# 3.4 Nature of Waves

F.5 Physics · Topic IIIB Wave Motion II · Lesson 1

What a wave is, transverse vs longitudinal, the five wave quantities, and reading y-x and y-t graphs.

## Figure

y-x gives λ (a photo of every particle at one instant); y-t gives T (one particle over time); both give A. Longitudinal: forward along the wave is +y.

## Key points

- A **wave** transfers energy from the source to the target **without a transfer of matter**. Waves that need a medium (water, string, sound) are **mechanical waves**.
- **Transverse wave ／ 橫波**: particles vibrate **perpendicular** to the direction of travel (crests and troughs; water waves, light, radio). **Longitudinal wave ／ 縱波**: particles vibrate **to and fro along** the direction of travel (compressions and rarefactions; sound).
- **Period** $T$: time for one complete cycle. **Frequency** $f = 1/T$ (Hz): cycles per second, fixed by the **source** only.
- **Wavelength** $\lambda$: distance between two consecutive crests (or C to C). **Amplitude** $A$: maximum displacement from the equilibrium position; more energy → larger $A$.
- **Wave speed** $v = f\lambda = \lambda/T$, fixed by the **medium** only. Spring: tension ↑ or mass per unit length ↓ → $v$ ↑.
- **Phase**: separation $n\lambda$ → **in phase**; $(n + \tfrac{1}{2})\lambda$ → **antiphase** (180° out of phase).
- Transverse particle motion: crest or trough → **momentarily at rest**; equilibrium position → **fastest**.
- Longitudinal: at a **compression ／ 密部** C or **rarefaction ／ 疏部** R the displacement is **zero** and the particle is fastest, at C **along** the wave, at R **against** it. Mid-way between C and R: largest displacement, momentarily at rest.
- Longitudinal graphs, forward $= +y$: on a **y-x** graph a **falling** zero is **C** and a **rising** zero is **R**; on a **y-t** graph the particle is **at C** where the graph **rises** through zero and **at R** where it **falls**.

## Must know

| Quantity | Symbol / unit | Formula or meaning | Read from |
|---|---|---|---|
| Period | $T$ / s | one complete cycle (one $\lambda$ passes a point) | y-t graph |
| Frequency | $f$ / Hz (= s^−1^) | $f = \dfrac{1}{T}$; source only | y-t graph |
| Wavelength | $\lambda$ / m | crest to crest, C to C | y-x graph |
| Amplitude | $A$ / m | maximum displacement from equilibrium | both graphs |
| Wave speed | $v$ / m s^−1^ | $v = f\lambda = \dfrac{\lambda}{T}$; medium only | neither: calculate |

A y-x graph is a photo of all the particles at one instant; a y-t graph follows one particle, and its slope is that particle's velocity.

## Common mistakes

- ✗ "Higher frequency → faster wave." ✓ The medium fixes $v$. Raising $f$ only shortens $\lambda$, since $v = f\lambda$.
- ✗ Amplitude = crest to trough. ✓ $A$ is measured from the **equilibrium position** to a crest; crest to trough is $2A$.
- ✗ Wavelength = crest to trough, or C to R. ✓ That is only $\lambda/2$. $\lambda$ = crest to crest, or C to C.
- ✗ Reading $T$ or $v$ off a y-x graph. ✓ y-x gives $\lambda$ and $A$; y-t gives $T$ and $A$. Neither gives $v$ directly: use $v = f\lambda$.
- ✗ On a longitudinal y-x graph, a crest is a compression. ✓ Crests are **mid-way** between C and R. With forward $= +y$, C is a **falling** zero and R a **rising** zero.

## Quick check

1. A source vibrates 10 times in each second. Find the period.
2. A y-x graph has consecutive crests at $x = 1$ m and $x = 3$ m and runs from $y = -0.2$ m to $y = +0.2$ m; the source vibrates at 5 Hz. Find $A$, $\lambda$ and $v$.
3. The source in Q2 is changed to 10 Hz. What happens to $v$ and $\lambda$?
4. Two particles on a transverse wave are $1.5\lambda$ apart. Are they in phase or in antiphase?
5. On the y-x graph of a sound wave travelling to the right (forward $= +y$), the curve **falls** through zero at P. Is P a compression or a rarefaction? Give the displacement and the direction of motion of the particle at P.

## Answers

1. $T = 1/f = 1/10 = 0.1$ s.
2. $A = 0.2$ m (not 0.4 m), $\lambda = 2$ m, $v = f\lambda = 5 \times 2 = 10$ m s^−1^.
3. $v$ stays 10 m s^−1^ (medium unchanged); $\lambda$ halves to 1 m.
4. Antiphase: $1.5\lambda$ is an odd multiple of $\lambda/2$.
5. A **compression** C. Displacement zero; the particle moves **to the right**, along with the wave, at its highest speed.

## Did you know?

Earthquakes send out both kinds of wave: longitudinal P-waves arrive first and transverse S-waves later. S-waves cannot travel through liquid, and their absence on the far side of the Earth is how scientists found that the Earth's outer core is liquid.