Frequency, Period, and Wave Speed

The video lesson for this topic is coming soon.

Essential Question

How fast does a wave travel?

Waves repeat over and over, and they travel from place to place. By the end of this lesson, you should be able to find the period and frequency of a wave and use v = fλ to solve for a wave’s speed, frequency, or wavelength.

Period

Time for one full cycle
Symbol T
Unit: seconds (s)

A graph of displacement versus time in seconds from 0 to 3. A blue sine wave makes three full cycles. A double arrow between the first two crests, at 0.25 s and 1.25 s, is labeled period T.

The period (T) is the time it takes for one full cycle of a wave to pass a point. On a graph of displacement versus time, you can measure it from one crest to the next crest. The wave on this graph has a period of 1.0 s. Period is a time, so it is measured in seconds.

Frequency

Cycles per second
Symbol f
Unit: hertz (Hz)
1 Hz = 1 cycle per second

Two waves drawn over the same 1 second window. The top blue wave makes 2 cycles and is labeled 2 Hz. The bottom red wave makes 4 cycles and is labeled 4 Hz. Caption: more cycles per second means higher frequency.

Frequency (f) is the number of cycles that pass a point each second. Its unit is the hertz (Hz), and 1 Hz means one cycle per second. In one second, the top wave completes 2 cycles, so its frequency is 2 Hz. The bottom wave completes 4 cycles, so its frequency is 4 Hz.

Frequency and Period Are Inverses

f = 1 / T and T = 1 / f

f = frequency, in hertz (Hz)
T = period, in seconds (s)

Two yellow equation boxes: f equals 1 over T, and T equals 1 over f. Below them, three cards: T equals 2.0 s gives f equals 0.50 Hz; T equals 0.50 s gives f equals 2.0 Hz; T equals 0.10 s gives f equals 10. Hz. Caption: longer period, lower frequency.

Frequency and period are inverses (reciprocals). If one cycle takes 0.50 s, then 2 cycles fit in each second, so the frequency is 2.0 Hz. A long period means a low frequency, and a short period means a high frequency. Since 1 Hz is one cycle per second, a hertz is the same as 1/s.

The Wave Equation

v = f λ

v = wave speed, in meters per second (m/s)
f = frequency, in hertz (Hz)
λ = wavelength, in meters (m)

The equation v = f lambda in a yellow box. Below it, three cards: v is wave speed in m/s, f is frequency in Hz, and lambda is wavelength in m.

The wave equation connects how fast a wave travels to how often it repeats and how long each cycle is. Check the units: hertz is 1/s, so Hz × m = m/s, which is a speed.

Why v = f λ

In time T, the wave moves λ
v = d / t = λ / T
1/T = f, so v = f λ

Two snapshots of the same short wave. In the top snapshot, labeled t = 0, the front crest is marked with a red dot. In the bottom snapshot, labeled t = T, the whole wave has shifted right and the red dot is one wavelength farther along. A double arrow between the two dot positions is labeled lambda. Caption: in one period, the crest moves one wavelength.

Follow one crest (the red dot). In one period, the next crest takes its place, which means the wave has moved forward exactly one wavelength. Speed is distance divided by time, so v = λ/T. Since 1/T is the frequency, this becomes v = fλ.

Speed Depends on the Medium

The medium sets the speed
Same medium, same speed
Raise f, and λ gets shorter

Three cards showing the speed of sound: in air, 343 meters per second; in water, about 1,500 meters per second; in steel, about 5,900 meters per second. Caption: speed of sound in different media.

The speed of a wave depends on the medium it travels through, not on how fast the source shakes. Sound moves at about 343 m/s in room-temperature air, but much faster in water and steel. In one medium, every frequency travels at the same speed. So if you raise the frequency, v stays the same and the wavelength gets shorter to keep v = fλ true.

Solving for Any Variable

Find speed: v = f λ
Find frequency: f = v / λ
Find wavelength: λ = v / f

A triangle split into three parts with v on top and f and lambda on the bottom. Caption: cover the one you want.

Rearrange v = fλ to solve for whatever the problem asks for. To find frequency, divide both sides by λ. To find wavelength, divide both sides by f. The triangle is a shortcut: cover the quantity you want, and what’s left shows the operation.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Bobbing Buoy

A buoy bobs up and down once every 4.0 s as waves pass. What is the frequency of the waves?

A red buoy floating on a blue wave, with a red double arrow showing it moves up and down. Label: one bob every 4.0 s.

Knowns: T = 4.0 s
Equation: f = 1/T
Substitute: f = 1 / 4.0 s
Answer: f = 0.25 Hz

Since one cycle takes 4 seconds, only a quarter of a cycle happens each second, which matches 0.25 Hz.

Example 2: Wave Speed on a Rope

You shake a rope at 2.5 Hz, and the waves have a wavelength of 1.6 m. How fast do the waves travel along the rope?

A blue wave on a rope with a double arrow from crest to crest labeled 1.6 m, and a label f = 2.5 Hz.

Knowns: f = 2.5 Hz, λ = 1.6 m
Equation: v = fλ
Substitute: v = (2.5 Hz)(1.6 m) = (2.5 1/s)(1.6 m)
Answer: v = 4.0 m/s

Example 3: Wavelength of a Sound

A speaker plays a note at 686 Hz. Sound travels through the air at 343 m/s. What is the wavelength of the sound?

A speaker on the left sending curved sound wave fronts to the right. Labels: f = 686 Hz and v = 343 m/s.

Knowns: f = 686 Hz, v = 343 m/s
Equation: v = fλ, so λ = v/f
Substitute: λ = (343 m/s) / (686 Hz). Hz is 1/s, so the seconds cancel, leaving meters.
Answer: λ = 0.500 m

Example 4: Ocean Waves

Ocean waves travel toward a pier at 6.0 m/s. The crests are 12 m apart. Find the frequency and the period of the waves.

Blue ocean waves moving right toward a pier post, with a double arrow from crest to crest labeled 12 m and an arrow labeled 6.0 m/s.

Knowns: v = 6.0 m/s, λ = 12 m
Frequency equation: f = v/λ
Substitute: f = (6.0 m/s) / (12 m) = 0.50 Hz
Period: T = 1/f = 1 / 0.50 Hz = 2.0 s

A crest hits the pier every 2.0 seconds.

Big Ideas

Period T: seconds per cycle
Frequency f: cycles per second (Hz)
f = 1/T and v = f λ
The medium sets the wave speed

Back to the essential question: a wave travels one wavelength every period, so its speed is v = fλ. The speed is set by the medium, so if the frequency changes, the wavelength changes to match. Next up, we’ll see what happens when two waves run into each other in Interference.