How to Operate the Simulator
This interactive laboratory reproduces the classical wave tank, optical bench, and acoustic probe experiments. Use the upper tabs to select the wave phenomenon you wish to inspect:
- Tab 1 (Rectilinear Propagation): Examine how plane or circular wavefronts radiate normal to their surface, maintaining constant wavelength in a homogeneous medium.
- Tab 2 (Reflection): Compare plane, concave, and convex reflective barriers. Observe plane waves converging toward a real focus F (where focal length f = R / 2) or diverging away from a virtual focal point behind a convex surface.
- Tab 3 (Refraction): Adjust refractive indices n₁ and n₂ or medium depths to verify Snell's Law. Watch wave velocity and wavelength alter abruptly while source frequency remains perfectly invariant.
- Tab 4 (Diffraction): Modulate aperture size w relative to wavelength λ to observe the Huygens stadium envelope transition from linear beams to circular arcs.
- Tab 5 (Interference): Contrast coherent synchronized wave emitters against drifting incoherent sources. When configured to sound mode, drag the yellow Microphone Probe across the field to observe spatial and temporal superposition traces on the dual-trace CRO monitor!
Scientific Theory & Working Principles
A mechanical or electromagnetic wave represents the propagation of energy through a medium (or vacuum) without the permanent displacement of mass. The periodic perturbation is governed by four core axioms:
1. Huygens-Fresnel Wavelet Principle
Every point on a primary wavefront acts as a secondary source of spherical wavelets. The envelope tangent to these secondary wavelets establishes the new wavefront. When passing an obstacle or aperture, wavelets that are unobstructed spread laterally into the geometrical shadow zone. When aperture width w is comparable to or smaller than wavelength λ (w ≤ λ), diffraction is pronounced and wave fronts emerge as pure semicircles. When w is significantly larger than λ, the central portion continues essentially straight as a plane wave, with bending confined to the edges.
2. Snell's Law and Refraction Invariance
When a wave crosses the interface between two media of differing optical or mechanical densities, its propagation speed changes from v₁ to v₂. Because wavefront continuity must be preserved at every node along the interface boundary, the oscillation frequency f cannot change. Consequently, the spatial wavelength must scale proportionally: λ₁ / λ₂ = v₁ / v₂ = n₂ / n₁. If incidence is oblique, the difference in phase velocity causes the wavefront to pivot according to Snell's law: n₁ sin(θ₁) = n₂ sin(θ₂).
3. Principle of Superposition & Interference
When two or more waves traverse the same region simultaneously, the resultant displacement at any coordinate is the algebraic vector sum of individual displacements: y_total = y₁ + y₂. For two mutually coherent sources operating at identical frequency with zero initial phase difference, constructive interference occurs wherever the path length difference Δr equals an integer multiple of wavelengths (Δr = m λ, where m = 0, ±1, ±2...). Destructive interference (nodal lines with zero amplitude) occurs where the path difference equals an odd number of half-wavelengths: Δr = (m + 0.5) λ.
Mathematical Foundations & Formulas
The fundamental relationships modeled throughout the laboratory include:
Wave Period: T = 1 / f
Wave Number: k = 2π / λ, Angular Frequency: ω = 2πf
Snell's Law of Refraction: (sin θ₁) / (sin θ₂) = v₁ / v₂ = λ₁ / λ₂ = n₂ / n₁
Spherical Reflector Focal Length: f_mirror = R / 2
Aperture Diffraction Condition: sin θ_min = λ / w
Interference Path Difference: Δr = |r₁ - r₂|
Constructive Antinodes: Δr = m × λ (m = 0, 1, 2, ...)
Destructive Nodes: Δr = (m + 0.5) × λ (m = 0, 1, 2, ...)
Real-World Engineering & Industrial Applications
The principles visualized in this simulator are integral to multiple high-tech disciplines:
- Medical Ultrasound Imaging (Sonar & Beamforming): Phased-array transducers emit acoustic waves with carefully calibrated time delays, using constructive interference to steer acoustic beams deep into human tissue and construct detailed real-time diagnostic cross-sections.
- Fiber-Optic Telecommunications: Fiber cables utilize total internal reflection and controlled refractive index profiles across the silica core and cladding to transmit terabits of data per second with minimal signal attenuation.
- Noise-Cancelling Headphones (Acoustic Destructive Superposition): Exterior microphones capture incoming ambient acoustic noise and generate an inverted anti-phase sound wave. When combined at the ear canal, destructive interference neutralizes low-frequency environmental rumble.
- Radar & Satellite Parabolic Dishes: Concave reflectors leverage geometric reflection to collect weak plane waves from distant satellites and focus them onto a focal receiver feed horn, magnifying signal intensity thousands of times.
Practice Problems & Diagnostic Tasks
Test your quantitative physics skills using the five senior school laboratory exercises below:
A ripple tank vibrator creates 24 complete ripples in a span of 8.0 seconds. A student measures the distance between 5 consecutive bright wave crests on the projection screen to be 60.0 cm. Calculate the frequency of the wave source, the wavelength of the water wave, and the wave propagation velocity in meters per second.
A water wave with velocity v = 1.60 m/s and frequency f = 2.00 Hz approaches an adjustable barrier opening of width w = 1.60 m. Describe the diffraction pattern observed on the other side. If the frequency of the source is subsequently doubled while the wave speed and slit width remain unchanged, determine the new wavelength and explain how the diffraction pattern changes.
A plane water wave moving through a deep region with velocity v₁ = 2.40 m/s enters a shallow region at an angle of incidence θ₁ = 35.0 degrees to the normal. In the shallow water, the wave slows down to v₂ = 1.50 m/s. Given that the source frequency is maintained at 1.20 Hz throughout, calculate the wavelength in deep water, the wavelength in shallow water, and the angle of refraction θ₂.
Two loudspeakers spaced 2.50 m apart emit in-phase coherent sound waves at a frequency of 680 Hz into an auditorium where the speed of sound is 340 m/s. A microphone positioned along an observation axis is 4.00 m from Speaker 1 and 4.75 m from Speaker 2. Determine the path difference at the microphone position, and state whether the microphone detects constructive reinforcement, complete destructive cancellation, or an intermediate signal.
An acoustic dish designed as a spherical concave reflector has a radius of curvature R = 1.80 m. Plane sound waves traveling at 340 m/s with a frequency of 850 Hz arrive parallel to the principal axis.
(a) Calculate the location of the focal point F measured from the vertex of the dish.
(b) Find the wavelength of the incident sound.
(c) How many full wavelengths are contained between the vertex of the reflector and its focus?
Click to Reveal Solutions and Step-by-Step Proofs
Solution to Task 1:
1. Frequency f = number of cycles / total time = 24 / 8.0 s = 3.0 Hz.
2. The distance between 5 consecutive crests spans 4 full wavelengths: 4 λ = 60.0 cm = 0.60 m. Therefore, λ = 0.60 m / 4 = 0.15 m (15.0 cm).
3. Velocity v = f × λ = 3.0 Hz × 0.15 m = 0.45 m/s.
Solution to Task 2:
1. Initial wavelength λ₁ = v / f₁ = 1.60 m/s / 2.00 Hz = 0.80 m.
2. Comparing slit width to wavelength: w / λ₁ = 1.60 m / 0.80 m = 2.0. Because w > λ, the wave passes through with minimal spreading in the center (maintaining plane wave fronts) with slight curved bending only at the edges.
3. When frequency is doubled to f₂ = 4.00 Hz: new wavelength λ₂ = 1.60 m/s / 4.00 Hz = 0.40 m.
4. Now w / λ₂ = 1.60 m / 0.40 m = 4.0. The aperture is now four times wider than the wavelength, so diffraction spreading decreases substantially. Waves pass through almost entirely as a straight rectilinear beam with sharp geometrical shadows.
Solution to Task 3:
1. Wavelength in deep water λ₁ = v₁ / f = 2.40 m/s / 1.20 Hz = 2.00 m.
2. Wavelength in shallow water λ₂ = v₂ / f = 1.50 m/s / 1.20 Hz = 1.25 m.
3. By Snell's Law: (sin θ₁) / (sin θ₂) = v₁ / v₂.
sin θ₂ = sin(35.0°) × (v₂ / v₁) = 0.57358 × (1.50 / 2.40) = 0.57358 × 0.625 = 0.35848.
θ₂ = arcsin(0.35848) = 21.0 degrees (the refracted wave bends toward the normal).
Solution to Task 4:
1. Wavelength of the sound λ = v / f = 340 m/s / 680 Hz = 0.50 m.
2. Path difference Δr = |r₂ - r₁| = 4.75 m - 4.00 m = 0.75 m.
3. Express path difference in terms of wavelength: Δr / λ = 0.75 m / 0.50 m = 1.5 = 3 / 2.
4. Since Δr = (1 + 0.5) λ (an odd multiple of half-wavelengths), the sound waves arrive exactly 180 degrees out of phase, creating complete destructive interference (a nodal quiet point).
Solution to Task 5:
1. For a spherical reflector with radius R = 1.80 m, focal length f_mirror = R / 2 = 1.80 m / 2 = 0.90 m from the vertex.
2. Wavelength λ = v / f = 340 m/s / 850 Hz = 0.40 m (40.0 cm).
3. Number of wavelengths between vertex and focus = f_mirror / λ = 0.90 m / 0.40 m = 2.25 wavelengths.