Mechanical waves

A hump travels along a rope, but a marker tied to it only rises and falls. What reaches the other end if the rope itself does not move there as a whole?

Move the end of a taut rope. Elastic forces make the displaced section pull its neighbour; that section moves and acts on the next one. The disturbance passes from section to section at a finite speed. This is a mechanical wave: the propagation of a disturbance and energy through a medium. Break the connection between sections and the transfer stops. The medium is not a road for a flying piece, but a system of interacting parts.

Tie a bright marker to the rope and watch only that point. In the simple model, it oscillates about its position while the hump travels onward. Wave speed v measures how fast a particular phase of the disturbance moves through the medium, not how fast the marker moves up or down. At an extreme, the marker can stop while the wave continues. Motion of matter and transfer of the wave pattern are different; real waves can have additional effects beyond this simple model.

Compare the direction of wave travel with the direction of particle displacement. On a taut string, transverse oscillations are perpendicular to propagation. In sound travelling through air, particles move along the propagation direction, creating alternating compressions and rarefactions: the wave is longitudinal. A sine curve representing sound often plots pressure, not winding molecular paths. A graph's shape alone does not identify the wave type.

Take an instantaneous snapshot of a periodic wave. The distance along propagation between the nearest points in the same phase is wavelength λ, measured in metres. For a sinusoidal wave, it is the distance between adjacent crests or troughs. A crest and its nearest trough are only λ/2 apart: their states are opposite. Wavelength describes spatial repetition of the pattern, not the length of the entire disturbed rope.

Now watch one marker with a clock instead of taking a snapshot. The time for its full cycle is period T. In that same time, the next crest reaches the previous crest's position, and the pattern advances by λ. Thus v = λ/T. Twice the spatial step in the same time means twice the speed; the same step in twice the time means half the speed. Period measures repetition in time, wavelength repetition in space.

Since ν = 1/T, the relation is also v = λν: metres multiplied by hertz give m/s. The source produces ν cycles per second, each corresponding to a spatial interval λ. The source sets the frequency, while the medium's properties—and in some media, frequency—determine the speed. Wavelength adjusts to these conditions. The formula relates the quantities; it does not say that changing ν always changes v.

Suppose wave speed in the chosen medium stays the same while the source oscillates twice as often. Crests are produced twice as often and travel at the same speed, so their spacing halves: λ = v/ν. At a stationary boundary between two media, frequency stays the same, while a speed change changes wavelength. If speed depends on frequency—dispersion—the simple rule 'twice the frequency, half the wavelength' needs an additional check.

Can this motion be transmitted through empty space? A mechanical wave requires particles and interactions between them; in a vacuum, there is no chain to transmit displacement. Liquids and gases carry longitudinal sound but do not support ordinary transverse elastic waves in their bulk. Surface waves on water are a separate, more complex case: gravity also provides a restoring effect, and particle motion does not fit this simple classification.

A wave travels at 4 m/s, and its source makes 2 oscillations per second. Find the wavelength.

The source makes 2 oscillations per second, so ν = 2 Гц. The wave pattern travels 4 m in one second.

These 4 m contain two spatial repetitions: λ = v/ν = 4/2 = 2 м.

Check using time: T = 1/ν = 0,5 с. In one period, a crest travels vT = 4 · 0,5 = 2 м, exactly one wavelength.

λ = 2 м is the spacing between adjacent crests. Every 0.5 s, the pattern advances by these 2 m.

In a mechanical wave, neighbouring parts of the medium pass on a disturbance; wave speed differs from the speed of an individual particle.

Period T measures repetition of one point's motion in time; λ measures repetition of the pattern in space.

v = λν follows from advancing one wavelength per period; frequency changes must be discussed together with the medium's properties.

Apply these ideas to audible waves and echoes.