produced when a vibrating source periodically disturbs the first particle of a medium. This creates a wave pattern that begins to travel along the medium from particle to particle.

## Dan Russell's Acoustics and Vibration Animations

The frequency at which each individual particle vibrates is equal to the frequency at which the source vibrates. Similarly, the period of vibration of each individual particle in the medium is equal to the period of vibration of the source.

In one period, the source is able to displace the first particle upwards from rest, back to rest, downwards from rest, and finally back to rest. This complete back-and-forth movement constitutes one complete wave cycle. The diagrams at the right show several "snapshots" of the production of a wave within a rope. The motion of the disturbance along the medium after every one-fourth of a period is depicted.

Observe that in the time it takes from the first to the last snapshot, the hand has made one complete back-and-forth motion. A period has elapsed. Observe that during this same amount of time, the leading edge of the disturbance has moved a distance equal to one complete wavelength. So in a time of one period, the wave has moved a distance of one wavelength.

Rearranging the equation yields a new equation of the form: Stan and Anna are conducting a slinky experiment.

## Sound Waves and Sources

They are studying the possible effect of several variables upon the speed of a wave in a slinky. Their data table is shown below.

• The Wave Equation
• Vibration and Structural Waves

Fill in the blanks in the table, analyze the data, and answer the following questions.The wave equation is a partial differential equation.

We discuss some of the tactics for solving such equations on the site Differential Equations. One of the most popular techniques, however, is this: choose a likely function, test to see if it is a solution and, if necessary, modify it.

Math A { October 13, «Viktor Grigoryan 7 The energy method Energy for the wave equation Let us consider an in nite string with constant linear density ˆand tension magnitude T. The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a lausannecongress2018.com is a three-dimensional form of the wave lausannecongress2018.com homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form: (∇ − ∂ ∂) = (∇ − ∂ ∂).

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum.

It is a three-dimensional form of the wave equation. The 2D wave equation Separation of variables Superposition Examples Representability The question of whether or not a given function is equal to a double Fourier series is partially answered by the following result.

Theorem If f(x,y) is a C2 function on the rectangle [0,a] ×[0,b], then. The wave equation is a linear second-order partial differential equation which describes the propagation of oscillations at a fixed speed in some quantity \(y\): A solution to the wave equation in two dimensions propagating over a fixed region .

Wave Equation, Wave Packet Solution