Progressive Waves
Waves carry energy from one place to another without transporting matter. Think of a ripple that spreads across a pond after you drop a pebble – the water itself doesn’t move far, but the disturbance travels outward. In physics we describe this disturbance with several key terms.
Key Terms & Analogies
- Displacement (𝑥) – the distance a point on the medium moves from its rest position. Imagine a slinky being stretched and released; each coil’s movement is its displacement. 🌈
- Amplitude (𝐴) – the maximum displacement from the rest position. It tells you how “tall” the wave is. In a slinky, it’s the farthest coil moves from its original spot. 🎈
- Phase Difference (Δϕ) – the difference in phase between two points on the wave. If one point is at the peak while another is at the trough, their phase difference is 180°. Think of two friends dancing: one is at the top of a jump while the other is at the bottom. 🕺💃
- Period (𝑇) – the time taken for one complete oscillation. For a pendulum, it’s the time for a full back‑and‑forth swing. ⏱️
- Frequency (𝑓) – how many oscillations occur per second. It’s the inverse of the period: 𝑓 = 1/𝑇. A drum that beats 5 times per second has a frequency of 5 Hz. 🎵
- Wavelength (λ) – the distance between two consecutive points in the same phase (e.g., peak to peak). In a string, it’s the length of one complete wave pattern. 📏
- Speed (𝑣) – how fast the wave travels through the medium. It’s the distance a point on the wave covers per unit time. For a wave on a string, it depends on the string’s tension and mass. ⚡
Important Relationships
| Quantity | Formula | Example |
|---|
| Speed | \$v = \lambda \, f\$ | A wave with λ = 2 m and f = 3 Hz travels at 6 m s⁻¹. ⚡ |
| Period | \$T = \dfrac{1}{f}\$ | If f = 4 Hz, then T = 0.25 s. ⏱️ |
| Phase Difference | \$Δϕ = 2π \dfrac{Δx}{λ}\$ | Two points 0.5 m apart on a λ = 2 m wave have Δϕ = π/2. 🔄 |
Calculating Wave Speed – Step‑by‑Step
- Measure the wavelength (λ) – the distance from one crest to the next. Use a ruler or a marked string. 📏
- Determine the frequency (f) – count how many crests pass a fixed point in one second. Use a stopwatch. ⏱️
- Apply the formula \$v = \lambda \, f\$ to find the speed. Multiply the two numbers. ⚡
- Check units: meters × hertz = meters per second (m s⁻¹). ??
Real‑World Example: Sound Waves in Air
Sound travels as longitudinal waves. In air at 20 °C, the speed of sound is about 343 m s⁻¹. If a tuning fork vibrates at 440 Hz (A4 note), the wavelength is:
\$λ = \dfrac{v}{f} = \dfrac{343}{440} \approx 0.78\,\text{m}\$
So each full wave cycle is roughly 78 centimetres long – about the length of a small table! 🎶
Quick Review Quiz
- What is the amplitude of a wave that reaches 0.3 m above and below its rest position? ➡️ 0.3 m.
- If a wave has a period of 0.2 s, what is its frequency? ➡️ 5 Hz.
- Two points on a wave are 1 m apart. The wavelength is 4 m. What is their phase difference? ➡️ \$Δϕ = 2π \dfrac{1}{4} = \dfrac{π}{2}\$.
Remember: speed = wavelength × frequency. Keep this rule handy when you tackle problems in your physics exams! 🚀