When a wave passes through a wall with two holes,
a striped interference pattern is created.
But, an interference pattern can also be created
even if the wave passes through a wall with just a single hole.
To understand why this is the case, we need to understand that
each wave can be thought of as the combination of
an infinite number of smaller waves that spread out in all directions.
When the wave hits a barrier with a small hole, only the portion of
the wave that is directly behind the hole is able to pass through.
This is why waves spread out when they pass through a barrier with a small hole.
If the hole is bigger, then more of the waves are able to pass through,
and these waves have the ability to interfere with each other.
To understand the interference pattern that these waves will produce,
let us first consider the interference pattern created by two small holes.
The waves coming out of each of the holes can be thought of as
sine waves that travel out in all directions.
The waves, as seen along a one dimensional line,
can be thought of as just a single sine wave.
At any given point, the interference pattern formed by the two waves from the holes
can be thought of as the sum of the two sine waves along the
intersecting lines coming from the two holes.
When the peaks of two sine waves occur at the same time,
we say that the two waves are “in phase.”
Since both sine waves are created by the same original wave
which passes through both holes at the same time,
the two sine waves are always in phase at the moment when they exit the holes.
The two sine waves are also in phase for any point for which
the distances from the two holes are equal.
The distances from the two holes will no longer be equal
if the angle of this yellow line changes.
The more the angle of this yellow line changes,
the greater the difference in distances from the two holes.
The greater the difference in the distances from the two holes,
the greater the difference in the phases of the two sine waves.
The strength of the interference pattern at each point
is the sum of these two sine waves.
A sine wave can be represented graphically like this.
The sum of two sine waves with the same frequency and phase
can be represented graphically like this.
The sum of two sine waves with the same frequency,
but different phases, can be represented graphically like this.
Therefore, the sum of two sine waves of the same frequency
is always another sine wave of the exact same frequency,
but with a different amplitude and phase.
The amplitude of the sum is represented by the length of this green line.
In the case of the sine waves coming from our two small holes,
the two waves have the same amplitude and frequency, but different phases.
As the angle of this yellow line changes, the difference in phases increases.
As the difference between the phases increases,
the sum of the two sine waves also changes.
The amplitude of the sum of the two sine waves
is represented by the length of this green line.
In some cases their sum has a very large amplitude,
and in other cases, their sum has an amplitude of zero.
As the angle of the yellow line changes,
the sum of the two sine waves increases and decreases,
as is shown by the length of the green line.
This is why a striped pattern is produced.
Now, instead of two small holes,
let us consider the case of just a single large hole.
The single large hole can be thought of as a number of small holes
that are right next to each other.
A sine wave comes out of each of these holes, and these sine waves add together.
As before, as the angle of this yellow line changes,
the differences in the phases of the sine waves increases.
As the differences in the phases of the sine waves increases,
this can be represented as shown.
Here, as the difference in phases increases,
the amplitude of the sum of all the sine waves goes up and down as before,
and this again creates a striped pattern.
This time, however, each peak in the sum of the sine waves
is smaller than the one before it.
This means that amplitude of the wave
becomes much smaller as the angle of this yellow line increases.
Now let’s consider another scenario where the hole is even bigger.
In this case, this can be thought of as
an even larger number of small holes next to each other.
As before, each of these holes generates a sine wave,
and all these sine waves add together.
As the angle of this yellow line changes,
the differences in the phases of the sine waves increases.
As the differences in the phases of the sine waves increases,
their sum can be represented as shown.
This time,
as the phase between each of the sine waves changes by even a very small amount,
the amplitude of the sum of all the sine waves goes down very rapidly.
This means that with a larger hole, if the angle of this yellow line changes,
the amplitude of the wave decreases very rapidly.
For this reason, when a wave passes through a large hole,
the amplitude of the wave is strong only directly in front of the hole.
The wave spreads out only when
the hole is small relative to the size of the wavelength.
If the hole is large relative to the size of the wavelength,
the wave does not spread out,
and instead looks like it is travelling in a straight line.
Much more information is available in the video titled
“Waves: Light, Sound, and the nature of Reality”,
and in the other videos on this channel.
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