Wave Superposition

§The model

Two sine waves travel along one string. Taking the phase of the first as the reference, their displacements are

$$y_1(x,t) = A_1 \sin(k_1 x - \omega_1 t), \qquad y_2(x,t) = A_2 \sin(k_2 x - \omega_2 t + \Delta\varphi),$$

and the string, which can only be in one place at a time, shows their sum, $y = y_1 + y_2$. That is all superposition is: displacements add.

Both waves travel on the same medium at the same speed $v$, so each angular frequency follows its wave number, $\omega = v\,k$. This is the non-dispersive case — a taut string, sound in air — and it is an assumption of the instrument, not a law. The wavelength is $\lambda = 2\pi/k$.

§What to look for

When the two wave numbers are equal, the sum is again a single sine wave, with amplitude

$$A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\Delta\varphi}.$$

It reaches $A_1 + A_2$ when $\Delta\varphi = 0$ (constructive interference) and falls to $|A_1 - A_2|$ when $\Delta\varphi = \pi$ (destructive); two equal waves in opposite phase cancel completely, and the string lies flat. When the wave numbers differ there is no single amplitude: the sum beats, and the pattern itself travels.

The highlight shades, at each point and instant, whether the two displacements pull the same way or opposite ways. It is honest for any pair of waves, equal wavelength or not.

Units are arbitrary. Nothing here is measured; it is the equations, drawn.