Multi-wave interference extends ordinary two-wave interference to many coherent sources. If each source produces a sinusoidal contribution
\[
y_i = A_i\sin(\omega t + \phi_i),
\]
then the total displacement is the sum
\[
y = \sum_{i=1}^{N} y_i.
\]
The resulting pattern can be much more structured than the two-wave case, especially when the phases are evenly spaced or when many sources act together like a grating.
A very useful viewpoint is the phasor method. Each wave is represented by a complex vector
\[
A_i e^{i\phi_i}.
\]
The phasor sum is
\[
\tilde Y = \sum_{i=1}^{N} A_i e^{i\phi_i}.
\]
The magnitude of this phasor determines the overall amplitude of the coherent superposition, and the corresponding intensity is proportional to
\[
I \propto |\tilde Y|^2.
\]
When all waves have the same amplitude and the same phase, the phasors line up in the same direction. Then the magnitude is large:
\[
|\tilde Y| = NA,
\]
so the intensity scales like
\[
I \propto (NA)^2.
\]
This is strong constructive interference.
On the other hand, if the phases are distributed uniformly around the circle, the phasors can cancel. A classic example is three equal waves separated by \(120^\circ\):
\[
\phi_1=0,\qquad \phi_2=\frac{2\pi}{3},\qquad \phi_3=\frac{4\pi}{3}.
\]
Their phasors form a closed triangle, so
\[
\tilde Y = 0.
\]
That means the net coherent amplitude is zero and the ideal intensity is zero as well.
A phase difference can also come from a path difference. If the wavelength is \(\lambda\) and adjacent sources differ in path by \(\Delta x\), then
\[
\Delta\phi = \frac{2\pi \Delta x}{\lambda}.
\]
This is exactly the relation used in diffraction-grating and multi-source interference problems. By changing \(\Delta x\), the sources move from reinforcement to cancellation and back again.
For the sample case of three equal waves with \(\Delta\phi=120^\circ\), the phasor construction predicts complete cancellation. In the time-domain sum, the waveform contributions also cancel at every instant in the ideal coherent model. This is an important illustration that complex patterns can emerge even from very simple equal-amplitude waves.
As the number of sources increases, the interference becomes sharper. This is why a grating with many slits produces narrow bright peaks and strong suppression elsewhere. The same idea also appears in phased arrays, laser arrays, antenna arrays, and acoustics.
At more advanced level, one studies partial coherence, unequal amplitudes, and finite-aperture effects. But the coherent equal-amplitude model used here is the essential starting point because it shows clearly how phase spacing controls the total phasor, the summed signal, and the final intensity.
This calculator visualizes the component waves, the total signal, and the phasor addition directly. It also lets you switch between direct phase spacing and path-difference spacing to connect the algebra with the physical geometry.
