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Physics · enrichment

Waves, Sound & Optics — Visual Lab

Module 4. Drop two stones in a pond at once and the ripples cross — reinforcing here, canceling there. Place two point sources, sweep the frequency, wavelength, and spacing, and watch the interference pattern breathe. The same idea explains beats, noise-canceling headphones, the colors in a soap bubble, and how a microscope resolves detail.

Interactive Lab Module 04 · Waves, Sound & Optics Enrichment · not TEKS-graded

A wave carries energy without carrying matter, and its three numbers are locked together by one rule: \(v = f\lambda\). The frequency \(f\) is how many crests pass per second, the wavelength \(\lambda\) is the crest-to-crest distance, and their product is the wave speed \(v\). When two waves overlap they simply add — that's superposition, and it produces the bright and dark bands of interference. Drive the lab below and keep the geometry, the algebra, and the physical readouts in perfect agreement.

A note on standards This is a Physics · enrichment module — not a TEKS-graded course. There is no Texas Essential Knowledge and Skills strand being assessed here and no pacing calendar to keep. It exists so curious scholars can see the physics that sits one layer beneath the math — explore freely, follow your questions, and use it however it serves you.
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Wave Interference Animator

Two point sources \(S_1\) and \(S_2\) emit identical waves. Sweep the frequency \(f\), the wavelength \(\lambda\), and the source separation \(d\), and watch the constructive (bright) and destructive (nodal) bands form. The readout reports the wave speed \(v = f\lambda\), the period \(T = 1/f\), and the number of interference maxima — all physically correct.

Orientation

What you're seeing

  • Two sources, one medium. \(S_1\) and \(S_2\) sit on the left wall and send out identical circular waves. Every point in the tank feels the sum of the two — that's superposition.
  • Bright bands are constructive. Where the two paths differ by a whole number of wavelengths (\(\Delta r = m\lambda\)), crest lands on crest and the waves reinforce.
  • Dark lines are destructive. Where the paths differ by a half-wavelength (\(\Delta r = (m+\tfrac12)\lambda\)), crest lands on trough and they cancel — a nodal line.
  • \(v = f\lambda\) ties it together. Change \(f\) or \(\lambda\) and the wave speed in the readout updates instantly; the spatial pattern depends on \(\lambda\) and the spacing \(d\), not on \(f\).
  • More spacing, more bands. Pull the sources apart (or shrink \(\lambda\)) and more bright bands fit between them — the readout counts them for you.
Investigation

Try this

  1. Shrink the wavelength. Slide \(\lambda\) down and watch the bands crowd together — shorter waves pack more maxima into the same tank. Read the rising "interference maxima" count.
  2. Hold \(\lambda\), change \(f\). The bright/dark pattern doesn't move — but \(v=f\lambda\) and the period \(T=1/f\) in the readout do. Geometry depends on \(\lambda\); speed depends on both.
  3. Pull the sources apart. Increase \(d\) and new nodal lines appear between the sources. Find the smallest \(d\) that produces even one dark line (\(d \ge \tfrac{\lambda}{2}\)).
  4. Aim for a single central band. Make \(d\) smaller than \(\lambda\) and the whole tank goes bright — the two sources never get a full half-wavelength out of step, so nothing cancels.
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Worked Examples

Three moves this module leans on most: the wave equation \(v=f\lambda\), the Doppler shift of a moving source, and Snell's law for light bending across a boundary.

Example 1 — Wave speed, period, and a missing wavelength

The wave equation v = fλ and T = 1/f

A guitar string vibrates at \(f = 256\) Hz, producing a wave that travels along the string at \(v = 410\) m/s. Find the wavelength and the period.

  1. Start from \(v = f\lambda\). Solve for the unknown wavelength: \(\lambda = \dfrac{v}{f}\).
  2. Substitute. \(\lambda = \dfrac{410}{256} \approx 1.60\) m.
  3. Period is the reciprocal of frequency. \(T = \dfrac{1}{f} = \dfrac{1}{256} \approx 0.0039\) s.
  4. Sanity check. One full cycle takes \(T\) seconds and covers \(\lambda\) meters, so speed \(= \dfrac{\lambda}{T} = \lambda f = v\). \(\checkmark\)
Answer: \(\lambda \approx 1.60\) m and \(T \approx 0.0039\) s. Set the lab's \(\lambda\) near \(1.6\) m to see the matching crest spacing.

Example 2 — The Doppler shift of an approaching siren

Moving source · observed frequency rises

An ambulance siren emits \(f = 600\) Hz and drives toward a stationary listener at \(v_s = 30\) m/s. Using the speed of sound \(v = 343\) m/s, find the frequency the listener hears.

  1. Pick the approaching-source formula. A source moving toward you compresses the wavefronts: \(f' = f\,\dfrac{v}{v - v_s}\).
  2. Substitute the numbers. \(f' = 600 \cdot \dfrac{343}{343 - 30} = 600 \cdot \dfrac{343}{313}\).
  3. Compute. \(\dfrac{343}{313} \approx 1.0958\), so \(f' \approx 600 \times 1.0958 \approx 657.5\) Hz.
  4. Direction check. Approaching raises the pitch, so \(f' > f\) — and \(657.5 > 600\). \(\checkmark\) (Receding would use \(v + v_s\) and lower the pitch.)
Answer: \(f' \approx 657.5\) Hz — noticeably higher. As the ambulance passes and recedes, the pitch drops to \(600 \cdot \tfrac{343}{373} \approx 552\) Hz.

Example 3 — Bending light with Snell's law

Refraction across a boundary

A ray of light travels from air (\(n_1 = 1.00\)) into glass (\(n_2 = 1.50\)), striking the surface at \(40^\circ\) from the normal. Find the angle of refraction inside the glass.

  1. Write Snell's law. \(n_1 \sin\theta_1 = n_2 \sin\theta_2\).
  2. Solve for \(\sin\theta_2\). \(\sin\theta_2 = \dfrac{n_1 \sin\theta_1}{n_2} = \dfrac{1.00\,\sin 40^\circ}{1.50}\).
  3. Evaluate. \(\sin 40^\circ \approx 0.643\), so \(\sin\theta_2 \approx \dfrac{0.643}{1.50} \approx 0.4285\).
  4. Take the inverse sine. \(\theta_2 = \sin^{-1}(0.4285) \approx 25.4^\circ\).
Answer: \(\theta_2 \approx 25.4^\circ\). Light slows and bends toward the normal entering the denser glass — that's why a straw looks bent in a glass of water.
Why it matters Waves are how the universe moves information without moving stuff. The same superposition you see in this tank is the reason noise-canceling headphones work (cancel the incoming wave with its mirror image), why a soap bubble shimmers (light reflecting off the front and back of the film interferes), how MRI, ultrasound, and sonar image hidden things, and why a passing siren drops in pitch. Sound, light, water, radio, and the quantum wavefunction all obey the same arithmetic — \(v = f\lambda\) and the principle that waves add.
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Key Vocabulary

The precise words a physicist uses to describe what the lab is doing.

Wavelength \(\lambda\)

The distance between two consecutive crests (or troughs) of a wave, in meters. Halve it and the interference bands crowd twice as close.

Frequency \(f\) & period \(T\)

Frequency is cycles per second (hertz); period \(T = \dfrac{1}{f}\) is the seconds per cycle. They are reciprocals.

Wave speed \(v\)

How fast a crest travels: \(v = f\lambda\). Set by the medium — \(343\) m/s for sound in air, \(3\times10^8\) m/s for light in vacuum.

Superposition

When two waves overlap, the medium's displacement is the sum of the two. This single rule produces all of interference.

Constructive & destructive

In phase (\(\Delta r = m\lambda\)) the waves reinforce into a bright band; a half-wave out of step (\(\Delta r = (m+\tfrac12)\lambda\)) they cancel into a nodal line.

Refraction & index \(n\)

Light bends when it changes speed crossing a boundary. The index \(n = \dfrac{c}{v}\) and Snell's law \(n_1\sin\theta_1 = n_2\sin\theta_2\) predict the new angle.

What this lab covers

Topics in this module

This enrichment lab explores the wave strand of introductory physics: the wave equation, sound and the Doppler effect, two-source interference, standing-wave harmonics, and geometric optics (refraction, total internal reflection, and thin lenses). Algebra-based, plug-and-chug — no calculus required.

v = fλ f = 1/T Beats Doppler Harmonics Snell's law Critical angle Thin lens

Enrichment only. This module is offered as exploration alongside the graded mathematics pathway; it is not aligned to a TEKS strand and carries no pacing obligation. Constants used: \(v_{\text{sound}} \approx 343\) m/s, \(c = 3\times10^{8}\) m/s.

Want the rest of the physics map? Head back to Physics, or if a step felt shaky, visit Student Support or warm up with the Module 4 Foundations page.

Physics enrichment content is offered alongside Dr. Ijezie's STEM Studio as exploratory material. It is not a TEKS-graded course and is not part of the assessed mathematics sequence.