Waves: separate amplitude, frequency, and speed

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A crest can move forward while one string point moves up and down. Use spatial and temporal records to keep those motions separate.

About 18 minutes, including the worked examples and checks.

Read the horizontal axis before choosing a wave equation

A photograph of a moving wave and a recording from one fixed sensor can look almost identical. The photograph shows what varies across space; the sensor shows what varies over time. Mixing their axes turns a wavelength into a period and gives a calculation with the wrong units. Periodic motion in AAMC category 4A and light/sound in 4D use this distinction.¹⁵

A wave carries a disturbance from one location to another. In a simple mechanical traveling wave, nearby parts of the medium interact and pass the disturbance onward while oscillating about their local equilibrium positions. The traveling pattern is not a group of particles riding unchanged from source to receiver. Mechanical waves require a medium; electromagnetic waves can propagate through a vacuum.²

Separate the size, spacing, and timing of the disturbance

Amplitude A is the maximum displacement from equilibrium in the displacement model used here. A crest at +3 mm and a trough at −3 mm give A = 3 mm, not 6 mm. The latter is the peak-to-peak displacement. A pressure wave could instead be described by a pressure amplitude, so keep the measured variable and its unit attached to the number.

Wavelength λ is the spatial distance between successive points at the same phase, such as two neighboring crests. Period T is the elapsed time between successive returns of one point to the same phase. Frequency f counts cycles per second: f = 1/T, with unit hertz (Hz). Returning to the same displacement is insufficient if the point is moving in the opposite direction; an equilibrium crossing happens twice per cycle.²

For a steadily traveling periodic pattern, one wavelength passes a fixed point during one period. Therefore v = λ/T = fλ. This v is the propagation speed. The local material velocity is a different quantity and can reverse direction while the pattern continues forward.³

Worked application: combine the photograph and the sensor

The plots and numbers are original illustrative data from an ideal sinusoidal wave moving right on a uniform string. The snapshot is taken at time zero. The sensor follows the string point at x = 0. The chosen wave begins with that point at its positive maximum.

In the snapshot, successive crests are at x = 0 and 0.80 m, so λ = 0.80 m. At the sensor, successive maxima occur at t = 0 and 0.20 s, so T = 0.20 s and f = 5.0 Hz. Combining the two views gives v = (5.0)(0.80) = 4.0 m/s. Dividing the 3 mm amplitude by the period would answer neither the pattern-speed question nor the particle’s average-speed question.

During half a period, the crest pattern advances (4.0 m/s)(0.10 s) = 0.40 m. The material point at x = 0 instead moves from +3 mm to −3 mm, a vertical travel distance of 6 mm. During a full period it travels 12 mm and returns to its starting displacement. Its average vertical velocity over that cycle is zero, even though its average speed is 12 mm ÷ 0.20 s = 60 mm/s.

These observations can coexist because they track different objects: a location in a pattern and a piece of the material. At the top or bottom of its oscillation, the material point is momentarily stationary vertically. That does not stop the crest pattern from propagating.

Transverse and longitudinal describe motion relative to travel

For the ideal string wave, the material displacement is perpendicular to the direction in which the wave travels: it is transverse. In a longitudinal wave, the displacement is parallel to propagation. A longitudinal sound wave in a fluid produces alternating compressed and expanded regions; fluid elements oscillate back and forth along the travel direction.²

The distance between successive compressions is a wavelength. A pressure-versus-position graph of that sound can be sinusoidal, but its vertical graph axis represents pressure, not a literal up-and-down path through the room. Graph orientation does not decide whether a wave is transverse or longitudinal.

Change one condition and keep the others explicit

In the small-amplitude, nondispersive string model, the medium and its mechanical conditions set the propagation speed. Tension and mass per unit length matter; increasing the driver’s frequency alone does not set a new speed.⁴ If the driver doubles from 5.0 to 10 Hz while v remains 4.0 m/s, the new wavelength is 0.40 m. Twice as many cycles fit into each second and each meter of the pattern.

Now keep a 5.0 Hz driver running while a transmitted wave enters a second stationary medium where its speed is 2.5 m/s. The boundary continues to be driven five times per second, so the transmitted frequency remains 5.0 Hz and λ becomes 0.50 m. The transmitted amplitude cannot be calculated from that speed change alone; reflection and the coupling between media also matter.

Amplitude, frequency, and propagation speed are therefore separate measurements. Their independence here is a model statement, not a claim about every wave: nonlinear media and dispersive propagation can require additional relationships. Start by identifying the stated conditions, extract a spatial repeat and a temporal repeat, and only then connect them with v = fλ.

Check your reasoning

Choose an answer before opening its feedback. Each option explains the reasoning behind that choice.

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Sources

1. AAMC: C/P Content Category 4A

2. OpenStax University Physics1: Traveling Waves

3. OpenStax University Physics1: Mathematics of Waves

4. OpenStax University Physics1: Wave Speed on a Stretched String

5. AAMC: C/P Content Category 4D