WAVES

All waves are unusual in that while individual parts move repetitiously, THE PARTS DON'T GO ANYWHERE!! They merely repeat their motion over and over again.

Waves come in many forms and include seismic waves, sound, light has wavelike behaviour, sea waves, soliton waves, standing waves, tsunami and probability waves ( quantum mechanics).

Understanding them is crucial to the telecommunication industry, physicists, seismic geologists, optics industry and many other scientific and technological industries.

We study two wave forms, Transverse Waves and Longitudinal Waves, the first - easy to get, the second nearly impossible to visualise. Fortunately we can get the gist of Longitudinal Waves through Transverse Waves.

The start of understanding waves is to get the idea of SIMPLE HARMONIC MOTION - hereafter SHM. Fourier showed that all waves can be made up of mixtures of the simplest waves - even "pulses" - bits of waves - ( digital communications including desktop computers uses pulses called "bits". )

SHM is defined as motion in which the displacement is proportional to the acceleration, but oppositely directed.

Mathematically s = -k a k is an arbitrary constant, s and a are vectors.

Oscillating springs and pendulum bobs closely follow this rule. The best model, however is to link the SHM to circular motion via the trig functions, sine or cosine.

In the above animation, the orange ball and the green ball are linked to the red circulating ball via sin and cos.

x_orange = r cosθ y_green = r sinθ If we make θ a function of time, thenθ = (ωt + phase number)

ω = 2π f = 2π / T ( the angular speed) f = frequency, T = period

then

x_orange = r cos (ωt + phase number) y_green = r sin (ωt + phase number)

The phase number is nothing more than an angle to get the starting point correct with respect to the x axis.

Using calculus, vel = dx/dt, acceleration = d²x/dt² it is pretty easy to show that

x_orange = -ω²a_orange and y_green = -ω²a_green.

In other words

• the orange and green balls are moving in pure SHM!
• SHM makes sine graphs in time

TOTAL WAVE EQUATION

Waves can be viewed as a string of particles moving in SHM BUT each particle moving slightly at a different time than its neighbour. Mathematically this is

Displacement = Asin(ωt + fn(x)) ( we could use cosine instead of sine )

Also, a wave in its simplest form has a sine shape if we take a photo of it - lock it in time - but has moved if we take a later photo.

This translates to

Displacement = Asin(kx + fn(t)) k = wave number = 2π /λ λ = wavelength

Graph of y= sinx

Graph of y = sin(x+1)

- the "+1" has moved the graph to the left - this is equivalent to a later photo of a wave moving to the left.

( Those of you who are doing maths up to plotting y = 3 sin(4x+π/2) should be pretty comfortable with this.)

Combining these versions of Displacement of a point from its mean position ( centre of SHM ) gives

Displacement = Asin(kx±ωt) This is often known as the GENERAL WAVE EQUATION.

( Again, sine can be replaced by cosine. We can even use displacement = Ae^{i(kx ± ωt)} )

The Plus moves the wave to the left, the Minus moves the wave to the left.

TRANSVERSE WAVES

These are the simplest to picture. The displacements of the individual particles ( doing SHM ) are at right angles to the line of the wave direction.

As a result we see the classic sine wave move past, closely resembling the sea waves we are familiar with.

Notice that each part is in SHM - a sine wave in time, but a photo ( ie holding it still) will reveal the sine wave in position.

Notice also that from crest to crest is a wavelength and that the wave takes one period to travel one wavelength.

The speed of the wave is therefore

v =λ / T =λf

This equation holds for all waves.

LONGITUDINAL WAVES

These are the hardest of the waves types to visualise as the SHM is along the line of the wave motion.

Also, one tends to get lost in trying to link to the speeds of the particles in time and in position.

Try to focus in the animation on the red wave particle which is oscillating around the red arrow. That arrow is the centre of the Simple Harmonic Motion of the red wave particle.

Once you have locked on to it, follow the red particle and notice that it is moving fastest when in a compression ( in this case moving to the right) with the wave AND in a rarefaction. This fastest motion occurs when over the arrow.( Zero displacement.)

In the COMPRESSION, it is moving to the RIGHT IN THE WAVE DIRECTION but in the RAREFACTION it is moving to the LEFT AGAINST THE WAVE DIRECTION!!