Travelling Waves

WAVES

THE LANGUAGE DESCRIBING WAVE MOTION

WAVE MOTION

Most information about our surroundings arrives as a wave: sounds are transpoted to our ears; light to our eyes and electromagnetic radiaiton to our mobile phones. Through wave motion, energy can be transferred from a source to a receiver without the transfer of matter between the two points.

A good visual example are the waves on the surface of water. When a stone is dropped into a lake, waves will be generate that travel outwards in expanding circles, with the centres as the source of the disturbance. The wave propagates, not the water.

Fig.1. Snap shot of the waves on the surface of water.

At each point, the water bobs up an down. The shape of the wave

can be approximated by a sine curve.

DESCRIBING WAVES

Consider a simple model for the propagation of a wave along the X axis which is represented pictorially as a sine function that depends both on time and position . The high points on the sine wave are called crests and the low points are called troughs as shown in figure 1.

The amplitude of the wave is the maximum disturbance of the wave from the mid-point between the crest and trough to either the top of the crest or to the bottom of a trough. The amplitude is a positive number. A loud sound has a large amplitude, whereas a weak radio signal has a small amplitude. Approximately the energy carried by a wave is proportional to the square of the wave amplitude.

The wavelength is the distance between two adjacent crests or two adjacent troughs or between any two successive identical parts of the wave.

The frequency of the wave is the number of vibrations each part of the wave undergoes in one second.

1 kHz = 10³ Hz (kilo) 1 MHz = 10⁶ Hz (mega) 1GHz = 10⁹ Hz (giga)

The period is the time interval for one complete vibration.

(1)

AM radio waves are broadcast in the kHz range

FM radio waves are broadcast with MHz frequencies

Microwaves have GHz frequencies

Audible sounds are generally in the range from ~ 1 Hz to < 20 kHz

The energy carried by a wave is approximately proportional to the square of the wave frequency. The higher the frequency the higher the transfer of energy in a given time interval.

The speed of a wave is related to its wavelength and its period (frequency). The wave advances 1 wavelength in a time interval of 1 period, therefore,

(2) propagation speed of the disturbance

This relationship holds true for all types of waves, whether they are water waves, sound waves, waves on strings or electromagnetic waves.

It is mathematically very convenient to define two other quantities in describing waves: the wave number or propagation constant and the angular frequency .

(3) wave number of propagation constant [ rad.m^-1 ]

(4) angular frequency [ rad.s^-1 ]

The shape of a sinusoidal wave is given by

(5a)

wave travelling to the right (+ X direction)

(5b)

wave travelling to the left (- X direction)

The symbol is used to describe the shape of the wave and is called the wave function which depends upon the two variables, position and time . This symbol is not commonly used the wave function is mostly given by the Greek letter (psi). N.B. could use cos instead of sin.

The term or its equivalents is called the phase of the wave. The velocity of the wave is often called the phase velocity, since it describes the velocity of the shape (phase) of the wave.

Equations 5a and 5b describe a travelling sinusoidal wave (harmonic wave). Because the wave function depends both on time and position, it is impossible to draw a simple graph of the wave function. The function must be animated or shown as a graph at a fixed time or a graph showing the variation with time at a fixed location.