Quantum Mechanics 101 · Unit 1: The Classical World Cracks

Quantum Academy · Chapter 2: Waves 101: The Mathematics of Wiggles

Amplitude, wavelength, interference — the vocabulary every quantum idea is written in

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Instructions. Work through these on paper before checking the answers. Use v = fλ and c = fλ for speeds, whole and half wavelengths for interference, I ∝ A² for intensity, and λₙ = 2L/n for standing waves. Take c = 3.00×10⁸ m/s and the speed of sound in air as 343 m/s unless a question says otherwise. Show units at every step. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Amplitude
2.Wavelength
3.Frequency
4.Period
5.Phase
6.Electromagnetic spectrum
7.Superposition
8.Interference
9.Path difference
10.Standing wave
11.Intensity
12.Phasor

A.The number of complete cycles per second; symbol f, measured in hertz.
B.The maximum displacement of a wave from its resting level; it sets the wave's intensity.
C.The position within a wave's cycle, measured as an angle; one full cycle is 360° or 2π radians.
D.The principle that overlapping waves add: the net displacement is the sum of the individual displacements.
E.The power a wave delivers per unit area, in W/m²; proportional to amplitude squared.
F.A confined wave pattern that oscillates in place, with fixed nodes and antinodes.
G.The distance over which a wave repeats, crest to crest; symbol λ (the Greek letter lambda), measured in metres.
H.The full range of electromagnetic waves, from radio to gamma rays, all travelling at c in vacuum.
I.The time for one complete cycle; symbol T, equal to 1/f.
J.A rotating arrow, written as a complex number, whose length is a wave's amplitude and whose angle is its phase.
K.The reinforcement or cancellation that occurs when superposed waves arrive in phase or out of phase.
L.The extra distance one wave travels, compared with another, to reach the same point.

Part B Questions

1.
Which quantity is set by the source and stays the same when a wave passes from one medium into another?Circle one
1. AWavelength
2. BSpeed
3. CFrequency
4. DAmplitude
2.
Two waves of equal amplitude A meet with a phase difference of 90°. What is the amplitude of their sum?Circle one
1. A2A
2. B0
3. C√2·A, about 1.41A
4. DA/2
3.
Which list is in order of increasing wavelength?Circle one
1. AGamma rays, X-rays, visible light, microwaves
2. BRadio, infrared, visible light, ultraviolet
3. CVisible light, gamma rays, radio, X-rays
4. DMicrowaves, visible light, infrared, gamma rays
4.
In a stadium wave, the fans travel around the stadium along with the wave.Circle one
TrueFalse
5.
If two equal waves, each of which would give intensity I on its own, meet exactly in phase, the resulting intensity is 4I.Circle one
TrueFalse
6.
A point on a standing wave that never moves is called a .Fill in the blank
7.
The rule that overlapping waves simply add their displacements is called the principle of .Fill in the blank
8.
Explain why a guitar string tuned to 110 Hz can ring at 220 Hz but not at 150 Hz.Short answer
9.
Why were the dark fringes in Young's experiments such strong evidence that light is a wave?Short answer
10.
You shake one end of a long rope 4.0 times per second and produce waves with crests 0.75 metres apart. Find the wave speed and the period.Show your working
Working
Answer:
11.
Neutral hydrogen atoms in interstellar space emit radio waves at a frequency of 1420.405751 MHz. What is their wavelength? (Use c = 2.998×10⁸ m/s.)Show your working
Working
Answer:
12.
Two loudspeakers, driven in phase, emit a 1,715 Hz tone in air. A listener is 4.30 m from one speaker and 4.80 m from the other. Is the sound loud or quiet there?Show your working
Working
Answer:
13.
A string 0.80 m long is fixed at both ends, and waves travel along it at 240 m/s. Find its fundamental frequency, and the frequency and wavelength of its third harmonic.Show your working
Working
Answer:
14.
Light from one laser is split into two coherent beams that overlap again. Each alone would deliver 2.0 W/m², and at a particular point their phase difference is 60°. What is the intensity at that point?Show your working
Working
Answer:

Part C Mad scientist challenge bonus

Measure the speed of light with a chocolate bar. Take the turntable out of a microwave oven, lay a flat bar of chocolate on a microwave-safe plate in the oven, and heat it in short bursts of about 15 to 20 seconds until a few separate spots just begin to melt. The label on the oven says 2.45 GHz. You measure 6.1 cm between the centres of neighbouring melted spots. (a) Explain why separate melted spots appear. (b) Calculate the speed of light from your data. (c) If your spacing could really be anywhere from 5.8 to 6.4 cm, what range of values for c does that give, and what does it tell you about the experiment? Safety: never run a microwave oven empty, keep the plate in it, and use only short bursts.

Working

Memory aid

Mnemonic Very Fast Llamas; whole waves win, half waves hush; add before you square. Picture a conga line of llamas trotting past a fence post. Very Fast Llamas is V = F × L: the line's speed, V, equals how many llamas pass the post each second, F, times the length of one llama, L for lambda. Whole waves win, half waves hush: a path difference of a whole number of wavelengths gives constructive interference, and an odd number of half-wavelengths gives silence or darkness. Add before you square: combine amplitudes as arrows first, then square the total to get intensity. Recite the three lines in order on any wave problem: the first finds a wavelength or frequency, the second decides loud or quiet, the third tells you how bright.

In one line Waves carry energy, not matter; they add amplitude by amplitude, their intensity is amplitude squared, and trapped waves allow only whole-number patterns.