The principle of superposition states that when two or more waves overlap, the total displacement at any point is the algebraic sum of the individual displacements. For two waves,
\[
y=y_1+y_2.
\]
This is one of the most important ideas in wave physics because it explains interference, beats, standing waves, and many other wave phenomena.
If the two waves have the same angular frequency and are sinusoidal,
\[
y_1=A_1\sin(\omega t+\phi_1),\qquad
y_2=A_2\sin(\omega t+\phi_2),
\]
then the total wave depends strongly on the phase difference
\[
\delta=\phi_2-\phi_1.
\]
The phase difference tells us how shifted one wave is relative to the other.
When the waves are in phase, \(\delta=0\), the peaks line up with peaks and the troughs line up with troughs. This produces constructive interference. If the amplitudes are equal, the resultant amplitude is the sum of the two amplitudes.
When the waves are exactly out of phase, \(\delta=\pi\), the peaks of one line up with the troughs of the other. This produces destructive interference. If the amplitudes are equal, the two waves cancel completely and the resultant displacement is zero at all times.
The general formula for the resultant amplitude of two same-frequency waves is
\[
A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\delta}.
\]
This formula shows that the resultant amplitude depends on both the individual amplitudes and the cosine of the phase difference. Full reinforcement happens when \(\cos\delta=1\), and maximum cancellation happens when \(\cos\delta=-1\).
In many interference problems, the phase difference comes from a path difference \(\Delta x\). If the wavelength is \(\lambda\), then
\[
\delta=\frac{2\pi\Delta x}{\lambda}.
\]
A path difference of one full wavelength gives a phase difference of \(2\pi\), which is equivalent to being in phase. A path difference of half a wavelength gives a phase difference of \(\pi\), which leads to destructive interference for equal amplitudes.
Consider the sample case
\[
y_1=\sin t,\qquad y_2=\sin(t+\pi).
\]
Since
\[
\sin(t+\pi)=-\sin t,
\]
the sum becomes
\[
y=\sin t + \sin(t+\pi)=\sin t - \sin t = 0.
\]
This is perfect destructive interference.
The calculator handles both direct phase inputs and path-difference-based phase relations. It reports the individual wave values, the total displacement, the phase difference, and the resultant amplitude. The interactive graph shows the two component waves and their sum, while the animation helps visualize how reinforcement or cancellation happens over time.
At more advanced level, superposition extends to many waves, wave packets, Fourier analysis, and interference in optics, acoustics, and quantum mechanics. But the two-wave case used here is the correct starting point because it clearly shows how phase controls the final result.
