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Wavelength — Definition, Formula & Examples

Wavelength

The period of a sinusoid. Note: This term is not typically used in a mathematics setting.

Key Formula

λ=2πk\lambda = \frac{2\pi}{k}
Where:
  • λ\lambda = Wavelength (the distance for one full cycle)
  • kk = Wave number, the coefficient of the spatial variable in the sinusoidal function

Worked Example

Problem: Find the wavelength of the wave described by y = 3 sin(4x).
Step 1: Identify the wave number k from the function. Here the coefficient of x is 4, so k = 4.
y=3sin⁡(4x)y = 3\sin(4x)
Step 2: Apply the wavelength formula.
λ=2πk=2π4=π2\lambda = \frac{2\pi}{k} = \frac{2\pi}{4} = \frac{\pi}{2}
Answer: The wavelength is π/2 units, meaning one complete wave cycle repeats every π/2 units along the x-axis.

Why It Matters

Wavelength is central in physics, where it describes electromagnetic waves, sound waves, and water waves. In mathematics, the same concept appears as the period of a sinusoidal function. Understanding wavelength helps you move between math models and real-world wave phenomena, such as calculating the frequency of light or the pitch of a sound.

Common Mistakes

Mistake: Confusing wavelength with amplitude.
Correction: Wavelength measures horizontal distance per cycle (how long the wave is), while amplitude measures the vertical distance from the midline to a peak (how tall the wave is). They describe completely different properties.

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