4.5 Learning to Stand Still
Formation of standing waves by superposition of sunusoidal waves travelling in opposite directions.
Now let’s look at a different superposition. Again we’ll add two sine waves but this time, travelling in opposite directions:
An important point, here, is that the wave velocity, v, is a scalar quantity. It is defined as the velocity in the direction of travel of the wave, whatever that may be. So if the wave is travelling backwards, v doesn't change sign: instead we'll have to modify the equation to correctly represent the direction change.
The outcome of this superposition is a standing wave: although there is amplitude oscillation, the positions of the peaks do not change:
TOP: wave travelling to right
MIDDLE: wave travelling to left
BOTTOM: Superposition is a standing wave. It does not travel sideways but amplitude oscillates with time.
This is reflected in the equation you have derived in Question 4-6:
y = 2Asin(2πx/λ)cos(2πvt/λ) IV-2
Unlike the equation we have established for a travelling wave, this one features separate time and space-dependent parts, which are just multiplied together. The sin(2πx/λ) term is the waveform: it defines the shape of the stationary wave, which remains constant in time. However the amplitude of this waveform oscillates in time, as you see in the picture above. This time-dependent oscillation is describes by the cos(2πvt/λ) term. The ability to separate these terms in the function describing a stationary wave will turn out to be immensely useful later, in our development of an equation to describe a matter wave.
Let's look at some snapshots of this superposition:
At position J, the sum of the amplitudes of the two waves that are superposed remains at zero at all times. This is a NODE in the standing wave.
At other positions, the amplitude varies from positive to negative and back again , following that cosine function. Position K is where the amplitude is at a maximum – in either the positive or negative sense – at all times. This is an ANTINODE in the standing wave.
As we’ve seen in Chapter 1, such waves are encountered commonly in physical situations, and crucially we are going to use them as our model for the matter waves that we are proposing underlie the quantum theory of - amongst many, many other things – electrons in atoms. The interesting revelation here is that such a wave is mathematically equal to a sum of two equal travelling waves, moving in opposite directions.
