Waves in the real world
Many things repeat in regular cycles: tides, daylight hours, temperatures through the year, a Ferris wheel ride, sound waves, and the motion of a pendulum or spring. These are periodic phenomena, and a sinusoidal model can describe them.
y = a sin(b(x − c)) + d or y = a cos(b(x − c)) + d
Building a model: four steps
| Step | Find | Formula |
|---|---|---|
| 1 | Amplitude a | (max − min) ÷ 2 |
| 2 | Midline d | (max + min) ÷ 2 |
| 3 | b from the period | b = 360° ÷ period (or 2π ÷ period) |
| 4 | Phase shift c | where a cycle starts (a max for cosine, midline going up for sine) |
Example 1: Ferris wheel
A Ferris wheel has a diameter of 40 m; its centre is 22 m above the ground. It turns once every 6 minutes. A rider boards at the bottom at t = 0.
- Max height: 42 m; min: 2 m
- a = 20, d = 22
- Period 6 min → b = 360°/6 = 60
- Starts at the minimum → use a negative cosine with no shift
h(t) = −20 cos(60t) + 22 (t in minutes, angle in degrees)
Check: h(0) = −20 + 22 = 2 ✓; h(3) = −20 cos 180° + 22 = 42 ✓.
Example 2: Tides
At a harbour, high tide is 8.2 m at 2:00 a.m. and low tide is 1.4 m about 6.2 hours later. The tidal period is about 12.4 hours.
- a = (8.2 − 1.4)/2 = 3.4
- d = (8.2 + 1.4)/2 = 4.8
- b = 360/12.4 ≈ 29.0
- Max at t = 2 → cosine with c = 2
D(t) = 3.4 cos(29.0(t − 2)) + 4.8 (t in hours after midnight)
Example 3: Daylight hours
A city has about 15.5 h of daylight on June 21 (day 172) and 8.5 h on December 21.
- a = 3.5, d = 12
- Period 365 days → b = 360/365 ≈ 0.986
- Max at day 172 → D(n) = 3.5 cos(0.986(n − 172)) + 12
Using a model
Once you have the model, you can:
- Predict: the tide height at 9 a.m.
- Solve: when the Ferris wheel rider is above 30 m.
- Interpret: the midline is the average; the amplitude is the variation.
Example: Ferris wheel: when is the rider first at 32 m?
−20 cos(60t) + 22 = 32 → cos(60t) = −0.5 → 60t = 120° → t = 2 min
Worked examples
Example 1: Temperatures range from −10 °C to 26 °C over a year. Amplitude and midline? 18 °C and 8 °C.
Example 2: A spring bobs with a period of 0.5 s. Find b (degrees). 360/0.5 = 720.
Example 3: Using the tide model, the depth at 8:20 a.m. (t ≈ 8.33)? cos(29 × 6.33) = cos(183.7°) ≈ −0.998 → D ≈ 1.4 m (near low tide).
Common mistakes
- Using the full range as the amplitude.
- Using the period as b.
- Mixing degree and radian modes.
- Starting a sine model at a max (that’s cosine).
Try it yourself
- Max 30, min 10. Amplitude and midline?
- Period 24 h. Find b in degrees.
- A wheel of radius 5 m, centre 6 m high, period 20 s, starting at the top. Write h(t).
- A model is y = 4 sin(30(x − 2)) + 10. What is the max value?
Answers: 1) 10 and 20 2) 15 3) h(t) = 5 cos(18t) + 6 4) 14
Quick recap
- A sinusoidal model fits a periodic phenomenon such as tides or daylight.
- Find amplitude, midline, b from the period, and the phase shift.
- Use the model to predict and solve real problems.