Properties of Waves

A wave is an oscillation that transfers energy and momentum through a medium or vacuum without the net transfer of matter. All waves—whether mechanical (sound, water) or electromagnetic (light, X-rays)—obey the fundamental wave equation.

v = f × λ

1. Fundamental Parameters

To fully describe a wave, we use several key quantities:

  • Frequency (f): The number of complete wave cycles passing a fixed point per unit time. Measured in Hertz (Hz).
  • Period (T): The time required for one complete cycle. T = 1/f.
  • Wavelength (λ): The minimum distance between two points that are vibrating in phase.
  • Amplitude (A): The maximum displacement from the equilibrium position.

2. Wave Intensity

Intensity is defined as the power per unit area transported by the wave. For a spherical wave source, intensity decreases with the square of the distance (inverse square law). Crucially, intensity is proportional to the square of the wave's amplitude.

Intensity (I) ∝ (Amplitude)²
Exam Tip: Be careful with unit prefixes. Modern Physics questions often give wavelength in nanometers (nm, 10⁻⁹ m) or frequencies in MHz (10⁶ Hz). Always convert to Base SI units (m, Hz, s, m/s) before plugging into v = fλ.

Deep Dive: Worked Examples

✅ Example 1: Speed of Light in a Vacuum

Visible red light has a wavelength of 700 nm. Calculate its frequency.

Step 1: Identify constants and convert units v = c = 3.00 × 10⁸ m/s, λ = 700 × 10⁻⁹ m
Step 2: Apply formula f = v / λ = (3.00 × 10⁸) / (700 × 10⁻⁹)
Step 3: Solve f ≈ 4.29 × 10¹⁴ Hz

✅ Example 2: Sound in Water

A sonar pulse has a frequency of 50 kHz. If it travels through water at 1500 m/s, what is its wavelength?

Step 1: Convert units f = 50,000 Hz, v = 1500 m/s
Step 2: Apply formula λ = v / f = 1500 / 50000
Step 3: Solve λ = 0.03 m (or 3 cm)

✅ Example 3: Intensity Change

A wave's amplitude is tripled. By what factor does the intensity increase?

Step 1: Use proportionality I₁ ∝ A₁²
Step 2: New State I₂ ∝ (3A₁)² = 9A₁²
Step 3: Compare I₂ / I₁ = 9. The intensity increases by a factor of 9.

✅ Example 4: Period from Oscilloscope

An oscilloscope shows 4 full cycles over a distance of 8 cm. The time base is set to 5 ms/cm. Find the frequency.

Step 1: Find time for 4 cycles Total time = 8 cm × 5 ms/cm = 40 ms
Step 2: Find Period (T) T = 40 ms / 4 cycles = 10 ms = 0.010 s
Step 3: Calculate frequency f = 1 / T = 1 / 0.010 = 100 Hz

✅ Example 5: Phase Difference in Radians

Two points on a wave are separated by 12 cm. If λ = 48 cm, find the phase difference in radians.

Step 1: Find fraction of wavelength Fraction = Δx / λ = 12 / 48 = 0.25 (or 1/4)
Step 2: Convert to radians φ = (Δx / λ) × 2π = 0.25 × 2π
Step 3: Solve φ = π/2 rad (or 1.57 rad)