Wave Motion: notes and previous year questions

1What is a wave?

Shake the end of a rope and a wave runs along it, but each bit of rope only moves up and down about its own place. A wave is a disturbance that travels, carrying energy from place to place without carrying the medium along.

• A leaf on a pond bobs as ripples pass; it is not carried across.
• Sound crossing a room: the air only moves back and forth a tiny distance; the disturbance crosses the room.

2Kinds of waves

Mechanical waves (sound, water waves, seismic waves) need a medium and cannot cross a vacuum. Electromagnetic waves (light, radio, X-rays) need no medium; in a vacuum they all travel at $c = 3 \times 10^8$ m/s. So we see lightning before we hear thunder, and we can see the Sun but not hear it.

In a transverse wave the medium moves at right angles to the wave (a string, light). In a longitudinal wave it moves along the wave, making compressions (crowded) and rarefactions (spread out): sound in air, a slinky pushed in and out, ultrasound in water.

For sound we can graph the displacement s of each air layer, or the extra pressure ΔP. The pressure is largest at a compression, where the layers on both sides have moved in and the displacement itself is zero: the pressure wave is a quarter wave (90°) out of step with the displacement wave.

3Describing a wave

• Wavelength $\lambda$: the distance between neighbouring points in the same step, such as crest to crest (m).
• Amplitude $A$: the largest displacement from the middle. The energy carried grows as $A^2$.
• Period $T$: the time for one full cycle of any particle, which does SHM.
• Frequency $f = 1/T$: waves passing each second (Hz); $\omega = 2\pi f$.

In one period the wave moves forward exactly one wavelength, so:

4The wave equation

Which way? At a crest the phase $kx - \omega t$ stays fixed, so as $t$ grows, $x$ grows: $kx - \omega t$ moves toward $+x$. With $kx + \omega t$, $x$ shrinks as $t$ grows: the wave moves toward $-x$. Any shape $f(x - vt)$ moves toward $+x$ at speed $v$, and $g(x + vt)$ toward $-x$; both satisfy the wave equation $\partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2$.

Example: $k = 4$ rad/m and $\Delta\phi = \pi/3$ give $\Delta x = \Delta\phi/k = \pi/12 \approx 0.26$ m (here $\lambda = \pi/2$ m).

5Particle velocity and wave velocity

Each particle moves up and down with a changing velocity: largest, $A\omega$, as it passes the middle, and zero at a crest or trough. The wave velocity $\omega/k$ is fixed by the medium. For a wave moving right, a particle just ahead of an arriving crest is moving up. Example: $y = 0.02\sin(30x - 400t)$ has largest particle speed $0.02 \times 400 = 8$ m/s but wave speed $400/30 \approx 13.3$ m/s.

6Wave speed in different media

The speed is set by the medium: a restoring stiffness over an inertia.

In air, $v \approx 331 + 0.6\,t$ m/s with $t$ in °C: about 343 m/s at 20 °C.

7Energy, power and intensity

Each metre of string holds on average energy $\tfrac{1}{2}\mu\omega^2A^2$, carried forward at speed $v$:

8Reflection and transmission

• Fixed end: the pulse comes back upside down, a phase change of $\pi$.
• Free end (a ring sliding on a pole): it comes back the right way up, with no phase change.
• In both, the amplitude stays the same.

At a join between media with speeds $v_1$ (incoming side) and $v_2$:

Into a heavier, slower string ($v_2 = v_1/2$, e.g. $\mu_2 = 4\mu_1$ at the same tension): $A_r = -A/3$ (inverted) and $A_t = 2A/3$. Into a lighter, faster string the reflection is upright. The frequency stays the same, so the wavelength changes with the speed: from air into water (about 4.4 times faster) $\lambda$ becomes about 4.4 times longer.