Schrödinger's Cormorant

4.3 Making Waves

Introduces the time dimension: equations for travelling sinusoidal waves with different amplitudes, wavelengths, velocities and phases.

A graph like one of those in section 4.2 would give you the correct form of a snapshot of a sinusoidal wave – such as the electric field component of an electromagnetic wave – at a particular instant in time. This can be called a waveform. What turns it into a wave is that it propagates – meaning that the peaks move to the left or right, as time passes. This is illustrated in the animation below for a wave travelling to the right.

Make sure you understand what’s happening in the picture: focus on a peak and you can follow it moving gradually to the right as time goes by: this is called a travelling wave.

Now focus on Cormorant, floating on the waveform: you can see that he doesn’t move to the side at all. He just oscillates up and down. This tells you that this is a transverse wave: the wave moves in a direction that’s perpendicular to the oscillation.

Question 4-2

If you worked through this question, you have now arrived at an important result: the general equation for a travelling wave.

y = Asin(2π(x-vt)/λ) IV-1

Nefertiti is not happy, though, about the stipulation we made in deriving this, which demanded that y =0 when x and t are equal to zero. “Can’t we do it properly?” she demands. Let’s see:

Question 4-3

This exercise has led us to an equation which is more general but also more cumbersome:

y = Asin(2π(x-vt)/λ + P) IV-2

The additional term, P, is called the phase shift. The size of P relates to the idea of one wavelength equating to 2π radians. So in the example of question 4-3, the phase shift of the orange curve is ⅛ of a wavelength = 2π/8 = p/4 radians to the left, so P = -π/4.

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