Describing travelling waves

Two graphs, two different slices of a wave

A travelling wave carries a disturbance and transfers energy between locations. In the usual small-amplitude mechanical model, a particle oscillates about its equilibrium position instead of accompanying the pattern across the whole medium. Amplitude describes its greatest displacement; wavelength is the shortest spatial repeat distance between points at the same phase. Speed describes the motion of that repeating pattern, not the instantaneous speed of one particle.

A displacement–distance graph is a snapshot of many locations at one instant. Adjacent crests on that graph are one wavelength apart. A displacement–time graph follows one location, so adjacent crests are one period apart. Frequency counts complete cycles per second and equals the reciprocal of period. Read units explicitly: milliseconds require conversion before using frequency in hertz.

During one period a travelling pattern advances one wavelength, which gives v = λ/T = fλ. Changing source frequency need not change speed in an unchanged approximately nondispersive medium; instead wavelength adjusts. In a dispersive medium different frequencies can propagate at different speeds, so identify which speed a question supplies. Prefixes are case-sensitive: MHz means 10⁶ Hz whereas mHz means 10⁻³ Hz.

f = 1/T; v = fλ; 1 kHz = 10³ Hz; 1 MHz = 10⁶ Hz

Symbols and units

• T is period in s; f is frequency in Hz = s⁻¹; λ is wavelength in m; v is propagation speed in m s⁻¹.

How the equation explains the idea

1. One complete cycle takes T, so the number of cycles in one second is f = 1/T. In one period a fixed phase travels one wavelength, giving v = λ/T = fλ.
2. At fixed wave speed, doubling frequency halves wavelength. A displacement–time graph gives T, whereas a simultaneous displacement–position graph gives λ.

Worked example

A surface-wave snapshot and trace

1. A crest-to-crest distance is 0.080 m and the period at one point is 0.20 s.
2. Frequency = 1/0.20 = 5.0 Hz.
3. Wave speed = 5.0 × 0.080 = 0.40 m s⁻¹.

The pattern moves at 0.40 m s⁻¹; this is not a particle drift speed.

Keep this idea. Use the graph axes first; a wavelength divided by a period gives propagation speed.

Compression, displacement and transverse motion

A transverse disturbance has displacement perpendicular to its propagation direction. A stretched string can move vertically while a pulse travels horizontally. A longitudinal disturbance has displacement parallel to propagation. In air, a sound wave produces regions of increased pressure and density called compressions, separated by rarefactions. Molecules oscillate to and fro; the compression pattern progresses even though there is no sustained transport of the same molecules from loudspeaker to ear.

Graphs can show either kind of wave as a sinusoid. For longitudinal sound, the vertical axis might be molecular displacement along the propagation direction, or pressure change above ambient pressure. It is not automatically a literal sideways photograph of the medium. A compression occurs where neighbouring displacements crowd molecules together. At its centre displacement from equilibrium can be zero even while pressure change is greatest; maximum displacement and maximum pressure do not occur at the same position.

For a small sinusoidal progressive sound wave, displacement and pressure-change snapshots differ by one quarter of a cycle in position. The sign of this phase relation depends on the propagation and displacement conventions; the important construction is whether neighbours converge or separate. Distinguish pressure variation from absolute pressure: a negative pressure change means below ambient pressure, not a physically negative absolute pressure. Use arrow descriptions if drawing a molecular diagram is difficult.

Transverse: displacement ⟂ propagation. Longitudinal: displacement ∥ propagation. Compression means increased local density.

Worked example

Follow three neighbouring parcels

1. At one instant the parcel on the left is displaced rightward and the parcel on the right is displaced leftward.
2. Their separation decreases, so the central region becomes denser.
3. This is a compression even if the centre parcel is momentarily at its equilibrium position.

Density depends on neighbouring relative displacements, not on one parcel’s displacement alone.

Keep this idea. Classify a wave using actual particle displacement, then read whether its graph shows displacement or pressure.

Core practical 4: sound speed from two traces

Connect a signal generator to a loudspeaker and compare its signal with a microphone output on a two-beam oscilloscope. Keep frequency steady and place the microphone on a ruler aligned with the sound path. Adjust the display so several complete cycles and both traces are clear. The electronic and acoustic signals can have an unknown fixed phase offset, so one apparently aligned position does not necessarily mark a whole wavelength from the speaker.

Record a microphone position at a chosen repeatable phase relation, then move it away through several successive returns to that same relation. If N full phase recurrences occur over displacement Δx, wavelength is Δx/N. Measure frequency from generator calibration or period from the oscilloscope time base, then calculate v = fλ. Alternatively use two microphones separated by a measured distance and a time delay sufficiently unambiguous to identify the same event.

Reflections from walls or benches can create additional waves and distorted traces. Use moderate sound level, a clear path and consistent microphone orientation; avoid unnecessarily loud sound. Measure several intervals and repeat the positioning rather than quoting only the ruler resolution. A large total distance reduces fractional endpoint uncertainty, but count the number of intervals rather than the number of marked positions. Record room conditions because the speed of sound depends on the air.

λ = Δx/N; v = fΔx/N; f = 1/(horizontal divisions × time per division)

Symbols and units

• Δx is microphone travel in m across N complete repeats of phase; N is a dimensionless integer.
• f is source frequency in Hz, λ is wavelength in m and v is sound speed in m s⁻¹.

How the equation explains the idea

1. Successive equal-phase positions are one wavelength apart. Across N complete intervals, Δx = Nλ; divide by N before using v = fλ.
2. For five intervals over 1.70 m at 1000 Hz, λ = 0.340 m and v = 340 m s⁻¹. Counting six positions gives five intervals, not six.

Worked example

Count intervals between phase matches

1. A microphone moves 1.020 m through three complete returns to the same phase at 1000 Hz.
2. Wavelength = 1.020/3 = 0.340 m.
3. Speed = 1000 × 0.340 = 340 m s⁻¹.

There are three wavelength intervals, even though four matching positions could be marked.

Keep this idea. Measure multiple phase recurrences and divide by the interval count, while checking reflections and time-base calibration.