Wave Explorer

A transverse wave traveling in the +x direction is described by a single formula that captures its shape and motion.

Each parameter controls a different aspect of the wave. A is the amplitude, the maximum displacement from equilibrium. k is the wave number, related to wavelength by $k = \frac{2\pi}{\lambda}$. ω is the angular frequency, $\omega = 2\pi f$. φ is the initial phase, which shifts the wave along the x-axis.

Every periodic wave obeys a single relationship that connects speed, frequency, and wavelength.

Speed is how fast the wave pattern moves through the medium. Frequency is how many complete cycles pass a fixed point per second, measured in hertz. Wavelength is the distance between consecutive crests (or any two corresponding points).

If you increase the frequency while the speed stays fixed, the wavelength must decrease, and vice versa.

When two waves overlap in the same region, their displacements add at every point. This is the principle of superposition.

Constructive interference happens when two waves are in phase. Their amplitudes add together, producing a larger combined wave.

Destructive interference happens when two waves are exactly out of phase. Their amplitudes cancel, and the combined displacement can reach zero.

When two waves have slightly different frequencies, the alternating constructive and destructive interference produces beats, a periodic rise and fall in the combined amplitude.

A standing wave forms when two identical waves travel in opposite directions and interfere. The result looks like it oscillates in place rather than traveling.

Nodes are points that never move. They occur where $\sin(kx) = 0$. Antinodes are points of maximum oscillation, located halfway between nodes.

Neighboring nodes are separated by exactly $\frac{\lambda}{2}$. Standing waves are the basis of musical instruments, where fixed boundary conditions select which wavelengths can exist.