Pick your class, read the steps, then try them right in the sandbox below.
Class 11 — Standing waves and harmonics
Concept: a string fixed at both ends only supports certain frequencies —
harmonics — where fn = n·v/(2L), n = 1, 2, 3…
- Switch to Standing waves mode.
- Set String length L to 1 m and Wave speed v to 50 m/s. Use the
Harmonic n stepper to select n = 1 (the fundamental) and note the value in the
Frequency stat (25.0 Hz).
- Step n up to 2, then 3. Watch the number of nodes increase and check the
Frequency stat exactly doubles (50.0 Hz), then triples (75.0 Hz), matching
fn = n·v/(2L).
Takeaway: higher harmonics mean more nodes and proportionally higher
frequency — this is exactly how a guitar string produces different overtones.
Class 11 — The Doppler effect
Concept: a moving sound source shifts the frequency you actually hear — higher
if it's approaching, lower if it's moving away — even though the source's own
frequency never changes.
- Switch to Doppler effect mode.
- Set Source frequency f to 500 Hz and Source speed to 0 m/s first —
confirm Observed freq. reads 500 Hz with no motion.
- Increase Source speed while the source moves toward the listener and watch
Observed freq. rise above 500 Hz.
- Let the source pass and move away — Observed freq. should now drop below 500 Hz.
Takeaway: this is exactly why a passing ambulance siren sounds higher-pitched
as it approaches and lower-pitched as it recedes.
Class 12 — Two-source interference
Concept: waves from two sources reinforce (constructive, bright) where their
path difference is a whole number of wavelengths, and cancel (destructive, dark) where
it's a half-integer number of wavelengths — the same idea behind Young's double-slit
experiment.
- Switch to Interference mode.
- Set Source separation d and Wavelength λ to any values, then
drag the detector around the stage.
- Find a spot the readout labels "constructive (bright)", then move the detector
slightly and find one labelled "destructive (dark)".
Takeaway: whether a point is bright or dark depends only on the
difference in distance from the two sources, measured in wavelengths.
Class 11 — A travelling wave on a string
Concept: a wave on a string travels at v = √(T ÷ μ), set by the string and not by how you shake it. Each piece of
string only moves up and down; it is the shape that travels.
- Open Wave on a string with T = 100 N and μ = 25 g/m. The speed is v = √(100 ÷ 0.025) = 63.25 m/s, and for λ = 1 m the
frequency is f = v ÷ λ = 63.25 Hz.
- Raise the tension to 200 N: v rises to 89.4 m/s (a factor of √2, not 2). Raise μ to 100 g/m instead and v falls to 31.6 m/s: a heavier
string is slower.
- Watch the orange particle and its red velocity arrow. It moves up and down, and it is fastest as it passes the middle: the greatest
particle speed is Aω = 0.02 × 397.4 = 7.95 m/s, very different from the wave speed 63.25 m/s.
- Two points 25 cm apart (a quarter of a wavelength) differ in phase by Δφ = kΔx = 90°.
- The power carried is P = ½μω2A2v = 49.9 W here. Double the amplitude and it becomes four times as large.
Takeaway: v = √(T ÷ μ) = fλ; the wave speed and the particle speed are different things.
Class 11 — Organ pipes: open and closed
Concept: the air in a pipe resonates when a standing wave fits. An open end is a displacement antinode and a closed end is a node.
- Open Organ pipes with an open pipe, L = 0.5 m, 20 °C. v = 331 + 0.6 × 20 = 343 m/s, and fn = nv ÷ 2L gives 343 Hz, 686 Hz,
1029 Hz … (every harmonic).
- Change to a closed pipe: f = (2m − 1)v ÷ 4L gives 171.5 Hz, then 514.5 Hz and 857.5 Hz. Only odd harmonics appear (1 : 3 : 5), and
the fundamental is half of the open pipe's: closing one end drops the note by an octave.
- Look at the N and A marks: in the closed pipe the closed end is a node (the air cannot move) and the open end is an antinode. The pressure
shading is the other way round: it changes most where the air moves least.
- Make the pipe wide: with an open pipe of L = 0.5 m and r = 25 mm the end correction 0.6r at each end makes it act like 0.53 m long, and the
fundamental falls from 343 Hz to 323.6 Hz.
- Warm the air to 30 °C: v = 349 m/s and an open pipe of 0.5 m plays 349 Hz, a little sharper.
Takeaway: open: f = nv ÷ 2L (all harmonics); closed: f = (2m − 1)v ÷ 4L (odd harmonics only).
Class 11 — The resonance tube: measuring the speed of sound
Concept: a tuning fork held over a tube of air makes a loud sound when the air column is an odd number of quarter-wavelengths long. Two
resonance lengths give the speed of sound without needing the end correction.
- Open Resonance tube with the 512 Hz fork, 20 °C and a tube radius of 20 mm. The wavelength is λ = 343 ÷ 512 = 67.0 cm.
- Slide the water level from short to long and find the first loud place. It is at l1 = λ ÷ 4 − e = 16.75 − 1.2 = 15.5 cm. The next
is at l2 = 3λ ÷ 4 − e = 49.0 cm and the next at 82.5 cm.
- Subtract: l2 − l1 = λ ÷ 2 = 33.5 cm, so v = 2f(l2 − l1) = 2 × 512 × 0.335 = 343 m/s. The end correction cancels.
- The correction itself is e = (l2 − 3l1) ÷ 2 = 1.2 cm = 0.6r.
- Change to the 256 Hz fork: λ doubles and the first resonance moves to l1 = 32.3 cm.
Takeaway: l2 − l1 = λ ÷ 2, so v = 2f(l2 − l1); the end correction e = 0.6r.
Class 11 — Beats
Concept: two notes of slightly different frequencies add to give a sound that swells and fades at the difference frequency |f1 − f2|.
- Open Beats with f1 = 256 Hz and f2 = 260 Hz. You hear a note of ½(f1 + f2) = 258 Hz whose loudness rises and falls
|f1 − f2| = 4 times a second: one beat every 250 ms. The dashed envelope in the bottom trace shows it.
- Move f2 to 258 Hz: the beats slow to 2 per second. At 256 Hz there are none.
- Move f2 to 276 Hz: 20 beats per second. That is too fast to hear as separate swells; it sounds rough.
- Lower the loudness of source 2 to 0.5: the sound no longer dies away completely. Imax ÷ Imin = ((1 + 0.5) ÷ (1 − 0.5))2 = 9.
- Tick “hear it” (with the volume low) to listen to the beats.
Takeaway: beat frequency = |f1 − f2|; Imax ÷ Imin = ((A1 + A2) ÷ (A1 − A2))2.
Class 12 — Young's double slit
Concept: light from two close slits gives bright and dark fringes. A bright fringe is where the path difference d sinθ = nλ, and the fringe width
is β = λD ÷ d.
- Open Double slit & diffraction, “Two narrow slits”, λ = 600 nm, d = 0.5 mm, D = 1.5 m. The fringe width is β = 600 × 10−9 × 1.5 ÷ 0.5 × 10−3 = 1.8 mm.
- Move the “look at the point y” slider to 1.8 mm: the path difference is 1.00 λ and the point is bright (first order). At 0.9 mm it is 0.50 λ: dark.
- Halve d to 0.25 mm: the fringes spread out to 3.6 mm. Double D to 3 m: also 3.6 mm. So β ∝ D ÷ d.
- Change the wavelength to 700 nm (red): β = 2.1 mm. At 400 nm (violet): 1.2 mm.
- Tick “white light”: the central fringe is white (every colour is bright where Δ = 0) and the others are coloured, because each colour has its own fringe width.
- Put the whole apparatus in water (n = 1.33): λ becomes 451 nm and β falls to 1.35 mm.
Takeaway: β = λD ÷ d; bright when Δ = nλ, dark when Δ = (n + ½)λ; the brightest fringes are 4I0.
Class 12 — Diffraction by a single slit
Concept: a slit spreads light out. The first dark band is where a sinθ = λ, and the central bright band has width 2λD ÷ a.
- Choose “One slit (diffraction)” with λ = 600 nm, D = 1.5 m and a = 50 µm. The first dark band is at y1 = λD ÷ a = 18 mm, so the central maximum is 36 mm wide.
- Halve the slit to 25 µm: the pattern doubles in width to 72 mm. A narrower slit spreads the light more.
- The central band is twice as wide as the side bands and much brighter; the first side band is only about 4.7% as bright.
- Now choose “Two slits of real width” with d = 0.5 mm and a = 50 µm. The double-slit fringes (β = 1.8 mm) sit inside the single-slit envelope, and about 2d ÷ a = 20 of them
fit in the central band. Where the envelope is zero a fringe is missing (every 10th order here).
Takeaway: single slit: yn = nλD ÷ a; double slit with wide slits = the two-slit fringes multiplied by the one-slit envelope.
Class 11 — Sound intensity and the decibel
Concept: a source spreads its power over a sphere, so the intensity is I = P ÷ 4πr2. The sound level in decibels is β = 10 log(I ÷ I0) with I0 = 10−12 W/m2.
- Open Sound intensity (dB) with P = 1 W and r = 10 m: I = 1 ÷ (4π × 100) = 7.96 × 10−4 W/m2, which is 89.0 dB.
- Drag the listener to 20 m: the intensity is a quarter and the level 83.0 dB (6 dB less). At 100 m it is 69.0 dB.
- Choose 2 sources: the intensity doubles and the level rises by 3.0 dB (to 92 dB), not by double. 10 sources add 10 dB.
- Compare the ladder: a whisper is 30 dB, conversation 60 dB, heavy traffic 80 dB and the threshold of pain about 120 dB.
Takeaway: doubling the distance costs 6 dB; doubling the number of sources adds 3 dB.