A Wave Crosses the Pond but the Floating Leaf Stays Put

Drop a stone into a pond and ripples race outward. A leaf floating on the water bobs up and down but does not ride along with them. What moves across the pond is not water but a disturbance — and the energy it carries.

Sound, light, the vibration of a string and earthquake tremors are all waves of this kind.

This part covers energy transport by waves, transverse and longitudinal waves, the progressive wave equation, and the relation between speed, frequency and wavelength.

In a wave, each particle of the medium only oscillates about its own mean position and passes the disturbance to its neighbour, so energy and momentum travel through the medium while the matter stays in place on average.

Worked example — wave speed versus particle motion. A ripple crest travels $1.5$ m across a pond in $3$ s, while a leaf on the surface moves up and down through only $1$ cm.

$$v_{wave}=\frac{1.5}{3}=0.5\text{ m/s}$$

The crest crosses the pond, yet the leaf ends up where it started. What reaches the far bank is energy, enough to rock a second leaf there.

Waves need:

- A source of disturbance
- A medium whose particles are linked, so each drags on its neighbour — except electromagnetic waves, which need no medium

An everyday example. A wave of spectators standing and sitting in a cricket stadium sweeps round the ground, though every person stays in their own seat.

The substance. Waves also carry information — speech, radio signals and phone calls all travel as waves.

In a transverse wave the particles oscillate perpendicular to the direction the wave travels, forming crests and troughs; in a longitudinal wave they oscillate along that direction, forming compressions and rarefactions.

- Transverse — waves on a stretched string, ripples on water; they need a medium that resists change of shape, so through a material body they travel only in solids
- Longitudinal — sound in air, pushes along a spring; they need a medium that resists change of volume, so they travel in solids, liquids and gases

Worked example — spacing of compressions. In a sound wave with wavelength $0.68$ m:

- Successive compressions are $0.68$ m apart
- A compression and the next rarefaction are $\frac{0.68}{2}=0.34$ m apart

For a transverse wave of wavelength $2$ m, a crest and the next trough are $1$ m apart.

An everyday example. An earthquake sends out both kinds — longitudinal waves that pass through rock and liquid, and slower transverse waves that cannot cross the liquid part of Earth's interior.

The substance. Liquids and gases cannot carry transverse waves through their bulk, because they do not resist being sheared.

**A progressive wave moving along $+x$ is $y(x,t)=a\sin (kx-\omega t+\varphi )$, where $a$ is the amplitude, $k=\frac{2\pi }{\lambda }$ the angular wave number, $\omega =\frac{2\pi }{T}$ the angular frequency, $(kx-\omega t+\varphi )$ the phase and $\varphi$ the initial phase.

A plus** sign between $kx$ and $\omega t$ means the wave moves along $-x$.

Worked example. $y=0.02\sin (2x-10t)$, in SI units.

$$a=0.02\text{ m}k=2\text{ rad/m}\Rightarrow \lambda =\frac{2\pi }{2}\approx 3.14\text{ m}$$

$$\omega =10\text{ rad/s}\Rightarrow T=\frac{2\pi }{10}\approx 0.628\text{ s},f\approx 1.59\text{ Hz}v=\frac{\omega }{k}=5\text{ m/s along }+x$$

Phase difference. Two points $0.5$ m apart differ in phase by $k\Delta x=2\times 0.5=1$ rad.

Particle speed. The largest speed of any particle is $a\omega =0.02\times 10=0.2$ m/s — much less than the wave speed of $5$ m/s.

An everyday example. Shaking one end of a rope tied to a pole in a steady rhythm sends a sine-shaped wave running along it.

The substance. Particle velocity and wave velocity are different things — one is $\frac{\partial y}{\partial t}$, the other is $\frac{\omega }{k}$.

**In one period $T$ a wave moves forward by one wavelength $\lambda$, so its speed is $v=\frac{\lambda }{T}=f\lambda =\frac{\omega }{k}$.

Worked example 1 — sound.** A $256$ Hz note in air at $340$ m/s:

$$\lambda =\frac{v}{f}=\frac{340}{256}\approx 1.33\text{ m}$$

Worked example 2 — radio. A $100$ MHz FM signal travelling at $3\times 10^8$ m/s:

$$\lambda =\frac{3\times 10^8}{1\times 10^8}=3\text{ m}$$

Worked example 3 — a string. A wave travels $12$ m along a string in $0.4$ s, and its crests are $0.6$ m apart:

$$v=\frac{12}{0.4}=30\text{ m/s}f=\frac{30}{0.6}=50\text{ Hz}$$

An everyday example. **Thunder arriving $3$ s after a lightning flash** means the storm is about $340\times 3\approx 1$ km away.

The substance. When a wave passes into a new medium, its frequency stays the same — set by the source — while its speed and wavelength change together.