# Waves: wavelength, frequency and speed

Canonical: https://duckyhelper.com/learn/physics/waves/
Updated: 2026-10-01

A wave is a disturbance that carries energy from place to place without carrying the material along with it. Its wavelength \(\lambda\) is the distance between two crests, its frequency \(f\) is how many waves pass each second, and its speed is \(v = f\lambda\). Transverse waves, like waves on a string, shake across the direction they travel. Longitudinal waves, like sound, shake along it.

## Key ideas

$$
v = f \lambda \qquad T = \frac{1}{f}
$$

- \(v\) is the wave speed, in meters per second (m/s).
- \(f\) is the frequency, in hertz (Hz). 1 Hz is one wave per second.
- \(\lambda\) (lambda) is the wavelength, in meters (m): crest to crest, or compression to compression.
- \(T\) is the period, the time for one full wave to pass, in seconds (s).
- **Amplitude** is how far the medium moves from its rest position. Bigger amplitude means more energy (louder sound, brighter light), but it does not change the speed.

The speed of a wave is set by the **medium** it travels through, not by how fast you shake it. Shake a rope faster and you get a higher frequency with a shorter wavelength, at the same speed. Sound in air at 20 °C travels about 343 m/s. Light in a vacuum travels \(3.00 \times 10^8\) m/s.

**Transverse vs longitudinal**

|  | Transverse | Longitudinal |
| --- | --- | --- |
| Medium moves | At right angles to the wave's travel | Back and forth along the wave's travel |
| Parts | Crests and troughs | Compressions and rarefactions |
| Examples | Waves on a string, light, the S waves of earthquakes | Sound, a pushed Slinky, the P waves of earthquakes |

## Worked examples

**Example 1: wavelength of a musical note**

Problem: The note A above middle C has a frequency of 440 Hz. In air where sound travels at 343 m/s, what is its wavelength?

1. Solve \(v = f\lambda\) for \(\lambda\).

   $$
   \lambda = \frac{v}{f} = \frac{343\ \text{m/s}}{440\ \text{Hz}} = 0.7795\ \text{m}
   $$
2. Round to 3 significant figures. That is about 78 cm, a bit less than a meter stick.

Answer: 0.780 m

**Example 2: an FM radio wave**

Problem: An FM station broadcasts at 98.1 MHz. Radio waves travel at \(3.00 \times 10^8\) m/s. What is the wavelength?

1. Convert megahertz to hertz: 1 MHz = \(10^6\) Hz.

   $$
   f = 98.1 \times 10^6\ \text{Hz}
   $$
2. Divide.

   $$
   \lambda = \frac{3.00 \times 10^8\ \text{m/s}}{98.1 \times 10^6\ \text{Hz}} = 3.058\ \text{m}
   $$

Answer: 3.06 m

**Example 3: water waves at a buoy**

Problem: A buoy bobs up and down 12 times in 30. s. The crests are 2.5 m apart. Find the frequency, the period and the wave speed.

1. Frequency is waves per second.

   $$
   f = \frac{12}{30.\ \text{s}} = 0.40\ \text{Hz}
   $$
2. Period is the flip of frequency.

   $$
   T = \frac{1}{0.40\ \text{Hz}} = 2.5\ \text{s}
   $$
3. Speed.

   $$
   v = f\lambda = (0.40\ \text{Hz})(2.5\ \text{m}) = 1.0\ \text{m/s}
   $$

Answer: 0.40 Hz, period 2.5 s, speed 1.0 m/s

## Common mistakes and how to fix them

- **Measuring wavelength from crest to trough.** That is only half a wavelength. Fix: crest to the next crest.
- **Measuring amplitude from trough to crest.** That is twice the amplitude. Fix: measure from the rest line to a crest.
- **Thinking a higher frequency travels faster.** In one medium, all frequencies of sound travel at the same speed. Fix: if \(f\) goes up, \(\lambda\) goes down.
- **Forgetting metric prefixes.** Fix: kHz is \(10^3\) Hz, MHz is \(10^6\) Hz, nm is \(10^{-9}\) m.

**Practice problems**

1. A wave has a frequency of 5.0 Hz and a wavelength of 0.60 m. What is its speed?
   A. 3.0 m/s
   B. 8.3 m/s
   C. 0.12 m/s
   D. 5.6 m/s

   Answer: 3.0 m/s. \(v = f\lambda = 5.0 \times 0.60 = 3.0\) m/s.

2. What is the period of a wave with a frequency of 50. Hz?

   Answer: 0.020 s. \(T = 1/f = 1 \div 50. = 0.020\) s.

3. What kind of wave is sound in air?
   A. Transverse
   B. Longitudinal

   Answer: Longitudinal. Air molecules move back and forth along the direction the sound travels, making compressions and rarefactions.

4. In the same medium, a wave's frequency doubles. What happens to its wavelength?
   A. It doubles
   B. It halves
   C. It stays the same
   D. It becomes 4 times as long

   Answer: It halves. The medium fixes the speed. With \(v\) constant, \(\lambda = v/f\), so doubling \(f\) halves \(\lambda\).

5. A tuning fork vibrates at 256 Hz. Sound travels at 343 m/s. What is the wavelength?

   Answer: 1.34 m. \(\lambda = v/f = 343 \div 256 = 1.340\) m, which is 1.34 m.

## Frequently asked questions

### Does a wave carry matter along with it?

No. Each bit of the medium moves back and forth around one spot, while the wave's energy moves forward. A duck on a pond bobs up and down as waves pass but stays in about the same place.

### Why does sound travel faster in water than in air?

Sound speed depends on how stiff the medium is and how dense it is. Water is far harder to squeeze than air, which more than makes up for its higher density. Sound travels about 1,480 m/s in water and roughly 343 m/s in air at room temperature.

### Can light travel without a medium?

Yes. Light is an electromagnetic wave: changing electric and magnetic fields that keep each other going. It needs no material, which is why sunlight crosses empty space. Sound is a mechanical wave and needs a medium, so there is no sound in space.

## Sources

- [OpenStax College Physics 2e, 16.9 Waves](https://openstax.org/books/college-physics-2e/pages/16-9-waves), accessed 2026-10-01
- [OpenStax College Physics 2e, 17.2 Speed of Sound, Frequency, and Wavelength](https://openstax.org/books/college-physics-2e/pages/17-2-speed-of-sound-frequency-and-wavelength), accessed 2026-10-01

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## Try asking Ducky

- "I measured the wavelength on this diagram. Did I go crest to crest or crest to trough?"
- "Check my unit conversion from MHz to Hz on number 2."
- "Quiz me on transverse versus longitudinal with real examples."

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