5.2 Constructive and destructive interference

Wave Interference Fundamentals

When two or more sound waves overlap in space, they combine to produce a single resultant wave. This process is called interference, and it's governed by the superposition principle: at any point, the total displacement equals the sum of the individual wave displacements. Whether the result is louder or quieter depends on how the waves align in phase.

Understanding interference is central to acoustics. It explains everything from dead spots in a lecture hall to how noise-cancelling headphones work, and it gives you the math to predict exactly where sound will be amplified or silenced.

Constructive vs. Destructive Interference

Constructive interference occurs when two waves arrive in phase, meaning their crests and troughs line up. The displacements add together, producing a wave with a larger amplitude (louder sound).

Destructive interference occurs when two waves arrive out of phase, meaning the crest of one aligns with the trough of the other. The displacements cancel, producing a wave with a smaller amplitude (quieter sound), or complete silence if the amplitudes are equal.

The key factor is the phase difference ($\Delta\phi$) between the two waves, measured in radians or degrees. A phase difference of $0$ means perfect constructive interference. A phase difference of $\pi$ radians (180°) means maximum destructive interference.

Conditions for Each Type

For constructive interference, all three of these must hold:

• The waves have the same frequency
• The path difference ($\Delta r$) is an integer multiple of the wavelength: $\Delta r = n\lambda$, where $n = 0, 1, 2, \ldots$
• Equivalently, the phase difference is an even multiple of $\pi$: $\Delta\phi = 0, 2\pi, 4\pi, \ldots$

For destructive interference:

• The waves have the same frequency
• The path difference is an odd multiple of half the wavelength: $\Delta r = (n + \tfrac{1}{2})\lambda$, where $n = 0, 1, 2, \ldots$
• Equivalently, the phase difference is an odd multiple of $\pi$: $\Delta\phi = \pi, 3\pi, 5\pi, \ldots$

Coherence is also required for a stable, observable interference pattern. Two sources are coherent if they maintain a constant phase relationship over time. If the phase relationship drifts randomly (as with two unrelated noise sources), the interference pattern shifts too fast to detect, and you just hear an average intensity. Lasers are a classic example of highly coherent sources; in acoustics, two speakers driven by the same signal are coherent.

Quantitative Analysis of Interference

Resultant Amplitude of Interfering Waves

The superposition principle tells you what to do (add the displacements), but you need a method to carry out that addition. There are two main approaches.

Trigonometric approach. Represent each wave as a sinusoidal function, then add them algebraically. For two waves of the same frequency:

$y_1 = A_1 \sin(\omega t)$ $y_2 = A_2 \sin(\omega t + \Delta\phi)$

The resultant $y = y_1 + y_2$ can be simplified using trig identities, but this gets tedious for more than two waves.

Phasor method. Represent each wave as a rotating vector (phasor) whose length equals the wave's amplitude and whose angle equals its phase. You then add the phasors tip-to-tail, just like adding vectors. The length of the resultant phasor gives the resultant amplitude. This is faster and scales well to many waves.

Both methods yield the same resultant amplitude formula:

$A_R = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2 \cos(\Delta\phi)}$

where $A_1$ and $A_2$ are the individual amplitudes and $\Delta\phi$ is the phase difference between them.

Check the extremes to build intuition:

• When $\Delta\phi = 0$ (constructive): $\cos(0) = 1$, so $A_R = A_1 + A_2$. The amplitudes add directly.
• When $\Delta\phi = \pi$ (destructive): $\cos(\pi) = -1$, so $A_R = |A_1 - A_2|$. The amplitudes subtract. If $A_1 = A_2$, you get complete cancellation: $A_R = 0$.
• When $\Delta\phi = \pi/2$ (quarter cycle offset): $\cos(\pi/2) = 0$, so $A_R = \sqrt{A_1^2 + A_2^2}$. This is the Pythagorean case, partway between full constructive and full destructive.

Path Difference and Interference Patterns

In many real situations, two waves start in phase but travel different distances to reach a listener. That difference in distance is the path difference ($\Delta r$), and it directly determines the phase difference:

$\Delta\phi = \frac{2\pi \, \Delta r}{\lambda}$

This equation is the bridge between geometry and interference. If you know the positions of two speakers and a listener, you can calculate $\Delta r$, convert it to $\Delta\phi$, and then use the resultant amplitude formula to find the sound level at that point.

Interference fringes are the spatial pattern that results. As you move through a room, $\Delta r$ changes continuously, so you pass through alternating regions of constructive interference (loud) and destructive interference (quiet). The spacing of these fringes depends on the wavelength and the geometry of the sources.

Standing waves are a special case of interference. When two waves of the same frequency travel in opposite directions (for example, a wave and its reflection), they produce a pattern of fixed nodes (points of zero amplitude) and antinodes (points of maximum amplitude). The nodes are spaced half a wavelength apart.