IB Physics HLTopic 3 — Oscillations & WavesPaper 1 & 2Amplitude, Period & Frequency~10 min read
Describing Oscillations
A swing going back and forth, a guitar string humming, a mass bouncing on a spring — all of these repeat the same motion over and over. Physicists call this an oscillation, and before we can do any calculations we need a shared vocabulary to describe one. This page sets up that toolkit: displacement, amplitude, period, frequency and angular frequency — the five words you’ll lean on for the whole of this topic.
📘 What you need to know
An oscillation is a repeated back-and-forth motion about a central equilibrium position
Displacement (x) is how far the object is from equilibrium, and in which direction — a vector
Amplitude (x0) is the maximum displacement from equilibrium
Time period (T) is the time for one complete oscillation, in seconds
Frequency (f) is the number of oscillations per second, in hertz, and f = 1/T
Angular frequency (ω) links the two: ω = 2πf = 2π/T, in rad s−1
What is an oscillation?
An oscillation is any motion that repeats itself, swinging back and forth around a fixed central point. That central point is the equilibrium position — the spot where the object would happily sit still because there’s no overall (resultant) force pushing it either way.
Pull the object away from equilibrium and something always tries to drag it back. It overshoots, gets pulled back again, and keeps repeating. We label the equilibrium position x = 0, so displacement is measured from there.
The bob swings between the two extremes A and B, passing through equilibrium O at the bottom. One trip out and all the way back is a single oscillation.
Think of equilibrium as “home base.” The object is never comfortable away from home — there’s always a pull dragging it back. But it’s moving fastest as it rushes through home, so it overshoots to the other side and the whole thing repeats. That restless back-and-forth is the heart of every oscillation you’ll meet in this topic.
Displacement and amplitude
Displacement (x) tells you how far the object has moved from its equilibrium position, and in which direction. Because direction matters, it’s a vector: it can be positive (one side of equilibrium) or negative (the other side), and it’s measured in metres.
The amplitude (x0) is the biggest displacement the object reaches — the distance from equilibrium out to either extreme. A bigger amplitude just means a wider swing. Amplitude is also measured in metres, and it’s always a positive number.
🎯 Displacement vs amplitude — don’t mix them up
Displacement changes constantly through the swing — it’s 0 at the centre and largest at the ends.
Amplitude is a single fixed value — the maximum that displacement ever reaches.
So amplitude is really just “the displacement at the extreme position,” frozen as one number for the whole motion.
Picturing the motion: the displacement–time graph
If we track the displacement of an oscillator over time and plot it, we get a smooth wave. For a simple oscillation with no energy loss, that wave is a sine curve. This single graph shows off three of our key quantities at once.
One full wave = one complete oscillation. Its height gives the amplitude x0; the time for one full cycle is the period T.
Time period and frequency
The time period (T) is the time taken for one complete oscillation — one full there-and-back trip. It’s measured in seconds. If every oscillation takes the same amount of time, the motion is called isochronous (from the Greek for “equal time”).
The frequency (f) is the flip side: it counts how many oscillations happen each second. It’s measured in hertz (Hz), where 1 Hz means one oscillation per second.
Period and frequency are two ways of saying the same thing, so they’re reciprocals of each other:
Frequency and periodf = 1 / T
Where f is frequency in hertz (Hz) and T is time period in seconds (s). If one oscillation takes 2 seconds, then you get half an oscillation per second — that’s 0.5 Hz. Slow motion means a long period and a low frequency; fast motion means a short period and a high frequency.
WE 1
A child on a swing completes 30 full swings in 45 seconds. Calculate the frequency of the oscillation, and the time period.
Step 1 — frequency is oscillations per secondf = 30 ÷ 45 = 0.667 HzStep 2 — period is the reciprocal, T = 1/fT = 1 ÷ 0.667 = 1.5 sf = 0.67 Hz, T = 1.5 sSense check: each swing takes 1.5 s, and 45 ÷ 1.5 = 30 swings. It fits.
Angular frequency
There’s a third way to describe how fast an oscillation goes, and it’s the one the SHM equations actually use: angular frequency (ω). It measures the rate of oscillation in radians per second instead of cycles per second.
Where does the “angular” part come from? An oscillation is deeply linked to going round a circle. One full oscillation matches one full trip around a circle, and one full circle is 2π radians. So to convert “cycles per second” into “radians per second,” we just multiply the frequency by 2π:
Angular frequencyω = 2πf = 2π / T
Where ω is angular frequency in rad s−1, f is frequency in Hz, and T is period in seconds. Both forms give the same answer, since f and 1/T are equal — use whichever quantity the question hands you.
As the point sweeps once around the circle, its height traces one full oscillation. That’s why we measure the rate in radians per second: ω = 2πf.
WE 2
A metronome arm swings with a time period of 0.50 s. Calculate its angular frequency.
Step 1 — use ω = 2π/Tω = 2π ÷ 0.50Step 2 — evaluateω = 12.566… rad s⁻¹ω = 12.57 rad s⁻¹Radians per second, not degrees — keep your calculator in radians for SHM.
WE 3
A buoy bobs up and down on the sea with a frequency of 0.25 Hz. Find its time period and its angular frequency.
Step 1 — period is the reciprocal of frequencyT = 1 ÷ 0.25 = 4.0 sStep 2 — angular frequency, ω = 2πfω = 2π × 0.25 = 1.571 rad s⁻¹T = 4.0 s, ω = 1.57 rad s⁻¹Check: ω = 2π/T = 2π/4.0 = 1.57 rad s⁻¹ too. Both routes agree.
Period T (seconds)
f = 1/T
Frequency f (hertz)
× 2π
Angular freq ω (rad s⁻¹)
The oscillation quantities at a glance
Quantity
Symbol
What it means
Unit
Displacement
x
Distance and direction from equilibrium (a vector)
m
Amplitude
x0
Maximum displacement from equilibrium
m
Time period
T
Time for one complete oscillation
s
Frequency
f
Oscillations per second
Hz
Angular frequency
ω
Rate of oscillation in radians per second
rad s−1
💡 Top tips
f = 1/T and ω = 2πf = 2π/T are the three you’ll reuse constantly — memorise them as a set.
Amplitude is one fixed number; displacement changes every instant. Don’t swap them.
Displacement is a vector — keep the plus or minus sign when you’re told a direction.
When measuring period from a graph, read across one full cycle (top of a peak to the top of the next).
Angular frequency is in radians per second — switch your calculator to radians mode for SHM work.
⚠ Common mistakes
Confusing amplitude (a single maximum value) with displacement (which varies through the swing)
Measuring period across only half a wave — one period is a complete there-and-back cycle
Reading amplitude from the very bottom of a trough to the very top of a peak — that’s twice the amplitude
Forgetting the 2π and writing ω = f — angular frequency is 2π times the frequency
Leaving the calculator in degrees when working with ω and the SHM equations
Quick recap: An oscillation repeats back and forth about equilibrium (x = 0). Displacement x is the vector distance from there; amplitude x0 is its maximum. One full cycle takes a time period T; frequency f = 1/T counts cycles per second; and angular frequency ω = 2πf = 2π/T measures the same thing in rad s−1.
You’ve now got the whole vocabulary for oscillations nailed down. The next step is to look at a very special kind of oscillation — one where the force always pulls back in proportion to the displacement. That’s Simple Harmonic Motion (SHM), and it’s where all these quantities start powering real equations. Head there next.
Starting oscillations & waves? Let’s build it right.
Book a free meeting and we’ll run through amplitude, period, frequency and angular frequency until they’re second nature.
