There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis

Saturday, March 12, 2016

Mathematical expressions for one dimensional traveling waves

All postings by author previous: Three types of waves: traveling waves, standing waves, and rotating waves up: Contents of this block of postings next: Standing waves in one dimension
This posting includes a flash animation showing the physics discussed. Most computers have a flash player already installed, but if yours does not, download the free Adobe flash player here.

Mathematical expressions for one dimensional traveling waves

In one dimensional wave systems, traveling waves can be expressed in real form as:


or in complex form as:

Fig. 4. Traveling wave with source and absorber. Mouse over the animation to see the action, mouse off to suspend it, and click on it to restart it. The lower object is a pipe with acoustical pressure waves propagating from the source to the absorber. The acoustical pressure is shown in color. The upper object can either be a graph of the acoustical pressure in the lower item or a depiction of waves on a string.


where ω is the angular frequency of the wave (ω = 2π times the regular frequency, f), the + sign (of the ±) is used for waves traveling in the positive x direction and the − sign for negatively traveling waves. The wavenumber κ is related to the wavelength λ by κ = 2π/λ . κ has units of radians/meter.

A, A1, and B are real constants determining the amplitude and phase of the waves. The C has a tilda over it to indicate that it is a complex constant and sets the amplitude and phase of the complex form of a traveling wave. As is normal for the complex form, a real operator, i.e. Re( ), is understood to be required to convert the complex form into waves that exist in the real world. The complex form is used because it greatly simplifies mathematical manipulations.

The frequency and wavenumber are related by:

   ,     (3)

where c is the phase velocity of traveling waves in this particular medium at the frequency of interest. Non-dispersive waves (such as waves on a string) have a constant phase velocity while dispersive waves (such as water waves) have a phase velocity that varies with frequency. There are no restrictions on possible values of the wave number κ other than they be real (although there are applications for complex κ such as for non-propagating evanescent waves).

In the case of waves on a string, the phase velocity c, i.e. the velocity of wave propagation, is given by:

   ,     (4)

where T is the tension on the string and μ is the linear mass density of the string (kg/m).

Energy and momentum in traveling waves

From Physics of Waves chapter 1, we get the following proportionality between the energy density E1 and momentum density gx for traveling waves on a string:

gx = E1/c    .        (5)

There is a controversy as to whether any momentum is carried by waves on a string. This is discussed in a later posting.

All postings by author previous: Three types of waves: traveling waves, standing waves, and rotating waves up: Contents of this block of postings next: Standing waves in one dimension