Wave Propagation: Speed, Wavelength, Frequency

A mechanical wave is a disturbance that propagates through a material medium (string, water, air) without carrying matter: only energy travels. Imagine a taut rope that you shake once: the bump moves along the rope, but each point of the rope merely goes up and then back down in place. This is the fundamental principle to remember before any calculation.

Three quantities describe a periodic wave. The period T is the duration of one complete cycle, expressed in seconds. The frequency f, in hertz, is its inverse: f = 1/T, the number of cycles per second. The wavelength lambda (written λ) is the distance between two identical points of the wave at a given instant, for example between two successive crests, expressed in meters.

The wave speed v is the speed at which the disturbance moves through the medium. It depends on the properties of the medium (tension and linear mass density for a string, compressibility and density for air) but absolutely not on the frequency of the source. The relation linking all this is central: v = λ × f, or equivalently λ = v × T. Always check the units: if v is in meters per second and f in hertz, then λ comes out in meters.

Let us take a simple numerical example. Sound travels through air at about 340 meters per second. For a frequency of 170 hertz, the wavelength is λ = v / f = 340 / 170 = 2 meters. If you double the frequency without changing the medium, the speed stays the same but the wavelength is halved: confusing frequency and wave speed is a classic pitfall, even though they do not depend on the same parameters.

We also distinguish two families of waves according to the direction of vibration relative to propagation: a wave is transverse when the disturbance is perpendicular to the direction of propagation (string, water waves), and longitudinal when it is parallel (sound in air, made of successive compressions and rarefactions).

wavefronts emitted by a source S (circular waves)

            . . . . .
         .             .
       .     . . .       .
      .    .       .       .
      .   .    S     .     .   -->  (direction of propagation)
      .    .       .       .
       .     . . .       .
         .             .
            . . . . .

distance between two consecutive fronts = lambda
v = lambda x f   (wave speed = wavelength x frequency)