Coaching Centre
Waves and thermodynamics review
A summary of Module 3 Waves and Thermodynamics, the key equations and how they connect, key words, and a mixed HSC-style test covering wave properties, wave behaviour, sound, the ray model of light and thermodynamics.
Use this page to check the whole module. Read the summary, make sure you can explain where each equation comes from, then do the test without looking back.
Module summary
- 3A Mechanical and EM waves. Waves transfer energy, not matter. Mechanical waves need a medium, which sets their speed; EM waves travel through a vacuum at $c = 3.00 \times 10^8$ m/s. Transverse: vibration at right angles to travel (crests, troughs). Longitudinal: parallel (compressions, rarefactions).
- 3B Wave graphs. Displacement–distance (snapshot): $\lambda$, $A$. Displacement–time (one particle): $T$, $A$. $v = f\lambda$, $T = 1/f$. Source sets $f$, medium sets $v$. Shift a snapshot to find particle motion.
- 3C Reflection, refraction, diffraction. $\theta_i = \theta_r$ from the normal. Refraction: speed changes, $f$ fixed, $\lambda$ changes, towards normal when slowing. Diffraction greatest when gap ≈ $\lambda$; no change in $v$, $f$, $\lambda$.
- 3D Superposition, standing waves, resonance. Displacements add. Standing waves: nodes and antinodes, $\lambda/2$ apart, no net energy transfer. Resonance: maximum amplitude when $f_{\text{driving}} = f_{\text{natural}}$; damping lowers and broadens the peak.
- 3E Sound. Longitudinal pressure wave; pressure a quarter-wavelength out of step with displacement. Pitch ↔ frequency, loudness ↔ amplitude. $I = P/4\pi r^2$, $I_1r_1^2 = I_2r_2^2$. Beats at $|f_2 - f_1|$ around the average pitch. Doppler $f' = f\frac{(v_{\text{wave}} + v_{\text{observer}})}{(v_{\text{wave}} - v_{\text{source}})}$.
- 3F Harmonics. String and open pipe: $f_n = nv/2L$, all $n$. Closed pipe: $f_n = nv/4L$, odd $n$. Resonance tube: $v = 2f(L_2 - L_1)$.
- 3G Mirrors and lenses. Standard rays; real (inverted, on a screen) vs virtual (upright). $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$, $M = \frac{h_i}{h_o} = -\frac{v}{u}$.
- 3H Snell and TIR. $n = c/v$, $n_1\sin\theta_1 = n_2\sin\theta_2$, $\sin\theta_c = n_2/n_1$. Fibres: core of higher $n$, angle above $\theta_c$. Dispersion: $n$ depends on colour. Light also obeys $I_1r_1^2 = I_2r_2^2$.
- 3I Temperature and specific heat. Temperature ∝ average particle KE; K = °C + 273.15. Heat flows hot to cold until thermal equilibrium. $Q = mc\Delta T$; mixtures: energy lost = energy gained. Electrical method $Q = VIt$.
- 3J Latent heat and transfer. $Q = mL$ at constant temperature. Heating curves. Conduction $Q/t = kA\Delta T/d$; convection by moving fluid; radiation by infrared EM waves through a vacuum.
Putting it together
The module is about two ways energy gets from one place to another. In waves (3A to 3H), a disturbance passes energy along while the medium stays put, or, for light, with no medium at all. One equation, $v = f\lambda$, and one rule, that the frequency is fixed by the source, explain refraction, the Doppler effect and harmonics. Superposition explains interference, beats and standing waves; standing waves that fit a length explain musical notes; and the same refraction that bends water waves at a beach gives Snell's law and optical fibres.
In heating (3I and 3J), energy flows because of a temperature difference. The particle model connects temperature to kinetic energy, explains why changes of state happen at constant temperature, and explains conduction. The two topics meet in radiation: infrared radiation is a transverse EM wave, obeys the inverse square law, and is one of the three ways energy is transferred by heating.
Key equations
| Equation | Use |
|---|---|
| $v = f\lambda$, $T = \frac{1}{f}$ | any wave |
| $I = \frac{P}{4\pi r^2}$, $I_1r_1^2 = I_2r_2^2$ | intensity from a point source (sound or light) |
| $f_{\text{beat}} = |f_2 - f_1|$ | beats |
| $f' = f\frac{(v_{\text{wave}} + v_{\text{observer}})}{(v_{\text{wave}} - v_{\text{source}})}$ | Doppler effect |
| $f_n = \frac{nv}{2L}$ / $f_n = \frac{nv}{4L}$ ($n$ odd) | string or open pipe / closed pipe |
| $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$, $M = \frac{h_i}{h_o}$ | lenses and curved mirrors |
| $n = \frac{c}{v}$, $n_1\sin\theta_1 = n_2\sin\theta_2$, $\sin\theta_c = \frac{n_2}{n_1}$ | refraction and TIR |
| $Q = mc\Delta T$, $Q = mL$ | temperature change, change of state |
| $\frac{Q}{t} = \frac{kA\Delta T}{d}$ | rate of conduction |
Key words
amplitude, antinode, beats, compression, conduction, convection, critical angle, damping, diffraction, dispersion, Doppler effect, electromagnetic wave, fundamental, harmonic, intensity, internal energy, latent heat, longitudinal, mechanical wave, medium, natural frequency, node, normal, period, progressive wave, radiation, rarefaction, ray, real image, refraction, refractive index, resonance, specific heat capacity, standing wave, superposition, temperature, thermal conductivity, thermal equilibrium, total internal reflection, transverse, virtual image, wavefront, wavelength.
Test yourself
A radio station broadcasts at 702 kHz. The wavelength of its signal is closest to
$\lambda = c/f = 3.00 \times 10^8 / 7.02 \times 10^5 = 427$ m.
Light travels from glass ($n = 1.50$) towards air at 45°. Which of these happens?
$\theta_c = \sin^{-1}(1/1.50) = 41.8°$; 45° is larger, so TIR.
A closed pipe 0.85 m long has a fundamental of 100 Hz. The speed of sound in it is
$v = 4Lf = 4 \times 0.85 \times 100 = 340$ m/s.
A 2.0 kg block of iron ($c = 450$) and a 2.0 kg block of aluminium ($c = 900$) each receive 18 kJ. The temperature rises are
$\Delta T = Q/mc$: iron $18\,000/900 = 20$ °C; aluminium $18\,000/1800 = 10$ °C.
A source moves away from a stationary observer at 20 m/s, emitting 680 Hz. With $v = 340$ m/s, the observer hears
$f' = 680 \times 340/(340 + 20) = 642$ Hz.
Which observation supports the idea that sound is a mechanical wave but light is not?
Removing the medium stops the sound but not the light.
Label the standing wave.
A water wave refracts into shallow water. What changes?
Electromagnetic waves are always . Two notes of 440 Hz and 444 Hz produce beats per second. A virtual image cannot be formed on a . While ice melts, its temperature stays . Energy reaches us from the Sun by .
A hot-water bottle contains 1.20 kg of water at 70 °C. (a) Calculate the energy it releases cooling to 30 °C. (b) A rubber bottle of the same size filled with a gel of specific heat capacity 2500 J kg⁻¹ K⁻¹ would release less. Explain why water is a good choice, using the particle model. (c) Its rubber wall is 2.0 mm thick with area 0.060 m² and $k = 0.16$ W m⁻¹ K⁻¹. When the water is at 60 °C and the outer surface at 35 °C, calculate the rate of conduction through the wall. (d) Evaluate the claim that "a hot-water bottle loses energy only by conduction."
(a) $Q = mc\Delta T = 1.20 \times 4180 \times 40 = 2.01 \times 10^5$ J.
(b) Water's specific heat capacity (4180) is much higher than the gel's (2500), so each kilogram stores and releases more energy per degree: here the water releases about 1.7 times as much for the same temperature drop. In the particle model, energy added to water does not all go into the molecules' kinetic energy; much of it goes into stretching and breaking the attractions (hydrogen bonds) between molecules. So a lot of energy is needed for each degree of temperature change, and the same energy is given back as it cools, keeping the bottle warm for longer.
(c) $\dfrac{Q}{t} = \dfrac{kA\Delta T}{d} = \dfrac{0.16 \times 0.060 \times 25}{2.0 \times 10^{-3}} = 120$ W.
(d) The claim is incomplete. Energy reaches the outer surface of the rubber by conduction (and by convection in the water inside). From the outer surface it then leaves by convection, as air next to it warms, becomes less dense and rises, and by radiation, since the warm surface emits infrared. When the bottle is pressed against a person, conduction into their body becomes the main route. So all three processes are involved, and conduction is only one of them.