KAHIBARO
Table of Contents
When waves meet
Interference is the pattern that appears when two or more waves overlap in the same place at the same time. Because waves obey the superposition principle, the total displacement is the sum of the individual displacements. Interference is the specific result of that addition when the waves combine in different ways at different points.
If two waves reinforce each other, the resulting displacement becomes larger. If they oppose each other, the resulting displacement becomes smaller, and in some cases can become zero. This is why interference is often described as reinforcement and cancellation.
Adding displacements
Suppose two waves produce displacements $y_1$ and $y_2$ at the same point. The total displacement is
$$
y = y_1 + y_2
$$
This simple rule creates many important patterns. The result depends mainly on how the waves are aligned in phase.
If the crests and troughs of the waves line up, the waves are in phase and tend to reinforce one another. If the crest of one lines up with the trough of the other, the waves are out of phase and tend to cancel.
The key rule of interference is that waves add by displacement, not by intensity or by shape directly.
$$
y_{\text{total}} = y_1 + y_2
$$
Phase difference and path difference
Interference depends on the phase difference between the waves. A phase difference tells us how much one wave is shifted relative to another.
For two sinusoidal waves of the same frequency, the phase difference $\phi$ is closely related to the path difference $\Delta r$, which is the difference in the distances traveled by the waves:
$$
\phi = \frac{2\pi}{\lambda}\Delta r
$$
where $\lambda$ is the wavelength.
If the path difference is a whole number of wavelengths, the waves arrive in phase. If the path difference is a half integer number of wavelengths, they arrive in opposite phase.
For waves with the same frequency,
$$
\phi = \frac{2\pi}{\lambda}\Delta r
$$
Constructive interference occurs when
$$
\Delta r = m\lambda
$$
Destructive interference occurs when
$$
\Delta r = \left(m + \frac{1}{2}\right)\lambda
$$
where $m = 0, 1, 2, 3, \dots$
Constructive and destructive interference
Constructive interference happens when overlapping waves have displacements in the same direction. Their amplitudes add, so the result is larger than either wave alone.
If two identical waves, each with amplitude $A$, interfere constructively, the resulting amplitude is
$$
A_{\text{result}} = 2A
$$
Destructive interference happens when the displacements are in opposite directions. If two identical waves are exactly opposite in phase, they can cancel completely:
$$
A_{\text{result}} = 0
$$
This does not mean the waves disappear permanently. It means that at that location and at that instant, their displacements sum to zero.
Interference of two sinusoidal waves
Consider two waves with the same amplitude $A$, same wavelength, and same frequency:
$$
y_1 = A\sin(kx - \omega t)
$$
$$
y_2 = A\sin(kx - \omega t + \phi)
$$
When these are added, the result is another sinusoidal wave with the same $k$ and $\omega$, but with a different amplitude:
$$
y = 2A\cos\left(\frac{\phi}{2}\right)\sin\left(kx - \omega t + \frac{\phi}{2}\right)
$$
So the combined amplitude is
$$
A_{\text{combined}} = 2A\cos\left(\frac{\phi}{2}\right)
$$
This formula shows that the amplitude depends on phase difference.
| Phase difference $\phi$ | Result |
|---|---|
| $0$ | maximum reinforcement |
| $\pi/2$ | partial reinforcement |
| $\pi$ | complete cancellation for equal amplitudes |
| $2\pi$ | maximum reinforcement again |
For two equal sinusoidal waves,
$$
A_{\text{combined}} = 2A\cos\left(\frac{\phi}{2}\right)
$$
Maximum interference occurs when $\cos(\phi/2)$ is $\pm 1$, and complete cancellation occurs when it is $0$.
What interference patterns look like
In many situations, interference creates regions where the wave is strong and regions where it is weak. For example, with water waves or sound waves, some places receive large oscillations while nearby places receive very small oscillations.
This happens because the path difference changes from point to point. At one location the waves may arrive in phase, while at another they may arrive out of phase.
The crossing regions of wavefronts suggest places where different kinds of interference can occur, depending on whether crest meets crest or crest meets trough.
Coherence
A stable interference pattern requires a stable phase relationship between the waves. Two sources that maintain a constant phase difference are called coherent sources.
If the phase difference changes randomly, the interference pattern does not remain fixed and usually becomes unobservable over time.
This is why interference is easiest to observe when the waves come from the same original source or from sources designed to stay synchronized.
A clear, lasting interference pattern requires coherent waves, meaning waves with the same frequency and a constant phase difference.
Interference is local
Interference does not mean that one wave permanently destroys another. The waves continue to propagate according to the medium and boundary conditions. The cancellation or reinforcement happens in the region where they overlap.
This is important. When two pulses meet on a string and cancel for a moment, they later continue onward after passing through each other.
Energy and interference
At first, destructive interference may seem to make energy disappear. It does not. Energy is redistributed. Regions of small amplitude are balanced by regions of larger amplitude elsewhere.
For mechanical waves, larger amplitude usually means greater energy carried by the wave. Since interference changes amplitude from place to place, it changes how energy is distributed in space.
A simple example
Imagine two identical sound waves reaching a listener. If they arrive in phase, the listener hears a louder sound. If they arrive out of phase, the sound becomes weaker, and at certain points can become very faint.
The same basic idea applies to water waves, waves on strings, and light waves. The detailed effects differ from system to system, but the interference rule comes from the same wave addition.
Summary table
| Situation | Condition | Result |
|---|---|---|
| Constructive interference | $\Delta r = m\lambda$ | amplitude increases |
| Destructive interference | $\Delta r = (m + 1/2)\lambda$ | amplitude decreases |
| Equal waves, in phase | $\phi = 0$ | amplitude $= 2A$ |
| Equal waves, opposite phase | $\phi = \pi$ | amplitude $= 0$ |
Final idea
Interference is one of the clearest signs that something behaves as a wave. Whenever waves overlap, their displacements add. Depending on phase difference, this addition can produce reinforcement, cancellation, or anything in between.