Formula of Frequency: f = 1/T and f = v/λ, Unit and Examples

The formula of frequency is $f = \frac{1}{T}$ , where T is the time period of one complete oscillation. For a travelling wave you can also find it from speed and wavelength, $f = \frac{v}{\lambda}$ . Frequency counts how many cycles happen in one second, and its SI unit is the hertz (Hz), equal to one per second.

A fan blade, a vibrating tuning fork, an alternating current and a radio signal all have a frequency. The two forms of the formula answer two kinds of question. Use 1/T when you are timing a repeating motion. Use v/λ when you know how fast a wave travels and how long each wave is. Both appear in Class 9 sound, Class 11 oscillations and waves, and the AP Physics 1 unit on waves.

Quick facts

Formula
$f = \frac{1}{T}$ or $f = \frac{v}{\lambda}$
SI unit
hertz (Hz), where 1 Hz = 1 s⁻¹
Dimensional formula
[M⁰ L⁰ T⁻¹]
Type
Scalar
From counting
$f = \frac{N}{t}$ : N oscillations in time t
Angular frequency
$\omega = 2\pi f$ in rad/s
Common symbols
f in most books, ν (nu) for light in Class 12 and modern physics

Symbols in the frequency formula physics students use

Letters used for frequency and their units
Symbol	Stands for	SI unit
$f$	Frequency: cycles completed per second	Hz
$T$	Time period: seconds taken by one cycle	s
$v$	Speed of the wave through the medium	m/s
$\lambda$	Wavelength: distance between two crests	m
$\omega$	Angular frequency	rad/s

Frequency and period are reciprocals. If one cycle takes 0.25 s, four cycles fit into a second and f = 4 Hz. A short period means a high frequency. The period formula page covers the reverse direction, along with the pendulum and spring results that give T directly.

What is the formula of frequency for a wave

A wave moves forward by one wavelength in each period. Its speed is therefore $v = \frac{\lambda}{T}$ . Replace 1/T with f and rearrange:

$v = f\lambda \quad \Rightarrow \quad f = \frac{v}{\lambda}$

This is the frequency of a wave formula. It is the wave velocity formula solved for f. The speed is fixed by the medium and the wavelength is what you measure, so the frequency follows. For light and other electromagnetic waves in vacuum the speed is c = 3.00 × 10⁸ m/s, giving $f = \frac{c}{\lambda}$ . Compare it with the formula of wavelength, which is the same relation solved the other way.

Frequency is set by the source. When light passes from air into glass, or sound passes from air into water, the frequency stays the same. Only the speed and the wavelength change.

Worked examples

Example 1: counting swings

A pendulum completes 30 oscillations in 45 s. Find its frequency and period.

$f = \frac{N}{t} = \frac{30}{45} = 0.667\ \text{Hz}$

$T = \frac{1}{f} = \frac{45}{30} = 1.5\ \text{s}$

Check: 0.667 × 1.5 = 1.0, so the two answers agree.

Example 2: a sound wave

A sound wave in air travels at 340 m/s and has a wavelength of 0.50 m. What is its frequency?

$f = \frac{v}{\lambda} = \frac{340}{0.50} = 680\ \text{Hz}$

That sits well inside the range of human hearing, roughly 20 Hz to 20 kHz.

Example 3: radio wave with a unit change

A radio transmitter sends out waves of wavelength 3.0 m. Find the frequency in megahertz.

Given: v = c = 3.00 × 10⁸ m/s, λ = 3.0 m.

$f = \frac{3.00 \times 10^{8}}{3.0} = 1.0 \times 10^{8}\ \text{Hz} = 100\ \text{MHz}$

Divide by 10⁶ to turn hertz into megahertz.

Example 4: a motor in rpm

An electric motor turns at 1200 revolutions per minute. Find f and T.

Convert to seconds: f = 1200 ÷ 60 = 20 Hz. Then T = 1 ÷ 20 = 0.050 s.

Units, prefixes and the dimensional formula

Frequency is a count per unit time. A count has no dimensions, so frequency has the dimensions of 1/time, [M⁰ L⁰ T⁻¹]. You can see every step on the dimensional formula of frequency page. The unit s⁻¹ is given the special name hertz. Large frequencies use prefixes:

Frequency prefixes you will meet
Unit	Value in Hz	Typical example
kHz (kilohertz)	10³	Sound from a whistle
MHz (megahertz)	10⁶	FM radio
GHz (gigahertz)	10⁹	Microwave ovens, phone signals
THz (terahertz)	10¹²	Infrared radiation

Angular frequency ω measures the same thing in radians per second. One cycle is 2π radians, so $\omega = 2\pi f$ . A wheel spinning at 5 Hz has ω = 10π ≈ 31.4 rad/s. Do not mix up f and ω in formulas such as $y = A\sin(\omega t)$ .

Frequency formula in physics: other forms

Where else frequency shows up
Situation	Formula
Simple pendulum	$f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$
Mass on a spring	$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
Stretched string, fundamental	$f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$
Photon energy	$f = \frac{E}{h}$
Circular motion	$f = \frac{v}{2\pi r}$

Each line is just 1/T for that system. Notice that the pendulum frequency does not depend on the mass of the bob, while the spring frequency does.

Traps to avoid

Reporting 1/T when T was given in milliseconds. Convert T to seconds first: 20 ms is 0.020 s, so f = 50 Hz.
Using ω where f is asked. If you get a number about 6.28 times too big, you have given angular frequency.
Thinking frequency changes on refraction. It does not; the colour of light depends on its frequency, which is why a red beam stays red under water.
Counting the wrong thing. N in f = N/t is the number of full cycles, not the number of times a pendulum passes the centre, which happens twice per cycle.

Quick check: the product f × T must always equal 1.

Frequently asked questions

What is the formula of frequency in physics?

Frequency is f = 1/T, the reciprocal of the time period. For a wave it is also f = v/λ, wave speed divided by wavelength.

What is the SI unit of frequency?

The hertz, symbol Hz. One hertz means one complete cycle per second, so 1 Hz = 1 s⁻¹.

What is the formula for frequency of a wave?

Divide the wave speed by the wavelength: f = v/λ. For light in vacuum, use v = c = 3.00 × 10⁸ m/s.

What is the difference between frequency and angular frequency?

Frequency f counts cycles per second in hertz. Angular frequency ω counts radians per second, and ω = 2πf because one cycle sweeps 2π radians.

Does frequency change when a wave enters a new medium?

No. The source sets the frequency, and it stays the same in every medium. The speed and the wavelength change together so that v = fλ still holds.