Frequency is a physical quantity that describes how often a periodic motion or wave repeats itself in a given time interval. It is defined as the number of complete oscillations or vibrations produced per second. The SI unit of frequency is hertz (Hz), where one hertz corresponds to one complete cycle per second.
Important Terminologies
1. Period (T)
Period is the time taken to complete one full cycle or oscillation. It is the reciprocal of frequency and is given by
where 'f' signifies frequency in Hertz.
E.g., if a wave boasts a frequency of 5 Hz, its period spans 1/5 seconds or 0.2 seconds.
2. Amplitude
Amplitude is the maximum displacement of a wave from its mean position. In sound waves, it determines loudness, while in electrical signals, it represents peak voltage or current.
3. Wavelength (λ)
Wavelength is the distance between two consecutive points in the same phase, such as two crests or troughs. It is inversely proportional to frequency and is given by
where 'c' represents wave speed.
4. Angular Frequency (ω)
Angular frequency is another form of frequency used in wave and harmonic motion. It is related to frequency by
where 'ω' denotes angular frequency in radians per second, and 'f' signifies frequency in Hertz.
Types of Frequency
1. Electrical Frequency
In electrical engineering, it refers to the number of AC cycles completed per second and is widely used in power systems and electronic circuits.
2. Rotational Frequency
It describes the rate of rotation or oscillation of an object and is measured in radians per second (rad/s). It is important in wave motion and harmonic motion.
Rotational Frequency (ω) is computed by dividing the angular displacement (θ) by the time interval (t):
ω = \frac{θ}{t}
3. Angular Frequency
Angular frequency indicates the rate at which an object rotates or oscillates in terms of radians per second. It is widely used in wave motion and simple harmonic motion.
\omega = \frac{d\phi}{dt}
4. Spatial Frequency
It represents the number of oscillations per unit distance and is mainly used in image processing and signal analysis.
f = \frac{1}{λ}
5. Sound and Light Frequency
In sound, frequency determines the pitch, while in light it determines the color. Higher frequency means higher pitch in sound and shifts light toward blue, whereas lower frequency produces lower pitch and red light.
How to Determine Frequency?
Step 1: Wave or signal nature
Determine whether the given wave is electrical, sound, or any other type.
Step 2: Evaluate period
Measure the time taken to complete one full cycle.
Step 3: Compute frequency
Leverage the frequency formula as follows:
For electrical signals:
f = \frac{1}{T} For sound waves:
f = \frac{c}{λ} For Angular Frequency
(ω) = 2π × f For Wave Speed
(v) = f × λ
Step 4: Unify units and conversion
Ensure correct unit conversion (Hz, kHz, MHz) and interpret the result according to the application.
Hz to kHz: Kilohertz (kHz) = Hertz (Hz) / 103
Hz to MHz: Megahertz (MHz) = Hertz (Hz) / 106
kHz to Hz: Hertz (Hz) = Kilohertz (kHz) × 103
MHz to Hz: Hertz (Hz) = Megahertz (MHz) × 106
Step 5: Interpretation
The calculated frequency should be interpreted according to the given application. E.g., in electrical signals, frequency helps in circuit design and waveform analysis.
Methods of Measuring Frequency
- Frequency is measured using a digital multimeter (DMM) in frequency mode.
- It is mainly used for measuring the frequency of AC signals.
- The auto range feature automatically selects the suitable frequency range.
- Recording MIN/MAX helps record frequency variations over a specific time.
Applications
- Physics & Engineering: Wave analysis, circuit, and communication design.
- Medicine & Biology: MRI, ultrasound, and EEG diagnostics.
- Geophysics: Seismic wave analysis and Earth structure study.
- Music & Environment: Audio production and climate pattern analysis.
Solved Problems
Question 1. Suppose you are grappling with an alternating current (AC) signal, and your measurements unveil a period of 0.02 seconds. Determine the frequency.
Solution: Frequency (f) = 1 / Period (T)
f = 1/0.02s
f = 50 Hz
The frequency of this AC signal amounts to 50 Hertz.
Question 2. The speed of sound in air is approximately 343 meters per second, and the wavelength of a specific note measures 0.7 meters. What is the frequency?
Solution: Frequency (f) = Speed of Sound (c) / Wavelength (λ)
f = [343 m/s] / 0.7 m
f = 490 Hz
The frequency of this musical note equates to 490 Hertz.
Question 3. A radio signal is identified by a wavelength of 3 meters. Find the frequency.
Solution: Frequency (f) = Speed of Light (c) / Wavelength (λ)
f = [3 x 10^8 m/s] / 3m
f = 100000000 Hz
f = 100 MHz
The frequency of this radio wave stands at 100 megahertz.
Question 4. Suppose you are provided a harmonic oscillator device of frequency 2 Hz. How would you find its angular frequency?
Solution: Angular Frequency (ω) = 2π × Frequency (f)
Frequency (f) = 2 Hz
ω = 2π × 2 Hz = 4π radians per second
The angular frequency of this harmonic oscillator is 4π radians per second.
Question 5. If the frequency and wavelength of the waves are 5 Hz and 2 m, respectively. Find the speed of waves.
Solution: Using the formula: Wave Speed (v) = Frequency (f) × Wavelength (λ)
Frequency (f) = 5 Hz
Wavelength (λ) = 2 meters
v = 5 Hz × 2 meters = 10 meters per second
speed of these water waves is 10 meters per second
Unsolved Problems
Question 1: An alternating current completes 150 cycles in 3 seconds. Find its frequency and time.
Question 2: A sound wave travels with a speed of 340 m/s and has a wavelength of 0.68 m. Calculate its frequency.
Question 3: A wave has an angular frequency of 16π rad/s. Determine its frequency.
Question 4: A radio wave is transmitted at a frequency of 100 MHz. Calculate its wavelength.
Question 5: A harmonic oscillator has a frequency of 5 Hz. Find its angular frequency.