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Waves — Frequency, Wavelength, and Speed

24 min Vibrations & Waves Math

Waves

A wave carries energy through space using the sine function. The relationship between frequency, wavelength, and speed is a simple proportion.

Wave Equation

$$v = f\lambda$$

$v$ = wave speed (m/s), $f$ = frequency (Hz), $\lambda$ = wavelength (m).

Key Wave Speeds

Example 1

Middle C (261.6 Hz) in air. Find the wavelength.

$\lambda = v/f = 343/261.6 = 1.31$ m.

Example 2

An FM radio station broadcasts at 101.1 MHz. Wavelength?

$\lambda = c/f = (3 \times 10^8)/(101.1 \times 10^6) = 2.97$ m.

Example 3

A wave has wavelength 0.5 m and speed 2 m/s. Find frequency and period.

$f = v/\lambda = 2/0.5 = 4$ Hz.

$T = 1/f = 0.25$ s.

1. Sound: $f = 440$ Hz in air. Wavelength?

2. Light: $\lambda = 500$ nm. Frequency?

3. A wave: $v = 10$ m/s, $\lambda = 2$ m. Find $f$.

4. What is the wavelength of a 1 GHz microwave?

5. Sound in water: $f = 500$ Hz. Wavelength?

6. If frequency doubles, what happens to wavelength (same medium)?

Show Answer Key

1. $343/440 \approx 0.78$ m

2. $f = 3 \times 10^8 / (500 \times 10^{-9}) = 6 \times 10^{14}$ Hz

3. 5 Hz

4. $3 \times 10^8 / 10^9 = 0.3$ m = 30 cm

5. $1480/500 = 2.96$ m

6. Wavelength halves ($\lambda = v/f$).