Clap your hands and you give the air a quick squeeze. That squeeze races outward as a moving pattern of "packed" and "spread-out" air, and when it reaches you it pushes your eardrum in and out — that is sound. Notice the trick: nothing actually flies from your hands to your ear except the pattern. Each little patch of air just jiggles back and forth in place, like fans doing a stadium wave while staying in their seats.
Three numbers describe that jiggle. How many times the air packs and unpacks each second is the frequency $f$, measured in hertz — a high $f$ sounds high-pitched. How far apart two "packed" regions sit is the wavelength $\lambda$. And how fast the whole pattern travels is the speed of sound $v_s$. One short rule ties them together.
The rule that links pitch, wavelength, and speed
$$v_s = f\,\lambda$$
In ordinary air the speed depends almost entirely on temperature: $v_s \approx 331 + 0.6\,T$ with $T$ in °C. On a 20 °C day that gives $v_s \approx 343$ m/s, so the 440 Hz "A" an orchestra tunes to has $\lambda = 343/440 \approx 0.78$ m — about the length of your arm.
Sound is a longitudinal wave: the air moves along the same line the wave travels, so it is cleaner to track pressure than position. The pressure swing is $\Delta P(x,t)=\Delta P_{\max}\cos(kx-\omega t)$, with $k=2\pi/\lambda$ and $\omega=2\pi f$. The energy carried (the intensity) grows as the square of the pressure, $I=\Delta P_{\max}^2/(2\rho v_s)$, and spreads over an ever-larger sphere, so $I\propto 1/r^2$. Because the ear judges ratios, not absolute amounts, we squeeze this huge range into decibels, $\beta=10\log_{10}(I/I_0)$ — every extra 10 dB means ten times the intensity. The sliders map straight onto these symbols: $f_1$ is $f$, ΔP is $\Delta P_{\max}$, the temperature sets $v_s$, and the observer distance sets $r$.
Try this in the sim above
(1) Drag the temperature slider and watch $v_s$ and $\lambda$ shift while the frequency readout stays fixed — warming the air stretches the wavelength. (2) Double the observer distance $r$ and confirm the dB readout falls by about 6, the inverse-square law in action. (3) Open the Beats tab and set $f_2$ just 2 Hz away from $f_1$; the loudness throbs once per second — exactly the wobble musicians listen for when tuning.
Sound is a longitudinal pressure wave. Air molecules oscillate parallel to propagation. Pressure and displacement waves are 90° out of phase: where pressure is maximum, displacement is zero (and vice versa). $\Delta P_\max=\rho v_s\omega s_\max$ where $s_\max$ is displacement amplitude.
2
Decibel scale. $\beta=10\log_{10}(I/I_0)$. Logarithmic because the ear has logarithmic sensitivity. Each 10 dB = ×10 intensity. Conversation (60 dB): $I=10^{-6}$ W/m². Pain threshold (120 dB): $I=1$ W/m². Doubling distance: −6 dB (inverse square law).
3
Beats. $f_1$ and $f_2$ close together produce audible amplitude modulation at $f_{\text{beat}}=|f_1-f_2|$. Mathematically: sum of two sinusoids = product of sum and difference frequencies. Used to tune instruments to fractions of Hz precision.
4
Pipe resonance. Standing waves in pipes. Open-open: all harmonics (rich timbre). Open-closed: odd harmonics only (hollow timbre, like clarinet). Temperature affects $v_s$ and thus resonant frequencies — instruments go sharp when warm.
They are 90° (quarter wavelength) out of phase. At a pressure antinode (maximum compression or rarefaction), particles are momentarily stationary — this is a displacement node. At a pressure node (atmospheric pressure), particles have maximum speed — this is a displacement antinode. This anti-correlation is why pressure microphones and displacement microphones measure different things.
Key takeaway: Pressure and displacement are 90° out of phase: pressure max ↔ displacement zero, and vice versa.
Ultrasound imaging (1–20 MHz — above hearing, high resolution), SONAR for ocean mapping and submarine detection, acoustic levitation (standing waves lift particles), noise-cancelling headphones (destructive interference), seismology, acoustic inspection of materials (finding cracks), bat echolocation (up to 100 kHz), and hearing aids.
Key takeaway: Sound waves enable ultrasound imaging, sonar navigation, material testing, and echolocation.
The flute is open at both ends: all harmonics present ($f_n=nv_s/(2L)$). The clarinet has one closed end (reed): only odd harmonics ($f_n=(2n-1)v_s/(4L)$). Missing even harmonics give the clarinet its hollow, woody timbre. The oboe (double reed) behaves like an open pipe despite having a closed end — because the conical bore compensates.
Key takeaway: Open pipe: all harmonics. Closed pipe: odd harmonics only. This determines timbre.
Intensity follows inverse-square law: $I\propto1/r^2$. Doubling $r$: $I$ drops by factor 4. $\Delta\beta=10\log_{10}(1/4)=-6$ dB. Moving from 1 m to 10 m (×10 in distance): $I$ drops by 100×, so $\Delta\beta=-20$ dB. This explains why concert sound engineers must compensate for distance.
Key takeaway: Doubling distance: −6 dB. Ten times distance: −20 dB. Rule: $\Delta\beta=-20\log_{10}(r_2/r_1)$.
Resources: Khan Academy; HyperPhysics (hyperphysics.phy-astr.gsu.edu); MIT OCW.
Section 05
Common Misconceptions
❌ Louder sounds have higher frequency.
✅ Loudness (perceived intensity, measured in dB) and pitch (perceived frequency, measured in Hz) are independent perceptual qualities. A loud bass drum is louder than a quiet flute but has lower frequency. The ear processes them via separate mechanisms (hair cells at different positions on the basilar membrane).
📖 HRW 10th Ed., §17-1.
❌ Sound travels at the same speed in all media.
✅ Sound speed varies enormously by medium: air (~343 m/s at 20°C), water (~1480 m/s), steel (~5100 m/s). Speed increases with stiffness and decreases with density: $v=\sqrt{B/\rho}$. Sound travels about 4× faster in water than air — important for SONAR calculations.
📖 HRW 10th Ed., §17-3.
❌ The decibel scale is linear — 60 dB is twice as loud as 30 dB.
✅ 60 dB is $10^3=1000$ times more intense than 30 dB (each 10 dB = ×10 in intensity). Perceived loudness (in phons/sones) is roughly logarithmic too: 60 dB sounds about 8× louder than 30 dB to the human ear, not 1000×.
📖 HRW 10th Ed., §17-2.
Misconception research: Arons — A Guide to Introductory Physics Teaching; Driver et al. — Making Sense of Secondary Science.