Frequency is the number of times that periodic changes are completed per unit time, and is a quantity describing the frequency of periodic motion. The unit of frequency is Hertz, abbreviated as “Hertz”, and the symbol is Hz. Every object has a frequency that is determined by its nature and has nothing to do with amplitude, called natural frequency.^{1}

When applied to ultrasonic resonators, “frequency” generally refers to the resonant frequency of a particular vibration mode. If not otherwise specified, “frequency” refers to the frequency of the primary mode of the resonator. For example, if the primary mode of a horn is the axial mode, then the statement “The horn frequency is 19852 Hz.” indicates that the frequency of the axial (primary) mode is 19852 Hz.

Frequency, also called wave frequency, is a measurement of the total number of vibrations or oscillations made within a certain amount of time. There are a few different ways to calculate frequency based on the information you have available to you. Keep reading to learn some of the most common and useful versions.

Table of Contents

### Frequency from Wavelength

#### Learn the formula

The formula for frequency, when given wavelength and the velocity of the wave, is written as **f = V / λ**

- In this formula, f represents frequency, V represents the wave’s velocity, and λ represents the wavelength of the wave.
- Example: A particular sound wave travelling in the air has a wavelength of 322 nm when the sound velocity is 320 m/s. What is the frequency of this sound wave?

#### Convert the wavelength into meters, if necessary

If the wavelength is given in nanometers, you need to convert it into meters by dividing it by the number of nanometers in a single meter. Note that when working with tiny numbers or huge numbers, it is generally easier to write scientific notation values. The deals will be shown in and out of their scientific notation forms for this example.

**Example:**λ = 322 nm- 322 nm x (1 m / 10
^{9}nm) = 3.22 x 10^{-7}m = 0.000000322 m

#### Divide the velocity by the wavelength

Divide the wave’s velocity, *V*, by the wavelength converted into meters, *λ*, to find the frequency, *f*.

**Example:**f = V / λ = 320 / 0.000000322 = 993788819.88 = 9.94 x 10^{8}

#### Write your answer

After completing the previous step, you will have completed your calculation for the frequency of the wave. Write your reply in Hertz, Hz, which is the unit for frequency.

- Example: The frequency of this wave is 9.94 x 10
^{8}Hz.

### Frequency from Time or Period

#### Learn the formula

Frequency and the time taken to finish a single wave oscillation are inversely proportional. As such, the formula for calculating frequency when given the time taken to complete a wave cycle is written as **f = 1 / T**. In this formula, f represents frequency, and T represents the period or amount of time required to complete a single wave oscillation.

**Example A:**The time for a particular wave to complete a single oscillation is 0.32 seconds. What is the frequency of this wave?**Example B:**In 0.57 seconds, a particular wave can complete 15 oscillations. What is the frequency of this wave?

#### Divide the number of oscillations by the time

Usually, you will be told how long it takes to complete a single oscillation, in which case, you would divide the number 1 by the period, T. If given a period for numerous oscillations, however, you will need to divide the number of oscillations by the overall period required to complete them.

**Example A:**f = 1 / T = 1 / 0.32 = 3.125**Example B:**f = 1 / T = 15 / 0.57 = 26.316

#### Write your answer

This calculation should tell you the frequency of the wave. Write your reply in Hertz, *Hz*, the unit for frequency.

**Example A:**The frequency of this wave is 3.125 Hz.**Example B:**The frequency of this wave is 26.316 Hz.

### Frequency from Angular Frequency

**Learn the formula**

When told the angular frequency of a wave but not the standard frequency of that same wave, the formula to calculate the standard frequency is written as **f = ω / (2π). **In this formula, *f* represents the frequency of the wave, and *ω* represents the angular frequency. As with any mathematical problem, *π* stands for pi, a mathematical constant.

**Example:**A particular wave rotates with an angular frequency of 7.17 radians per second. What is the frequency of that wave?

#### Multiply pi by two

To find the denominator of the equation, you need to double the value of pi, 3.14.

**Example:**2 * π = 2 * 3.14 = 6.28

#### Divide the angular frequency by the double of pi

Divide the angular frequency of the wave, given in radians per second, by 6.28, the doubled value of pi.Example: f = ω / (2π) = 7.17 / (2 * 3.14) = 7.17 / 6.28 = 1.14

#### Write your answer

This final bit of calculation should indicate what the frequency of the wave is. Write your reply in Hertz, *Hz*, the unit for frequency.

**Example:**The frequency of this wave is 1.14 Hz.