Relate wavelength, frequency and wave speed

Count how far one cycle travels

Suppose a crest advances one wavelength during one period. If wavelength is 2 m and period is 1 s, the pattern travels at 2 m/s. Wave speed c = λ/T = λf, where λ is wavelength in metres, T is period in seconds and f = 1/T is frequency in cycles per second, or hertz (Hz). The symbol c here means this wave’s pattern speed; it does not mean the speed of light.

Read the relation in words: metres per cycle times cycles per second gives metres per second. Use this dimensional check to distinguish frequency from period. A frequency of 2 Hz means two cycles each second; its period is 0.5 s.

Hold the medium’s speed fixed

The browser fixes rightward pattern speed at 2 m/s for every wavelength. Compare three settings at time zero:

• λ = 1 m gives f = 2 Hz and T = 0.5 s.
• λ = 2 m gives f = 1 Hz and T = 1 s.
• λ = 4 m gives f = 0.5 Hz and T = 2 s.

Capture at least two settings. Longer wavelength produces fewer cycles in the fixed spatial window and fewer cycles per second at the marked point. Amplitude remains 0.050 m throughout. Changing wavelength selects a different steady wave; it does not describe abruptly changing a source while an older wave is still propagating.

The time slider advances in 0.25 s steps. At wavelength 1 m, these steps are half a period apart, so the marked point can appear at zero displacement at every offered time even though it is moving between them. Its velocity alternates direction, and the history curve, sampled every 0.025 s, shows the intervening peaks. A sparse set of observations can hide motion; inspect the full history rather than concluding that the point is stationary.

For a source driving an ideal nondispersive medium whose conditions remain fixed, changing frequency changes wavelength to maintain the same speed. This is a conditional statement. Real materials can be dispersive, and changing tension, density or other properties can change wave speed. This model does not calculate those material effects.

Separate pattern speed from material velocity

Pattern speed describes a phase feature moving horizontally. The material point at x = 2 m moves vertically in this idealization. Its velocity changes sign and reaches zero at displacement extremes even though the pattern speed stays 2 m/s.

At wavelength 2 m and time zero, the marked point crosses equilibrium downward at approximately −0.314159 m/s. At 0.25 s it is at a trough with velocity zero and upward acceleration approximately +1.973921 m/s². The whole pattern has continued rightward during that interval. Zero velocity at an instant does not imply zero acceleration.

At wavelength 1 m, the same time-zero point has transverse velocity approximately −0.628319 m/s. At wavelength 4 m, it has +0.157080 m/s: changing wavelength also changes the phase at x = 2 m, so a sign change here is not evidence that the wave reversed direction. Compare full cycles or peak speeds when investigating frequency effects. The peak transverse speed is 2πfA, proportional to frequency at fixed amplitude.

Optional: interpret the derivative relation

For the rightward wave y = A sin(kx − omega t), k = 2π/λ is angular wave number in radians per metre and omega = 2πf is angular frequency in radians per second. At fixed x, transverse velocity is −A omega cos(kx − omega t). At fixed t, spatial slope is A k cos(kx − omega t). Thus transverse velocity equals −c times spatial slope for this rightward pattern, with c = omega/k. This does not equate material velocity with wave speed; the slope factor and different directions matter.

What can be concluded about transport?

The local point returns to the same position after each cycle in this model. A moving pattern therefore does not require that point to travel with the crest. Mechanical waves can transfer energy through interactions between neighboring parts of a medium, but this prescribed-displacement model does not calculate energy, power, forces or how a source supplies them. Do not infer an energy flux from the displacement plots alone.

Transfer beyond the control range

A wave has wavelength 3 m and frequency 2 Hz. Calculate its speed. If a different medium supports the same driving frequency at half that speed, what wavelength would be expected, assuming a steady transmitted wave with that frequency?

Check the wave speed transfer

The first speed is 3 × 2 = 6 m/s. Half that speed is 3 m/s, so wavelength becomes 3/2 = 1.5 m. The frequency is specified as unchanged in this question; the calculation does not determine transmission amplitude, reflection or material properties. These conditions are outside the browser's fixed 2 m/s setup.

OpenStax mathematics of waves develops the wave function and the distinction between propagation velocity and the velocity of the medium. The browser examples and values here are synthetic teaching cases.