3.3.1.1 - Progressive Waves

What a progressive wave is

A progressive wave transfers energy from one place to another without carrying matter along with it. In a mechanical wave, the particles of the medium oscillate about fixed equilibrium positions. In an electromagnetic wave, it is the electric and magnetic fields that oscillate. A marked piece of string moves locally while a crest passes it: the crest is a moving pattern, not a piece of material. Electromagnetic waves need no material medium.

To describe a wave properly, we start with the motion of a single particle. The displacement of a particle is its distance and direction from equilibrium at a particular instant. The largest displacement reached is called the amplitude.

The next key distance is the wavelength. For a progressive wave, points one wavelength apart are in the same stage of vibration, so they have the same displacement and the same direction of motion at the same time.

For a transverse wave, wavelength can be measured from crest to crest or trough to trough. Amplitude is measured from the equilibrium line to a crest or to a trough, not from crest to trough. In the diagram below, notice that amplitude is a vertical measurement from the equilibrium line to one crest, while wavelength is the horizontal distance between two matching points on successive cycles.

[DIAGRAM: asset_name: 3.1.1 - Progressive Waves - Diagram 1; asset_slug: 3.1.1 - Progressive Waves - Diagram 1; recommended_method: retained_png; caption: Amplitude and wavelength on a displacement–distance snapshot; description: A transverse wave with the horizontal axis labelled distance and the vertical axis labelled displacement. Mark the equilibrium line, the amplitude A from the equilibrium line to a crest, and the wavelength lambda between two adjacent crests.]

Frequency, period and wave speed

The timing of the oscillation is described by frequency and period. Frequency tells you how many complete oscillations happen each second, while period tells you how long one complete oscillation takes.

Because one quantity counts oscillations per second and the other gives the time for one oscillation, they are reciprocals of each other.

Here, $f$ is the frequency in hertz (Hz) and $T$ is the period in seconds (s). A large frequency means a small period, and a small frequency means a large period.

Wave speed is the distance travelled by the wave pattern each second. In one period, the wave advances by one wavelength, so the speed is wavelength divided by period.

Combining that idea with $f = 1/T$ gives the standard wave equation used throughout this topic.

In this equation, $c$ is the wave speed in $\mathrm{m\,s^{-1}}$, $f$ is the frequency in Hz, and $\lambda$ is the wavelength in m. This equation applies to all progressive waves.

For example, if 40 complete cycles pass in $0.50\,\mathrm{s}$, then $f=40/0.50=80\,\mathrm{Hz}$ and $T=1/80=0.0125\,\mathrm{s}$. If their wavelength is $0.25\,\mathrm{m}$, the disturbance moves at $20\,\mathrm{m\,s^{-1}}$. This is not the speed of an oscillating particle: its speed changes during each cycle.

When a source changes frequency while the medium and wave speed remain unchanged, $\lambda=c/f$ decreases as $f$ increases. Do not assume speed is constant if the medium itself changes.

Phase and phase difference

To say where a particle is in its oscillation, we use phase. This is more precise than just saying "moving up" or "moving down" because it identifies the particle's position within the cycle.

Two points are in phase if they have the same displacement and are moving in the same direction at the same time. Two points are in antiphase if they are half a cycle apart, so when one has maximum positive displacement the other has maximum negative displacement.

One full cycle is $2\pi$ rad, $360^\circ$, or 1 cycle. So a quarter of a cycle is $\pi/2$ rad, $90^\circ$, or 1/4 of a cycle. A half-cycle difference is $\pi$ rad, $180^\circ$, or 1/2 of a cycle.

For two points separated by a distance $d$ on a wave of wavelength $\lambda$, the phase difference is:

The formula gives the phase change accumulated over the stated distance. Phase repeats every $2\pi$: $270^\circ$ is equivalent to $-90^\circ$ for a directed comparison. If asked for the smallest angular separation, use $90^\circ$ in that example. This relation is useful because it converts a measured distance along the wave into an angular difference. If $d = \lambda$, the phase difference is $2\pi$ rad, so the points are in phase again. If $d = \lambda/2$, the phase difference is $\pi$ rad, so the points are in antiphase. In the diagram below, notice how each quarter-wavelength step changes the phase, so O and S line up in phase again while O and Q are in antiphase.

[DIAGRAM: asset_name: 3.1.1 - Progressive Waves - Diagram 2; asset_slug: 3.1.1 - Progressive Waves - Diagram 2; caption: Equal quarter-wavelength steps correspond to quarter-cycle phase changes; description: Exact cosine snapshot with O, P, Q, R and S at distance 0, one quarter, one half, three quarters and one wavelength. The vertical axis is displacement, with the same amplitude throughout.]

Equal displacement alone does not prove equal phase. Points P and R in the graph are both at equilibrium but will move in opposite directions. At an extremum, compare the stage of the cycle rather than requiring a nonzero velocity.

Reading space and time graphs

A displacement–distance graph is a snapshot of many particles at one instant. Its horizontal repeat distance is $\lambda$. A displacement–time graph follows one particle; its horizontal repeat interval is $T$. Both show amplitude on the vertical axis, but only the first shows the shape of the string.

[DIAGRAM: asset_slug: progressive-space-time; caption: Distance gives wavelength, while time gives period; description: Exact displacement–distance and displacement–time plots for a right-moving wave with amplitude 2 mm, wavelength 0.80 m and period 0.040 s. The time trace follows the particle at x = 0.]

The two plots describe the same wave: $f=25\,\mathrm{Hz}$ and $c=0.80/0.040=20\,\mathrm{m\,s^{-1}}$. Read the axis before choosing an equation. A steep slope on the time graph means a large particle velocity; a slope on the distance graph is the shape of the string, not the wave speed.

For a pattern travelling right, the part of the pattern just to a particle's left arrives next. Thus a particle on a rising section of the snapshot moves down next; on a falling section it moves up. Reversing the travel direction reverses this conclusion.

The same space–time idea can measure the speed of sound. Place two microphones a measured distance $x$ apart along the direction of a short sound pulse. Record their signals on two oscilloscope channels, identify the same feature and measure its delay $\Delta t$. Then $c=x/\Delta t$. Repeat for several separations and plot delay against separation: gradient $=1/c$. Larger separation reduces the relative timing uncertainty, provided both pulses remain clear and reflections are excluded.