Physics › Waves › Progressive waves
Progressive waves
A wave is a disturbance that travels, but the stuff it travels through goes nowhere. Once you separate what moves from what merely oscillates, the whole vocabulary of waves reduces to reading two graphs correctly.
IN THIS TOPIC
- State what a progressive wave transfers, and what the particles of the medium do instead.
- Define amplitude, wavelength, frequency, period and phase difference, and read each from the correct graph.
- Use c = fλ and f = 1/T together to move between speed, frequency, wavelength and period.
WHAT YOU PROBABLY THINK
The wave carries the water along with it.
What travels, and what stays
Watch a gull sitting on sea swell. The waves march steadily towards the beach, but the gull only bobs up and down. It ends the minute where it started. A progressive wave transfers energy through a medium without transferring the medium itself. Each particle oscillates about a fixed rest position, hands its motion to the next particle a moment later, and settles back.
The displacement of a particle is its distance from that rest position, with a direction, so it can be positive or negative. The amplitude A is the maximum displacement, measured from the rest position to a crest, never from trough to crest.
The wavelength λ is the distance between one point and the next point moving identically, crest to crest being the easiest pair to spot. The frequency f is the number of complete oscillations a particle makes each second, measured in hertz, and the period T is the time one oscillation takes. Each is the reciprocal of the other:
In one period, the pattern advances by exactly one wavelength. Distance over time gives the wave speed, and dividing λ by T is the same as multiplying it by f:
Two graphs that look identical
Almost every dropped mark on this topic comes from one habit: not reading the horizontal axis. A wave can be graphed two ways. Plot the displacement of every particle at one instant and you get a displacement-distance graph, a photograph of the wave. The repeat length on it is the wavelength.
Plot the displacement of one particle at every instant instead and you get a displacement-time graph, a diary of a single point. The curve looks the same, but the repeat along it is now the period. Before taking any reading, say to yourself what the horizontal axis is. If it is in metres you can read λ; if it is in seconds you can read T; neither graph hands you both.
Phase
Two points on a wave generally reach their crests at different moments. The phase difference between them measures the mismatch as a fraction of a cycle, expressed in radians, with one whole cycle counting as 2π rad. Points separated by a whole wavelength move identically and are in phase; points separated by half a wavelength always move oppositely and are in antiphase, a phase difference of π rad.
For two points a distance d apart along the same wave, the phase difference is the fraction d/λ of a full cycle:
THE EXAM BIT
- Read the axis before the graph. If a question gives displacement against time, the interval between crests is T, and quoting it as λ scores nothing.
- Amplitude is measured from the rest position. Halve a trough-to-crest reading before writing it down.
- In c = fλ, frequency must be in Hz. Convert kHz and MHz first, and check the answer's size: sound in air near 340 m s−1, light at 3.0 × 108 m s−1.
- Phase difference answers belong in radians unless the paper asks for degrees. A quarter of a cycle is π/2 rad, not “90” with no unit.
- One mark often rides on the definition: a progressive wave transfers energy without transferring the medium.
CHECK YOURSELF
A wave on a rope has a period of 0.020 s and a wavelength of 8.0 m. Find (a) its frequency and (b) its speed.
Show a hint
Which equation links frequency to period? Use its answer in the second equation.
Show the answer
(a) f = 1/T = 1 / 0.020 = 50 Hz.
(b) c = fλ = 50 × 8.0 = 400 m s−1.
The rope itself never travels at 400 m s−1. That is the speed of the pattern, and of the energy it carries; each piece of rope just oscillates fifty times a second about the place it started.
The wave moves on.
The particles only oscillate.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.