Waves & Simple Harmonic Motion
Explore oscillations, wave properties, and harmonic motion through interactive visualizations.
Simple Harmonic Motion (SHM) is periodic motion where the restoring force is directly proportional to the displacement from equilibrium. Examples include pendulums, springs, and vibrating strings.
Restoring Force
where k is the spring constant and x is displacement (Hooke's Law)
Position in SHM
where A is amplitude, ω is angular frequency, and φ is phase constant
Velocity and Acceleration
Period and Frequency
For a mass-spring system
Simple Pendulum
Period depends only on length L and gravity g (for small angles)
Energy in SHM
Total energy remains constant
Period
2.84 s
Frequency
0.35 Hz
Current Angle
0.0°
Time
0.00 s
Energy Conservation: Watch the energy bars as the pendulum swings. At the highest points, energy is all potential (blue). At the lowest point, energy is all kinetic (orange). Total energy remains constant throughout the motion.
Waves are disturbances that transfer energy through a medium or space. They can be mechanical (requiring a medium) or electromagnetic (can travel through vacuum).
Wave Equation
where A is amplitude, k is wave number, ω is angular frequency
Wave Speed
where f is frequency and λ is wavelength
Wave Number and Angular Frequency
Types of Waves
Transverse waves: Oscillation perpendicular to wave direction (light, water waves)
Longitudinal waves: Oscillation parallel to wave direction (sound, compression waves)
Wave Speed on a String
where T is tension and μ is linear mass density
Sound Wave Speed
where B is bulk modulus and ρ is density
In air at 20°C: v ≈ 343 m/s
Wave Speed
200 px/s
Period
1.00 s
Wave Number
0.031 rad/px
Angular Frequency
6.28 rad/s
Wave Equation: The wave follows y = A sin(kx - ωt). Notice how particles (orange dots) oscillate vertically while the wave pattern moves horizontally. This demonstrates that waves transfer energy, not matter. The relationship v = fλ is always maintained.
When waves overlap, they interfere with each other. This can produce constructive interference (waves add) or destructive interference (waves cancel). Standing waves form when waves reflect and interfere with themselves.
Principle of Superposition
The total displacement is the sum of individual waves
Standing Wave on a String
Formed by two identical waves traveling in opposite directions
Harmonics (String Fixed at Both Ends)
where n = 1, 2, 3, ... and L is string length
Nodes and Antinodes
Nodes: Points of zero amplitude (destructive interference)
Antinodes: Points of maximum amplitude (constructive interference)
Distance between adjacent nodes or antinodes: λ/2
Beats
Periodic variation in amplitude when two waves of slightly different frequencies interfere
Doppler Effect
Frequency shift due to relative motion between source and observer
Use + when approaching, - when receding (observer in numerator, source in denominator)
Fundamental
Wavelength
1200 px
Frequency
0.08 Hz
Wave Speed
100 px/s
Nodes
2
Nodes (red): Points of zero displacement - destructive interference
Antinodes (green): Points of maximum displacement - constructive interference
Standing Waves: For the nth harmonic, there are n+1 nodes and n antinodes. The wavelength is λ = 2L/n, and the frequency is f = nv/(2L). Higher harmonics have shorter wavelengths and higher frequencies. Increasing tension increases wave speed and all frequencies proportionally.