Define wavelength, frequency, period, and amplitude, and use the wave equation v = f(lambda) to relate the speed, frequency, and wavelength of a wave (MA STE Introductory Physics, Waves, HS-PS4-1).

What this topic is asking

This topic opens the Waves reporting category (HS-PS4) of the Massachusetts Introductory Physics MCAS. You must define the basic wave properties, wavelength, frequency, period, and amplitude, and use the wave equation $v = f\lambda$ to relate a wave's speed, frequency, and wavelength. The wave equation is on the reference sheet. The crosscutting idea is patterns: a wave repeats in space (wavelength) and in time (period), and the wave equation ties those repeats to how fast the wave travels.

What a wave is

This is the central idea, and the MCAS tests it directly: a wave moves energy, not material. A cork bobbing on a pond rises and falls as ripples pass but does not move across the pond with them. Sound carries energy through air without the air blowing from the speaker to your ear. Light carries energy across empty space. Understanding that the medium stays put while energy travels through it underlies every wave topic.

The four properties

These four show up on diagrams and in calculations:

• Wavelength is measured from one point on the wave to the next identical point: crest to crest, or trough to trough.
• Frequency and period are inverses: a wave with a period of $0.5$ s has a frequency of $2$ Hz. High frequency means short period.
• Amplitude is independent of the other three. It sets the energy (and, for sound, the loudness; for light, the brightness), but it does not change the speed, frequency, or wavelength.

The wave equation

The reference-sheet formula is

where $v$ is the wave speed (m/s), $f$ is the frequency (Hz), and $\lambda$ is the wavelength (m). The most important consequence the MCAS tests is the inverse relationship at fixed speed: in one medium, the speed does not change, so a wave with a higher frequency must have a proportionally shorter wavelength. This is why high-pitched sounds have shorter wavelengths than low-pitched ones in the same air, and why blue light has a shorter wavelength than red light.

Worked example

Reference-sheet note

The reference sheet prints the wave equation as $v = f\lambda$. What you recall are the definitions of wavelength, frequency, period, and amplitude, that frequency and period are inverses ($T = 1/f$), that amplitude sets the energy but not the speed or frequency, and that at fixed speed frequency and wavelength are inversely related.

Try this

Q1. A wave has a frequency of $25$ Hz and a wavelength of $2.0$ m. Calculate its speed. [2]

• Cue. $v = f\lambda = (25)(2.0) = 50$ m/s.

Q2. A wave travels at $12$ m/s with a frequency of $4.0$ Hz. Calculate its wavelength. [2]

• Cue. $\lambda = \dfrac{v}{f} = \dfrac{12}{4.0} = 3.0$ m.