Waves Add Up Where They Overlap
A wave carries crests and troughs as it travels through space, and when two waves meet at the same place their displacements simply add. This is the principle of superposition. The speed is set by v = fλ, two waves can reinforce or cancel, and slightly different frequencies produce beats. Here we work with real traveling waves, not standing waves.
A traveling wave slides sideways while keeping the shape of its crests and troughs. The wavelength λ is the distance from one crest to the next, and the frequency f is how many crests pass a fixed point each second. Their product v = fλ is the speed at which the wave moves. In a real medium the speed v is set by the medium itself and stays nearly constant, so raising the frequency shortens the wavelength by the same factor. Drag the λ and f sliders and watch how the speed v is set.
When two waves of the same frequency overlap, the real wave at each point is the sum of the two displacements. When the phase difference Δφ is zero, crests line up with crests and the wave doubles in constructive interference. When Δφ is π, one crest meets the other trough and they cancel completely to a flat line. Noise-cancelling headphones use exactly this destructive interference, generating a wave that is π out of phase with the incoming noise to erase the sound at your ear. Turn the Δφ slider and watch the sum curve change.
When two waves of slightly different frequency overlap, they drift in and out of step, so the size of the sum slowly grows and shrinks. This slow swelling is a beat. The beat frequency equals the difference of the two frequencies, |f₁ − f₂|. When tuning an instrument, you sound two notes together and adjust until the beats disappear, which means the two frequencies have become exactly equal. Move the f₂ slider to bring the two frequencies closer and see how much the beats slow down.
Check the superposition principle at a single point. Drag the vertical probe left and right, and at that spot you see the height of each wave and the value of the two added together. At every position, the sum is always just the two heights added. Because the sum is exactly the two heights at every point, after the two waves pass through each other they carry on in their original shapes as if the other had never been there. Move the probe and confirm that the sum of the two values matches the real sum curve exactly.
Superposition holds the same way no matter how a wave shakes the medium. In a transverse wave the medium moves perpendicular to the direction of travel, giving the curve shape we usually draw; in a longitudinal wave the medium moves back and forth along the direction of travel, making crowded and sparse regions. The two pictures are just different drawings of the same wave. Light is a transverse wave and sound in air is a longitudinal wave, yet the superposition principle applies to both in exactly the same way. Press the toggle to see the same shaking appear in two forms.