Jupiter2U School
Chapter 12

Waves & Sound

High School
At a glance
Core ideaA wave carries energy without transporting matter, and obeys v = fλ.
Key termResonance — a large response when a system is driven at its natural frequency.
You can…Measure the speed of sound with a resonance tube and a tuning fork.
Watch outSound needs a medium — it cannot cross a vacuum; low pitch means long wavelength.
Theory

Oscillations and travelling waves

A wave transfers energy through a medium (or space) without transporting matter. Key quantities: wavelength λ, frequency f (Hz), period T = 1/f, amplitude A, and speed:

v = f λ the wave equation

Transverse waves oscillate ⟂ to travel (light, string waves); longitudinal waves oscillate along travel (sound — compressions and rarefactions).

rest wavelength λ amplitude A v = f λ
Wavelength λ is the distance between repeats; amplitude A is the peak displacement from rest. Wave speed v = fλ.

Many oscillators obey simple harmonic motion (SHM), where restoring force ∝ displacement: F = −kx, giving a = −ω²x with ω = 2πf. A mass on a spring has T = 2π√(m/k).

Wave phenomena

Waves reflect, refract (bend on changing speed), diffract (spread past obstacles/gaps ~λ) and interfere. Superposition: overlapping waves add — constructive (in phase) or destructive (antiphase). Standing waves on a string of length L have harmonics fn = nv/2L.

Explanation

Resonance, beats and the Doppler effect

Every system has natural frequencies. Drive it at one and amplitude grows dramatically — resonance. It's why a pushed swing goes higher only with correctly-timed pushes, why a wine glass shatters at its pitch, and why bridges must avoid marching soldiers' step frequency (Tacoma Narrows is the cautionary tale).

The Doppler effect shifts observed frequency when source and observer move relative to each other: an approaching ambulance's siren is raised in pitch (waves bunch up), then drops as it passes. For light, the same effect red-shifts receding galaxies — the observational bedrock of the expanding universe.

Sound facts. Sound travels ~343 m·s⁻¹ in air at 20 °C, ~1480 m·s⁻¹ in water, ~5000 m·s⁻¹ in steel — faster in stiffer, denser-bonded media. It cannot travel through vacuum: no medium, no sound.
Practical

Experiment — measuring the speed of sound with resonance

Use a tube closed at one end (a graduated cylinder of water) and a tuning fork of known frequency f to find the speed of sound.

  1. Strike the fork and hold it over the open top. Raise the tube (lower the water) until the sound is loudest — the first resonance.
  2. At the first resonance the air column length L₁ equals a quarter-wavelength: L₁ = λ/4 (closed-pipe fundamental).
  3. Find the next loud point L₂ = 3λ/4. Then L₂ − L₁ = λ/2, which cancels the end-correction error.
  4. So λ = 2(L₂ − L₁). Example: fork f = 512 Hz, L₁ = 0.163 m, L₂ = 0.498 m ⟹ λ = 2(0.335) = 0.670 m.
  5. Speed: v = fλ = 512 × 0.670 = 343 m·s⁻¹. ✓ Matches the accepted value.

Using the difference L₂ − L₁ rather than L₁ alone is the clever part — it removes the systematic "end correction" where the antinode sits slightly outside the tube.

Q&A
A radio station broadcasts at 100 MHz. What is its wavelength? (c = 3.0×10⁸ m·s⁻¹)

λ = c/f = 3.0×10⁸ / 100×10⁶ = 3.0 m. Radio waves are metres long — hence the size of antennas.

Why can you hear around a corner but not see around it?

Diffraction is significant when the gap or obstacle is comparable to the wavelength. Sound wavelengths are ~metres, similar to doorways, so sound diffracts strongly and bends around corners. Light's wavelength is ~10⁻⁷ m, far smaller than everyday objects, so it diffracts negligibly and travels in apparent straight lines.

Two speakers play the same 340 Hz tone. At a point 1.0 m from one and 1.5 m from the other, is it loud or quiet? (v = 340 m·s⁻¹)

λ = v/f = 340/340 = 1.0 m. Path difference = 0.5 m = λ/2 — an odd number of half-wavelengths, so the waves arrive antiphase: destructive interference, a quiet spot.

A car horn sounds at 400 Hz. It approaches you at 30 m·s⁻¹. Roughly what frequency do you hear? (v = 340 m·s⁻¹)

Approaching source: f' = f·v/(v − vs) = 400 × 340/(340−30) = 400 × 340/310 = 439 Hz. You hear a higher pitch; it drops to 400 × 340/370 = 368 Hz once it passes and recedes.

Concept mind map

How the ideas connect

Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.

WavelengthFrequencyv = f*lambdaResonanceBeatsDoppler effectWaves & Sound
Infographic

The key facts, visualised

v = f*lambda
Wave speed equals frequency times wavelength
343 m/s
Speed of sound in air at about 20 C
Beats
Slow throb at the difference of two frequencies
Doppler
Pitch shift from relative motion of source
Solved examples

Worked problems, step by step

Follow each solution line by line, then try to reproduce it on paper before moving on.

Example 1A wave has frequency 500 Hz and wavelength 0.68 m. Find its speed.

  1. v = f * lambda
  2. v = 500 * 0.68

Example 2A resonance tube first resonates at 0.25 m for a 340 Hz tuning fork. Estimate the speed of sound.

  1. First resonance is a quarter wavelength: lambda = 4*0.25 = 1.0 m
  2. v = f*lambda = 340 * 1.0
Practice problem set

Now you try

Work each one out first, then tap to reveal the worked answer.

1What is the frequency of a wave with speed 340 m/s and wavelength 2 m?
f = v/lambda = 340/2 = 170 Hz.
2Two tuning forks at 256 and 260 Hz sound together. What beat frequency?
260 - 256 = 4 Hz.
3Does a siren sound higher or lower as it approaches you?
Higher, because the waves are compressed to a shorter wavelength.
4What is the difference between transverse and longitudinal waves?
Transverse vibrates across the travel direction; longitudinal vibrates along it, like sound.
5If frequency doubles at fixed speed, what happens to wavelength?
Wavelength halves, since v = f*lambda is constant.
6Why does an organ pipe resonate at certain lengths?
Only wavelengths that fit standing wave patterns in the pipe reinforce, giving resonance.