Waves Teaching Guide

Conceptual Foundations

Generation of mechanical waves

A mechanical wave is a vibration propagating through a medium. Requirements: a source + a medium. The particles do not travel with the wave; they only vibrate about their equilibrium positions.

Basic relations

$v = \lambda f = \frac{\lambda}{T}$

• Wave speed $v$: set by the medium
• Frequency $f$: set by the source
• Wavelength $\lambda = v/f$: set by both together

Standing-wave condition (string fixed at both ends)

$L = n \cdot \frac{\lambda_n}{2},\quad f_n = \frac{nv}{2L} \quad (n=1,2,3,\ldots)$

Interference

• Constructive: $\Delta r = n\lambda$
• Destructive: $\Delta r = (n+\frac{1}{2})\lambda$

Diffraction

Condition for pronounced diffraction: slit width $d \lesssim \lambda$.

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Using the Builder

Open the wave builder (nav bar → Builder → Waves). There are two modes:

• 2-D wave field: a top-down water-ripple simulation; observe interference and diffraction patterns
• 1-D wave on a string: transverse waves on a one-dimensional string; observe standing waves, harmonics and reflection

Gestures

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Detailed Entity Reference

Components of the 2-D wave field mode

Point source (source)

A point source that sends out circular waves in all directions.

Use cases:

• Single source: watch circular waves spread outward
• Two sources (same frequency): observe interference fringes
• Several sources: complex interference patterns

Demo notes:

• Two sources must have the same frequency to produce a stable interference pattern
• Different frequencies → beats (the fringes are not stable)
• Dragging a source changes the interference pattern in real time

Obstacle / barrier (barrier)

An obstacle that blocks the wave's progress. It can form slits.

Building a slit: place two barriers with a gap between them. The slit width is the distance between the two barriers.

Use cases:

• Single-slit diffraction: leave a gap between two barriers
• Double-slit interference: three barriers form two slits
• Bending: the wave curls round the edge of a barrier

Demo notes: with slit width $d \gg \lambda$ diffraction is weak (the wave passes nearly straight through); with $d \approx \lambda$ diffraction is most pronounced.

Wave-speed region (region)

Sets a different wave speed inside a rectangular region.

Priority: wave speed is defined in 2 places: the global default wave speed and the speed of each wave-speed region. Inside a region the wave travels at the region's value; anywhere no wave-speed region covers it travels at the global default. When several wave-speed regions overlap, the later declaration wins.

Use case: model a wave going from deep water into shallow water (wave speed drops → wavelength shortens → refraction).

Probe (probe2d)

Records how the vibration of one point changes with time (a u-t vibration graph).

Use case: place it in regions of constructive/destructive interference and compare the amplitudes. At a node the amplitude is zero.

Boundary (boundary, scene level)

What happens when the wave reaches the canvas edge (default "Absorbing"):

• Absorbing: the wave leaves the boundary without reflecting, modelling open space (use this for interference/diffraction experiments)
• Reflecting wall: a fixed wall; reflection inverts phase (a crest comes back as a trough)
• Free end: reflection in phase (a crest comes back still a crest)

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Components of the 1-D wave-on-a-string mode

Sinusoidal driver (sine)

Applies a continuous sinusoidal drive at one point of the string.

Use case: produces a continuous travelling wave; when the frequency satisfies $f_n = nv/(2L)$ a standing wave forms.

Demo notes:

• When the frequency fails the standing-wave condition, the string shows complicated beats (the amplitude grows and shrinks)
• That is itself a valuable observation - it shows that only certain frequencies resonate

Pulse wave (pulse)

Applies a single pulse at one point of the string.

Use case: watch a single wave packet travel and reflect. Reflection at a fixed end inverts the pulse; reflection at a free end does not.

Standing-wave mode (mode)

Sets the string straight to the initial state of one standing-wave mode (skipping the transient).

Use case: observe the shape of the $n$-th standing-wave mode directly, with no waiting for the transient to build up.

Demo suggestions:

• $n=1$: one antinode, two nodes (the ends)
• $n=2$: two antinodes, three nodes
• $n=3$: three antinodes, four nodes
• Check: the spacing of adjacent nodes = $\lambda/2 = L/n$

Wave-speed segment (segment)

Sets a different wave speed on one section of the string.

Priority: wave speed is defined in 2 places: the whole-string speed and the speed of each wave-speed segment. Inside a segment the wave travels at the segment's value; outside all segments it travels at the whole-string speed. When several wave-speed segments overlap, the later declaration wins.

Use case: a wave passing from one medium into another (a thick string → a thin string, say); observe transmission and reflection.

Physics: different wave speed → different wavelength (the frequency is unchanged); at the interface the wave partly reflects and partly transmits.

Probe x (probe1d)

Records the displacement-time curve of one point on the string.

Use cases:

• At an antinode: maximum amplitude
• At a node: zero amplitude
• Two probes $\lambda/2$ apart: the curves are in antiphase
• Read the period $T$ off the $y$-$t$ curve

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Using the Lab

Click "Run it" to enter the lab:

• Play/Pause: watch the wave propagate and superpose
• Speed: 0.25× / 0.5× / 1× / 2× (use slow playback for high-frequency waves)
• Step forward: frame-by-frame stepping; good for judging the direction in which the particles move
• Render modes:
  • Waveform rendering: the shape of the string / 2-D colour map (red = crest, blue = trough)
  • Particle mode: shows the vibration direction of each particle (proves the particles do not migrate)
  • Components (1-D): the incident wave + the reflected wave + the resultant wave shown separately
• Probe curve: the $y$-$t$ graph on the right
• Parameter scan: sweep the driving frequency to find the resonance peaks

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Common Problems and Cautions

• Standing waves do not travel: normal behaviour. Switch to "Components" to see the incident wave and the reflected wave each in motion.
• Frequency mismatch: unless $f_n = nv/(2L)$ holds, no stable standing wave forms and beats appear.
• A probe reads zero: it may sit on a node. Move it half a wavelength and you will see maximum vibration.
• Diffraction is weak: with slit width $d \gg \lambda$ the wave passes almost straight through. Narrow the slit or increase the wavelength (lower the frequency).
• Particles do not migrate: use particle mode to observe - each particle only vibrates up and down; it does not advance with the wave.
• The 2-D wave field stutters: lower the playback speed or shrink the region.

Teaching Suggestions

• Wave-speed equation: fix the wave speed, change the frequency, measure the wavelength, verify that $v = \lambda f$ stays unchanged.
• Standing waves: use standing-wave mode to set $n=1,2,3$ directly, count the nodes and antinodes, verify $L = n\lambda/2$.
• Resonance scan: parameter-scan the driving frequency; amplitude peaks appear at $f_n$.
• Interference: two sources at the same frequency; observe the constructive/destructive fringes and verify the path-difference condition.
• Diffraction: vary the slit width; compare the effects for $d \gg \lambda$ and $d \approx \lambda$.
• Reflection: send a pulse at a fixed end (inverted) and at a free end (not inverted).

Related Experiments

• Transverse waves and standing waves on a string: verify $L = n\lambda/2$
• Interference of waves: two-source interference fringes
• Single-slit diffraction: the relation between slit width and wavelength
• Doppler effect: move the source and observe wavelengths being compressed/stretched
• Reflection and transmission of waves: behaviour at a boundary between different wave speeds