Waves

6.1

Wave motion

PMDC: Describe the meaning of wave motion as illustrated by vibrations in ropes and springs.

A wave is a travelling disturbance. Shake a rope: the hump runs along, the rope itself only bobs. A slinky can carry a side-to-side hump or a push–pull pulse. Energy and the shape travel. The medium mostly stays.

6.2

Mechanical vs electromagnetic waves

PMDC: Demonstrate that mechanical waves require a medium while electromagnetic waves do not.

Sound, water waves, a rope: they need something to shake. No air, no sound. Light, radio, X-rays are electromagnetic — oscillating E and B fields. They cross vacuum at c.

• mechanical: needs a medium
• EM: travels in vacuum at $c=3$ × 10⁸ m s⁻¹

6.3

Wave words

PMDC: Define and apply: medium, displacement, amplitude, period, compression, rarefaction, crest, trough, wavelength, velocity.

Medium: what the wave travels through. Displacement: how far a particle is from rest. Amplitude A: the maximum of that. Period T: time for one full cycle. Frequency $f=1/T.$

Transverse: crest (top) and trough (bottom). Longitudinal: compression (bunched) and rarefaction (spread). Wavelength $\lambda$: the repeat distance, crest to crest or compression to compression. Wave speed is how fast the pattern travels.

• $T=1/f$
• $\lambda$: crest to next crest
• v: speed of the pattern, not of a particle

6.4

Wave speed

PMDC: Solve problems using v = f λ.

In one second the wave sends f crests past you, each a distance $\lambda$ apart, so the front advances f $\lambda$.

• $v=f\lambda$
• $v=\lambda /T$

6.5

Energy in a progressive wave

PMDC: Describe that energy is transferred due to a progressive wave.

A progressive (travelling) wave carries energy from the source into the medium. The particles do not make that journey. Brighter / louder means more energy per second, not a new colour / pitch.

6.6

Transverse vs longitudinal

PMDC: Compare transverse and longitudinal waves.

Transverse: particles shake perpendicular to the travel direction. Rope, light, water (roughly). Longitudinal: particles shake along the travel direction. Sound in air, a slinky push.

• transverse: displacement ⊥ v
• longitudinal: displacement ∥ v

6.7

Newton’s formula for the speed of sound

PMDC: Explain that the speed of sound depends on the properties of the medium, and describe Newton’s formula.

Sound is faster in a stiffer, lighter medium. Newton treated air as changing slowly enough that temperature stays put (isothermal) and wrote $v=\surd (P/\rho$).

• $Newton:v=\sqrt{P/\rho }$
• stiffer / less $dense\Rightarrow faster$ sound

6.8

Laplace’s correction

PMDC: Describe the Laplace correction in Newton’s formula for the speed of sound in air.

Sound compressions happen too fast for heat to leak. The process is adiabatic, not isothermal. Laplace replaced P by $\gamma$ P, where $\gamma =C_p/C_v$ ≈ 1.4 for air. That lifts the predicted speed to match experiment.

• $Laplace:v=\sqrt{\gamma P/\rho }$
• $\gamma {}_{a}ir\approx 1.4$
• $v_{air}$ ≈ $340\mathrm{m}$ s⁻¹ at room temperature

6.9

Factors affecting speed of sound in air

PMDC: Identify the factors on which the speed of sound in air depends.

Temperature: hotter air, faster molecules, higher v. Humidity: moist air is less dense, slightly faster sound. Wind: adds as a vector. Pressure at constant temperature: P and $\rho$ rise together, so v does not change.

• $v\propto \sqrt{T}(Tinkelvin)$
• v independent of pressure (isothermal)
• humidity ↑ ⇒ v slightly ↑

6.10

Superposition

PMDC: Describe the principle of superposition of two waves from coherent sources.

Where two waves meet, the total displacement is the sum of the two displacements. Coherent sources keep a fixed phase difference, so the sum is stable — a lasting interference pattern.

• $y=y_1+y_2$

6.11

Interference of sound

PMDC: Describe the phenomenon of interference of sound waves.

Two notes of the same frequency: where crests meet crests you get a loud place (constructive). Crest meets trough: a quiet place (destructive). Walk around two loudspeakers and the loudness rises and falls.

• constructive: path $difference=n\lambda$
• destructive: path $difference=(n$ + ½) $\lambda$

6.12

Stationary waves (graphical)

PMDC: Explain the formation of stationary waves using a graphical method.

A travelling wave hits a wall and comes back. Forward and back waves add. At some points they always cancel (nodes). At others they always add at maximum (antinodes). The pattern stands still — a stationary (standing) wave. Draw two opposite waves at $t=0$, T/4, T/2 to see the nodes freeze.

• standing $wave=two$ equal waves, opposite directions

6.13

Nodes and antinodes

PMDC: Define the terms node and antinode.

Node: a point that never moves. Displacement stays zero. Antinode: a point that swings with the full amplitude. Neighbour node-to-node is $\lambda$ /2.

• node to next $node=\lambda$ / 2
• antinode to next $antinode=\lambda$ / 2

6.14

Modes of a stretched string

PMDC: Describe modes of vibration of strings.

A string fixed at both ends must have a node at each end. The longest wave that fits is a half wavelength (fundamental). Next: two halves, three halves, … Harmonics are integer multiples of the fundamental frequency.

• $\lambda {}_n=2L/nn=1,2,3,\dots$
• $f_n=nv/(2L)=n{f}_1$
• $v=\surd (T/\mu$) (tension, mass per length)

6.15

Organ pipes (air columns)

PMDC: Describe formation of stationary waves in vibrating air columns.

Open pipe: antinode at both ends. Fundamental is $\lambda$ /2, same pattern as the string. Closed pipe (one end closed): node at the closed end, antinode at the open end. Fundamental is $\lambda$ /4. Only odd harmonics fit in a closed pipe.

• open pipe: $f_n$ = n v / (2 L) $n=1$, 2, 3, …
• closed pipe: $f_n$ = n v / (4 L) $n=1$, 3, 5, …

6.16

Principle of superposition (again)

PMDC: Explain the principle of superposition.

Same law as 6.10, now used for standing waves: two displacements add at every point. That is all standing waves, beats, and interference are.

• $y_{total}=y_1+y_2$

6.17

Simple harmonic motion

PMDC: Explain SHM and the characteristics of SHM.

SHM is a special oscillation: the restoring force (and so the acceleration) is toward the centre and proportional to the displacement from that centre. A mass on a spring (no friction) is the picture. Displacement, velocity and acceleration are all sine/cosine in time, $90^{\circ }$ out of step.

• $F=-kx$
• $a=-\omega {}^{2}x$
• $T=2\pi \sqrt{m/k}$ (mass–spring)
• $T=2\pi \sqrt{l/g}$ (simple pendulum, small angle)
• $x=A\sin (\omega t+\varphi )$
• $E=\frac{1}{2}$ k $A^2=\frac{1}{2}$ m $\omega$ ² A² (total, conserved)

6.18

Circular motion and SHM

PMDC: Describe that when an object moves in a circle, the motion of its projection on a diameter is SHM.

A particle goes around a circle at constant speed. Watch its shadow on a diameter: the shadow speeds up in the middle and pauses at the ends — exactly SHM. That is why we write $x=A$ cos($\omega$ t) with $\omega$ the same as the circular motion.

• $x=A\cos (\omega t)$
• $v=-A\omega \sin (\omega t)$
• $a=-A\omega {}^2\cos (\omega t)=-\omega {}^{2}x$