Periodic Waves: Study Guide (Physics 2 14.2)

Waves that repeat

In Topic 14.1 you met the pulse, a single disturbance. Now the source keeps going. A vibrator attached to a string moves up and down again and again at a steady rate. Every piece of the string copies that motion, a little later than the piece before it. The result is a periodic wave: a pattern that repeats in time at any one place, and repeats in space at any one moment.

Two numbers describe the repetition in time:

• The period T is the time for one complete oscillation. Unit: seconds (s).
• The frequency f is the number of oscillations per second, the rate at which the wave repeats. Unit: hertz (Hz), where 1 Hz = 1 s⁻¹.

They are reciprocals:

T = 1/f

If a loudspeaker cone vibrates 1500 times in 3.0 s, f = 1500 ÷ 3.0 s = 500 Hz and T = 3.0 s ÷ 1500 = 2.0 × 10⁻³ s (2.0 ms).

One number describes the repetition in space:

• The wavelength λ is the distance between two successive matching points on the wave, such as one crest and the next crest, or one trough and the next trough. Unit: metres (m).

And, as in Topic 14.1, the amplitude A is the maximum displacement from equilibrium.

Amplitude does not depend on the period or the frequency. You can shake a rope gently or hard at the same rate (same f, different A), or quickly or slowly with the same size of shake (same A, different f). The two are set separately.

Two graphs that look alike

A periodic wave is described by a displacement y that depends on two things: where you look (x) and when you look (t). A graph has only one horizontal axis, so there are two ways to draw it.

• Displacement–position graph (snapshot). Freeze time and plot y against x. This is a photograph of the whole wave. The repeat distance on the horizontal axis is the wavelength λ.
• Displacement–time graph (history). Stay at one location and plot y against t. This is a record of one point moving up and down. The repeat time on the horizontal axis is the period T.

Both graphs show the amplitude as the height of a crest above the axis. The curves can look exactly the same, so always read the label on the horizontal axis first.

Equations for a sinusoidal wave

The curves in Figure 1 are cosine curves. A sinusoidal wave can be written as an equation in two ways, one for each graph.

At one location, as a function of time:

x(t) = A cos(ωt) = A cos(2πft)

Here x(t) means the displacement of the medium at that one location at time t, written the same way as for an oscillating object. It is not a position along the wave. ω = 2πf is the angular frequency, in rad/s. Because f = 1/T, the same equation can also be written A cos(2πt/T).

At one instant, as a function of position:

y(x) = A cos(2πx/λ)

Here x is the position along the wave and y(x) is the displacement there, at the instant of the snapshot.

The two equations use x for different things, so always say what each symbol means. In this guide, from here on, we use y for displacement in both cases, so the time equation is written y = A cos(2πft).

How to read them:

• The number in front, A, is the amplitude.
• The angle 2πft increases by 2π (one full cycle) each time t increases by one period, T = 1/f. So the equation repeats every period.
• The angle 2πx/λ increases by 2π each time x increases by λ. So the snapshot repeats every wavelength.
• Cosine fits a wave that is at +A when t = 0 (or x = 0), as in Figure 1. If the displacement is zero and increasing at the start, use sine instead: y = A sin(2πft). The amplitude, period and wavelength are the same.

Use radians when you evaluate these on a calculator. A quick check: at t = T/4 the cosine form gives cos(π/2) = 0, and at t = T/2 it gives cos(π) = −1, so y = −A. That matches the bottom graph in Figure 1.

Speed, frequency and wavelength

In one period, each point of the medium completes one oscillation, and the pattern moves forward by exactly one wavelength. So the wave speed is

v = λ/T, and since 1/T = f:

v = fλ

This links the three quantities, but they are not all free:

• The medium sets v (Topic 14.1: tension and mass per length for a string, the material and its temperature for sound).
• The source sets f. Every piece of medium oscillates at the frequency of whatever is driving it.
• λ follows: λ = v/f.

So, for a periodic wave:

• λ is proportional to v at a fixed frequency. If the same source sends waves into a faster medium, the waves are longer.
• λ is inversely proportional to f at a fixed speed. Double the frequency in the same medium and the wavelength halves.

Pitch, loudness and energy

For sound, the properties of the wave match what you hear:

• Frequency ↔ pitch. A higher frequency is heard as a higher note. Two notes an octave apart have frequencies in the ratio 2 : 1, and they sound very alike.
• Amplitude ↔ loudness (Topic 14.1). A larger pressure amplitude sounds louder.

These are independent. A quiet high note and a loud low note are both possible.

The energy carried by a wave depends on both:

• It increases with amplitude (Topic 14.1).
• It increases with frequency. To make a rope oscillate through the same amplitude more often each second, your hand has to move faster, so you must supply energy at a greater rate.

Worked example 1: reading the two graphs

Question. Figure 1 shows a wave on a long string. (a) State the amplitude, wavelength and period. (b) Find the frequency and the wave speed. (c) Write the equation for the displacement at x = 0 as a function of time, and use it to find the displacement at x = 0 when t = 0.0025 s. (d) How far does the wave travel in 0.020 s?

1. (a) From the top graph: A = 3.0 cm = 0.030 m; crest to crest is λ = 0.80 m. From the bottom graph: crest to crest is T = 0.020 s.
2. (b) f = 1/T = 1 ÷ 0.020 s = 50 Hz. v = fλ = 50 Hz × 0.80 m = 40 m/s. (Check: λ/T = 0.80 m ÷ 0.020 s = 40 m/s.)
3. (c) At x = 0 the displacement is +A at t = 0, so use cosine with f = 50 Hz: y = (0.030 m) cos(2π × 50t), which is the same as (0.030 m) cos(2πt/0.020 s). At t = 0.0025 s: 2πft = 2π × 50 × 0.0025 = 2π × 0.125 = π/4 rad. y = 0.030 × cos(π/4) = 0.021 m (2.1 cm).
4. (d) Distance = vt = 40 m/s × 0.020 s = 0.80 m, exactly one wavelength in one period.

Interpretation and check. At t = 0.0025 s, one eighth of a period has passed. The point has moved down from 3.0 cm but is still above zero: cos(π/4) ≈ 0.71, so it is at 71% of the amplitude, which agrees with the curve. Part (d) is the reasoning behind v = fλ.

Worked example 2: one note, changing conditions

Question. A tuning fork vibrates at 686 Hz in air where sound travels at 343 m/s. (a) Find the period and the wavelength of the sound. (b) Later the room is warmer and sound travels at 349 m/s. The same fork is struck. What happens to the frequency, the pitch and the wavelength? (c) A second fork sounds the note one octave higher, in the original air. Find its wavelength.

1. (a) T = 1/f = 1 ÷ 686 Hz = 1.46 × 10⁻³ s (1.46 ms). λ = v/f = 343 m/s ÷ 686 Hz = 0.500 m.
2. (b) The fork, not the air, sets the frequency, so f stays 686 Hz and the pitch is unchanged. The speed rises, so λ = 349 ÷ 686 = 0.509 m, slightly longer (by about 0.009 m).
3. (c) One octave higher means double the frequency: f = 1372 Hz. The air is the same, so v = 343 m/s and λ = 343 ÷ 1372 = 0.250 m, half the original.

Interpretation and check. Part (b) uses λ ∝ v at fixed f: a speed increase of 349/343 ≈ 1.017 gives the same factor on λ. Part (c) uses λ ∝ 1/f at fixed v: doubling f halves λ, with no new calculation needed.

Common misconceptions

• Mixing up λ and T. A crest-to-crest distance on a graph is λ only if the horizontal axis is position. If the axis is time, it is T.
• Taking crest to trough as the wavelength, or as the amplitude. Crest to the next trough is half a wavelength horizontally, and twice the amplitude vertically.
• “Higher frequency means a faster wave.” In one medium all frequencies travel at the same speed. The wavelength changes instead.
• “Changing the medium changes the frequency.” The source sets f. When the speed changes, λ changes.
• “A louder sound has a higher pitch.” Loudness depends on amplitude; pitch depends on frequency. They are independent.
• “Amplitude gets smaller when the frequency goes up.” Amplitude is independent of frequency and period.
• Reading the x in x(t) = A cos(2πft) as a position. In that equation x is the displacement of one point of the medium. Position along the wave appears only in the snapshot equation, y(x) = A cos(2πx/λ).
• “The displacement–time graph is a picture of the wave.” It is a record of how one point moves. The shape of the wave at one instant is the displacement–position graph. A point on a transverse wave moves only up and down; it never travels along the curve of either graph.
• Forgetting that f is fixed by the source. If a question says the same source or the same vibrator is used, keep f the same and let λ adjust.
• Calculator in degrees. cos(2πft) needs radians. In degree mode, cos(π/4) gives the wrong value.

Where this leads

Topic 14.3 asks what happens when a periodic wave reaches a boundary between two media. Speed changes, frequency stays the same, and so the wavelength changes: exactly the v = fλ reasoning from Worked example 2. Continue with Boundary Behavior of Waves and Polarization. Before that, test yourself with the practice questions, then use the revision notes and the topic checklist to consolidate.