Traveling Waves

"The essence of science: ask an impertinent question, and you are on the way to a pertinent answer"
Jacob Bronowski

Imagine a sequence of particles undergoing identical Simple Harmonic Motion, such that each particle begins to move slightly after the one before it. The result is a traveling "Wave Motion". If all the particles are connected, for example, in a string, the motion is described as a continuous "Sine Wave".
The Sine Wave is the simplest of all possible waves. A periodic wave is one in which the shape of the wave is repeated "periodically" - at regular fixed intervals.

Five basic properties which describe periodic waves.

Wavelength (lambda) - Distance after which the wave begins to repeat (Units: metres).
Frequency (f) - Number of waves passing a fixed point in one second (Units: Hertz).
Wave period (T) - Time taken for one wave to pass a given point (Units: seconds).
Wave velocity (v) - Distance travelled by the wave per second also called the phase velocity (Units: m/s).
Amplitude (y_m) - Maximum displacement of particle which comprise the wave from their equilibrium position (Units: metres).

Frequency, wavelength and velocity are related by,

There are two basic types of traveling waves.

TRANSVERSE : Motion of the constituent particles is at right angles to the wave direction, e.g. waves on a string, "stadium" wave, electromagnetic waves.

Animation courtesy of Dr. Dan Russell, Pennsylvania State University

LONGITUDINAL : Motion of the constituent particles is back and forth in the direction of motion of the wave, e.g. sound waves, the " Slinky " spring.

Animation courtesy of Dr. Dan Russell, Pennsylvania State University

Comparison of SHM and Waves

The amplitude of the SHM of the particles which comprise a Sine Wave is the same as the amplitude of the wave.
The frequency of the SHM of the particles which comprise a Sine Wave is the same as the frequency of the wave.
Particles which comprise the wave typically do not move at the speed of the wave, e.g. molecules of air do not move at the speed of sound in air.

Miscellaneous important facts :

A wave "travels" from A to B. The particles that comprise the wave do not move from A to B, they oscillate about their fixed (equilibrium) points.
A wave transmits energy from one point to another.
Transmission of Electromagnetic waves does not require a medium - there are no "particles". These waves are comprosed of oscillating electric and magneic fields and are quite happy to propagate in a vacuum.
Water waves appear to be transverse - boats bob up and down due to water waves. However, a detailed study shows that the molecules of water actually perform a circular motion, which can be considered as a combination of transverse and longitudinal wave motion.

Animation courtesy of Dr. Dan Russell, Pennsylvania State University

As we will see shortly, Fourier's theorem states that any periodic wave can be decomposed into a sum of sine waves with differing frequencies and amplitudes. This means that an analysis of the properties of sine waves can be applied to any periodic wave. The mathematical representation of a traveling sine wave moving from left to right is

Note that if the wave is moving from right to left we simply replace the negative sign by a positive sign. Using the above expression for velocity in terms of frequency, wavelength and period we find

which can be written

where we define the wave number, tw5

and the angular frequency, tw6

so that the phase velocity of the wave can be written

Be careful do not confuse "k" the wave number with "k" the spring constant.

Initial conditions

The above expression describing a traveling wave demands that when t = 0 and x = 0 then y = 0. This does not necessarily have to be the case. In order to allow for differing initial conditions we introduce a phase constant, Ø, such that

Wave Velocity on a Stretched String

We have seen the relationship between the velocity of a wave and its wavelength and frequency, but what physically determines the velocity of a wave ? For example, a wave on a stretched string.

The elasticity of the string creates the restoring force which causes the string to oscillate. Elasticity is a function of the tension in the string - the larger the tension the larger the restoring force will be and the larger the velocity. However, the inertia of the string determines the response of the string to the restoring force - the larger the inertia the slower the string will respond and the smaller the velocity. Inertia is proportional to the linear density of the string, also known as the mass per unit length. Considering all of these facts it can be shown that for a traveling wave on a stretched string,

where T is the tension in the string, m is its mass and tw10

is its length.

Where does bad light end up? Answer: In a prism!

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu