Waves: separate frequency, wavelength and travel time

A wave has both a time pattern and a spatial pattern

Frequency counts cycles per second at a fixed location and is measured in hertz. Period is the time for one cycle, so T = 1/f. Wavelength is the distance between corresponding points one cycle apart, such as successive crests on an ideal transverse wave. Wave speed describes how quickly the disturbance propagates. These quantities satisfy v = fλ for a periodic wave, where λ denotes wavelength. Amplitude describes the size of the oscillation and is a different property. Increasing amplitude does not, by definition, mean increasing frequency.

Describe the medium and the motion separately

For an ideal transverse wave on a string, local points move perpendicular to the direction in which the disturbance travels. For an ordinary sound wave in air, the local pressure and density variations are associated with longitudinal oscillation. Wave propagation is not the same as the entire medium moving from source to receiver at the wave speed. The medium's properties help determine speed. The source establishes the frequency of the sustained oscillation. These distinctions matter when a question changes the source, the medium, or both.

Worked example 1: connect space and time

A periodic sound wave travels at 330 m/s in a stated medium and has frequency 220 Hz. Find its wavelength and period. The speed is supplied as a model value, so no assumption about a universal sound speed is needed. Writing hertz as cycles per second helps explain the units rather than treating them as labels added at the end.

• Rearrange v = fλ to λ = v/f. Thus λ = 330/220 = 1.5 m.
• The period is T = 1/220 s, approximately 0.00455 s, or 4.55 milliseconds.
• Check the relationship another way: one wavelength, 1.5 m, travels in one period. Dividing 1.5 m by 1/220 s recovers 330 m/s.
• A frequency of 220 Hz means 220 cycles pass a fixed location each second. It does not mean the wave travels 220 metres per second.

Worked example 2: cross into a different medium

A steady source produces a 50 Hz wave. In the first medium, wave speed is 100 m/s. After transmission across a stationary boundary, the speed is 75 m/s. Assume a linear wave model and no frequency-changing process at the boundary. Find the wavelength on each side. The source's temporal rhythm continues across the boundary, while a different propagation speed changes the spatial spacing.

• In the first medium, wavelength is 100/50 = 2 m.
• The transmitted frequency remains 50 Hz under the stated conditions. It is not replaced by the new speed.
• In the second medium, wavelength is 75/50 = 1.5 m.
• The speed and wavelength both decrease by a factor of 0.75, while their ratio remains the same frequency. The amplitude may also change at a boundary, but that cannot be calculated from these data alone.

Travel-time questions need a complete route

For a pulse traveling at constant speed, distance equals speed times elapsed time. An echo includes travel from the source to a reflecting surface and back. If the outward and return distances are equal, the source-to-surface distance is half the total travel distance. This factor comes from the route, not from a special sound formula. A one-way signal between two different locations does not automatically need the same division. Read what event starts and stops the clock before substituting into d = vt.

Misconception: a higher frequency always travels faster

Within a model where wave speed is fixed by the medium, increasing frequency shortens wavelength rather than increasing speed. Some real media are dispersive, so speed can depend on frequency, but that relationship must be supplied or justified; it should not be assumed from the equation alone. Also distinguish wavelength from amplitude. A picture with taller peaks can represent greater amplitude while keeping the distance between peaks and the frequency unchanged.

Sources & further reading