🌊 Waves Calculators

What Is Wave Physics?

A wave is a disturbance that transfers energy through a medium (or through space, for electromagnetic waves) without permanently moving the medium itself. Wave physics describes oscillations in sound, light, water, seismic activity, and quantum mechanics. Understanding frequency, wavelength, wave speed, and amplitude is essential for fields ranging from acoustics and optics to telecommunications and structural engineering.

Core Wave Formulas

Step-by-Step Example: Wave Speed

A sound wave in air has a frequency of 440 Hz (concert A note) and a wavelength of 0.773 m. What is its wave speed?

1. Apply v = f × λ: v = 440 × 0.773 = 340.1 m/s

This matches the known speed of sound in air at 20°C (~343 m/s), confirming the calculation.

Step-by-Step Example: Pendulum Period

A grandfather clock uses a pendulum of length 0.994 m. What is its period (time for one full swing)?

1. Apply T = 2π × √(L/g): T = 2π × √(0.994/9.81)
2. Calculate inside the root: 0.994/9.81 = 0.1013; √0.1013 = 0.3183
3. Multiply: T = 2π × 0.3183 = 2.00 seconds

A pendulum of ~1 m length swings exactly once per second, which is why grandfather clocks use this length — the "tick-tock" marks half-periods of 1 second each.

Step-by-Step Example: Spring Constant

A spring stretches 0.05 m when a 2 N force is applied. What is the spring constant?

1. Rearrange Hooke's Law: k = F / x = 2 / 0.05 = 40 N/m

Real-World Applications

• Music and Acoustics: Every musical note corresponds to a specific frequency; v = fλ determines how sound travels in concert halls
• Telecommunications: Radio, Wi-Fi, and cellular signals are all electromagnetic waves described by f = c/λ
• Seismology: Earthquake magnitude is analyzed using wave speed and frequency data from seismographs
• Medical Imaging: Ultrasound uses high-frequency waves (1–20 MHz) to image tissue; wavelength determines resolution
• Mechanical Engineering: Springs and vibration dampers are designed using Hooke's Law to prevent resonance failures

Historical Context

The mathematical study of waves began with Christiaan Huygens' wave theory of light (1678) and was formalized by Leonhard Euler, Jean d'Alembert, and Daniel Bernoulli in the 18th century. The pendulum's isochronous property — that period is independent of amplitude for small swings — was discovered by Galileo Galilei around 1602, leading to the first accurate mechanical clocks. Robert Hooke stated his spring law in 1676: "as the extension, so the force."

Frequently Asked Questions

What is the relationship between frequency and wavelength?

Frequency and wavelength are inversely proportional when wave speed is constant: λ = v/f. In a given medium, a higher frequency means a shorter wavelength and vice versa. This is why high-pitched sounds (high frequency) have shorter wavelengths than low-pitched sounds (low frequency) in the same air.

What is the difference between transverse and longitudinal waves?

In a transverse wave, the medium oscillates perpendicular to the direction of wave travel (e.g., waves on a string, light waves). In a longitudinal wave, the medium oscillates parallel to the direction of travel (e.g., sound waves, compression waves in a spring).

Why does a pendulum's period not depend on its mass?

In the formula T = 2π√(L/g), mass cancels out because both the restoring force and the inertia scale with mass proportionally. A heavier pendulum requires more force to restore but also has more inertia resisting acceleration — these effects exactly cancel, leaving only length and gravity as factors.

What is resonance?

Resonance occurs when an external force is applied at a system's natural frequency, causing oscillation amplitude to grow dramatically. Famous examples include the Tacoma Narrows Bridge collapse (1940), where wind-induced resonance caused catastrophic failure, and the wine glass shattering from a singer's voice at the glass's resonant frequency.