Understanding Frequency: A Complete Physics Guide
Frequency is one of the most fundamental concepts in physics, describing how often a repeating event occurs within a specific time interval. Whether we're discussing sound waves, electromagnetic radiation, mechanical vibrations, or alternating electrical currents, frequency provides the crucial measure of how rapidly these phenomena oscillate. From the low-frequency hum of power lines at 50 or 60 Hz to the incredibly high frequencies of gamma rays exceeding 10^19 Hz, frequency spans an enormous range across different physical systems.
The concept of frequency connects intimately to our everyday experience. The pitch of a musical note directly corresponds to sound wave frequency. The color of light we perceive depends on electromagnetic frequency. Radio stations broadcast at specific frequencies, and our cell phones operate within allocated frequency bands. Even our hearts beat at a measurable frequency, and the rhythms of brain activity are characterized by frequency bands like alpha, beta, and theta waves.
The Fundamental Frequency Equation
The most basic relationship in frequency analysis connects frequency (f) to period (T), where period represents the time required for one complete cycle of an oscillation:
Frequency equals the reciprocal of period
This elegantly simple equation tells us that frequency and period are inversely related. If a pendulum completes one swing every 2 seconds (T = 2 s), its frequency is 0.5 Hz, meaning it completes half a cycle per second. Conversely, if a tuning fork vibrates at 440 Hz (the musical note A above middle C), each vibration takes only 1/440 of a second, or about 2.27 milliseconds.
Units of Frequency
The SI unit of frequency is the hertz (Hz), named after Heinrich Hertz who first conclusively proved the existence of electromagnetic waves. One hertz equals one cycle per second. For different applications, various prefixes scale the unit appropriately:
| Unit | Symbol | Value in Hz | Example Applications |
|---|---|---|---|
| Millihertz | mHz | 10⁻³ Hz | Earth tides, seismic waves |
| Hertz | Hz | 1 Hz | Heart rate, AC power |
| Kilohertz | kHz | 10³ Hz | Audio frequencies, AM radio |
| Megahertz | MHz | 10⁶ Hz | FM radio, TV broadcast |
| Gigahertz | GHz | 10⁹ Hz | WiFi, microwave, CPUs |
| Terahertz | THz | 10¹² Hz | Infrared, molecular spectroscopy |
Angular Frequency
In physics and engineering, angular frequency (ω, omega) often proves more convenient than ordinary frequency, particularly when dealing with circular motion, wave equations, and AC circuit analysis. Angular frequency measures the rate of rotation in radians per second:
Angular frequency in radians per second
The factor of 2π arises because one complete cycle corresponds to 2π radians of rotation. Angular frequency appears naturally in the mathematical description of simple harmonic motion, where position varies as x = A cos(ωt + φ), and in the impedance calculations for capacitors and inductors in AC circuits.
The Wave Equation: Frequency, Wavelength, and Velocity
For any traveling wave, frequency connects to wavelength through the wave velocity:
Wave velocity equals frequency times wavelength
This equation can be rearranged to solve for any variable: f = v/λ or λ = v/f. For electromagnetic waves in vacuum, v equals the speed of light (approximately 3 × 10⁸ m/s), giving us a direct relationship between the frequency and wavelength of light, radio waves, X-rays, and other electromagnetic radiation.
Speed of Various Wave Types
| Wave Type | Medium | Approximate Speed |
|---|---|---|
| Electromagnetic waves | Vacuum | 299,792,458 m/s |
| Sound | Air (20°C) | 343 m/s |
| Sound | Water | 1,480 m/s |
| Sound | Steel | 5,960 m/s |
| Seismic P-waves | Earth's crust | 5,000-8,000 m/s |
Worked Examples
Example 1: Frequency from Period
Problem: A mechanical oscillator completes one full cycle every 0.02 seconds. What is its frequency?
Solution:
Period T = 0.02 s
f = 1/T = 1/0.02 = 50 Hz
The oscillator vibrates at 50 cycles per second.
Example 2: Period from Frequency
Problem: An FM radio station broadcasts at 98.7 MHz. What is the period of these electromagnetic waves?
Solution:
f = 98.7 MHz = 98.7 × 10⁶ Hz
T = 1/f = 1/(98.7 × 10⁶) = 1.013 × 10⁻⁸ s
T ≈ 10.13 nanoseconds
Example 3: Angular Frequency
Problem: Calculate the angular frequency of the standard electrical power supply in the United States (60 Hz).
Solution:
f = 60 Hz
ω = 2πf = 2 × π × 60 = 120π
ω ≈ 377 rad/s
Example 4: Wavelength of Radio Waves
Problem: An AM radio station broadcasts at 1000 kHz. What is the wavelength of these radio waves?
Solution:
f = 1000 kHz = 1 × 10⁶ Hz
v = c = 3 × 10⁸ m/s (speed of light)
λ = v/f = (3 × 10⁸)/(1 × 10⁶) = 300 meters
Example 5: Frequency of Visible Light
Problem: Green light has a wavelength of approximately 550 nm. What is its frequency?
Solution:
λ = 550 nm = 550 × 10⁻⁹ m
v = c = 3 × 10⁸ m/s
f = v/λ = (3 × 10⁸)/(550 × 10⁻⁹)
f = 5.45 × 10¹⁴ Hz ≈ 545 THz
The Electromagnetic Spectrum
Electromagnetic radiation spans an enormous range of frequencies, from extremely low frequency (ELF) radio waves below 3 Hz to gamma rays exceeding 10¹⁹ Hz. Each region of the spectrum has distinctive properties and applications:
| Region | Frequency Range | Wavelength Range |
|---|---|---|
| Radio waves | 3 Hz - 300 GHz | 1 mm - 100,000 km |
| Microwaves | 300 MHz - 300 GHz | 1 mm - 1 m |
| Infrared | 300 GHz - 400 THz | 750 nm - 1 mm |
| Visible light | 400 - 750 THz | 400 - 750 nm |
| Ultraviolet | 750 THz - 30 PHz | 10 - 400 nm |
| X-rays | 30 PHz - 30 EHz | 0.01 - 10 nm |
Sound and Acoustic Frequencies
Human hearing typically spans frequencies from about 20 Hz to 20,000 Hz (20 kHz), though this range narrows with age. Sound below 20 Hz is called infrasound, while frequencies above 20 kHz constitute ultrasound. Musical instruments produce fundamental frequencies and harmonics (integer multiples of the fundamental) that give each instrument its characteristic timbre.
In music, the relationship between frequency and pitch follows a logarithmic scale. Each octave represents a doubling of frequency. The note A above middle C is standardized at 440 Hz. The next A up is 880 Hz, and the A below is 220 Hz. The twelve semitones in an octave are equally spaced on a logarithmic scale, meaning each semitone step multiplies the frequency by the twelfth root of 2 (approximately 1.0595).
Resonance and Natural Frequency
Every physical system capable of oscillation has one or more natural frequencies at which it tends to vibrate. When driven at its natural frequency, a system experiences resonance, leading to large amplitude oscillations. This phenomenon has both useful applications (musical instruments, radio tuning) and dangerous consequences (bridge collapse, building damage from earthquakes).
For a simple mass-spring system, the natural frequency is:
Where k is the spring constant and m is the mass
For a simple pendulum of length L in gravitational field g:
Frequency in Electronics
Electronic circuits operate with frequencies ranging from DC (zero frequency) through radio frequencies up to optical frequencies in photonic systems. The clock frequency of modern computer processors typically ranges from 2 to 5 GHz, meaning they execute billions of basic operations per second.
In AC circuit analysis, the behavior of capacitors and inductors depends directly on frequency. A capacitor's impedance decreases with increasing frequency (Z_C = 1/jωC), while an inductor's impedance increases (Z_L = jωL). This frequency dependence enables the design of filters, tuned circuits, and frequency-selective networks.
Doppler Effect
When a wave source or observer moves relative to the medium, the observed frequency differs from the emitted frequency. This Doppler effect is familiar from the changing pitch of a passing siren. For sound waves:
Upper signs for approaching, lower for receding
The Doppler effect applies to electromagnetic waves as well, though relativistic corrections become necessary at high velocities. Doppler radar uses this principle to measure wind speeds and vehicle velocities. Astronomers use the redshift of spectral lines to measure the recession velocity of distant galaxies.
Beat Frequency
When two waves of slightly different frequencies interfere, they produce beats - periodic variations in amplitude at the beat frequency:
Musicians use beat frequencies to tune instruments. When two notes are slightly out of tune, the beats are audible as a "wobbling" in the combined sound. As the frequencies approach each other, the beat frequency decreases until it becomes imperceptible when the notes match perfectly.
Related Calculators
- Wavelength Calculator - Calculate wavelength from frequency
- Inductance Calculator - Calculate inductive reactance
- Capacitance Calculator - Calculate capacitive reactance
- Frequency Converter - Convert between frequency units
Frequently Asked Questions
What is the difference between frequency and angular frequency?
Frequency (f) counts complete cycles per second, measured in hertz. Angular frequency (ω) measures the rate of phase change in radians per second. They're related by ω = 2πf. Angular frequency is often more convenient in mathematical physics because many equations simplify when using radians.
Why does frequency remain constant when a wave changes medium?
Frequency is determined by the source and represents how many oscillations per second the source produces. When a wave enters a different medium, its speed changes (and therefore its wavelength changes), but the frequency must stay constant to maintain continuity at the boundary between media.
Can frequency be negative?
In standard physics, frequency is always positive or zero. However, in signal processing and quantum mechanics, negative frequencies sometimes appear in mathematical representations. A negative frequency typically represents a wave rotating in the opposite direction or a complex conjugate component in Fourier analysis.
What determines the frequency of sound from a vibrating string?
The fundamental frequency of a vibrating string depends on its length (L), tension (T), and linear mass density (μ): f = (1/2L)√(T/μ). Shorter strings, higher tension, and lower mass density all produce higher frequencies. This is why guitar strings are tuned by adjusting tension and why different strings have different thicknesses.
How is frequency related to energy?
In quantum mechanics, the energy of a photon is directly proportional to its frequency: E = hf, where h is Planck's constant (6.626 × 10⁻³⁴ J·s). This explains why higher frequency radiation (like X-rays and gamma rays) carries more energy per photon than lower frequency radiation (like radio waves).