KAHIBARO
Table of Contents
When waves cancel
Destructive interference happens when two waves meet in such a way that the displacement from one wave opposes the displacement from the other. Instead of adding to make a bigger result, they partially or completely cancel.
This idea follows directly from the superposition principle. At each point, the total displacement is the algebraic sum of the individual displacements. If one displacement is positive and the other is negative, the sum becomes smaller, and it may even become zero.
If two identical waves arrive at the same place with equal amplitude but opposite displacement, the cancellation is complete. If their amplitudes are not equal, the cancellation is only partial.
The basic condition
For the clearest case, consider two sinusoidal waves with the same amplitude and frequency. If they differ in phase by $\pi$ radians, or $180^\circ$, then crest meets trough and trough meets crest.
We can write
$$
y_1 = A \sin(kx - \omega t)
$$
and
$$
y_2 = A \sin(kx - \omega t + \pi)
$$
Since $\sin(\theta + \pi) = -\sin\theta$, this becomes
$$
y_2 = -A \sin(kx - \omega t)
$$
So the total displacement is
$$
y = y_1 + y_2 = 0
$$
everywhere the two waves overlap perfectly.
Complete destructive interference requires waves of the same frequency and equal amplitude, with a phase difference of $\pi$ radians.
Path difference and phase difference
Very often, destructive interference is described using path difference. If one wave travels a little farther than the other, that extra distance creates a phase shift.
For destructive interference, the path difference must be an odd half multiple of the wavelength:
$$
\Delta L = \left(m + \frac{1}{2}\right)\lambda
$$
where $m = 0, 1, 2, 3, \dots$
This corresponds to a phase difference
$$
\Delta \phi = (2m+1)\pi
$$
The smallest such path difference is
$$
\Delta L = \frac{\lambda}{2}
$$
Destructive interference condition:
$$
\Delta L = \left(m + \frac{1}{2}\right)\lambda
$$
or equivalently
$$
\Delta \phi = (2m+1)\pi
$$
Partial destructive interference
In real situations, waves do not always cancel completely. If the amplitudes are different, the result is a smaller wave, not zero.
Suppose two waves have amplitudes $A_1$ and $A_2$ and are exactly out of phase. Then the resultant amplitude is
$$
A_{\text{result}} = |A_1 - A_2|
$$
So if $A_1 = 5$ units and $A_2 = 3$ units, the remaining amplitude is $2$ units.
This is still destructive interference because the combined effect is reduced.
Comparing constructive and destructive interference
| Type of interference | Phase difference | Path difference | Result |
|---|---|---|---|
| Constructive | $2m\pi$ | $m\lambda$ | Larger amplitude |
| Destructive | $(2m+1)\pi$ | $\left(m+\frac{1}{2}\right)\lambda$ | Reduced or zero amplitude |
Physical meaning
Destructive interference does not mean the waves stop existing before they meet. Each wave continues to propagate according to the properties of the medium. The cancellation is about the combined displacement at a particular place and time.
For example, if one wave pushes a string upward by $2 \, \text{cm}$ and another pushes it downward by $2 \, \text{cm}$ at the same point, then the string at that point does not move, because the net displacement is zero.
Simple example
Imagine two identical pulses moving toward each other on a rope. One is an upward pulse and the other is a downward pulse. When they fully overlap, the upward displacement and downward displacement cancel.
If they overlap perfectly, the rope becomes flat at that moment.
Destructive interference in sound
With sound, destructive interference means the pressure variations from one sound wave oppose those from another. This can reduce the loudness heard at some positions. Noise canceling headphones use this idea by producing a wave that is approximately out of phase with unwanted sound.
This does not remove sound everywhere in space. The cancellation is strongest where the phase relation is correct.
Important cautions
Destructive interference is not the same as energy disappearing. The wave energy is not simply lost because the displacement at one point becomes zero. In broader wave systems, energy is redistributed in space.
Also, complete cancellation usually requires ideal conditions. In practice, slight differences in amplitude, frequency, or phase often make the cancellation incomplete.
Destructive interference reduces the resultant amplitude. It is complete only under ideal matching conditions.
Summary
Destructive interference occurs when overlapping waves combine to produce a smaller displacement than either wave alone. In the strongest case, equal waves that are out of phase by $\pi$ radians cancel completely. The standard condition is
$$
\Delta L = \left(m + \frac{1}{2}\right)\lambda
$$
or
$$
\Delta \phi = (2m+1)\pi
$$
This idea is central to understanding why waves can cancel at certain positions even though each individual wave continues to exist.