Key Takeaways
- Frequency and period are reciprocals: f = 1/T (doubling frequency halves the period)
- Angular frequency omega = 2 times pi times f is essential for wave equations and AC circuits
- Wave equation: v = f times lambda connects frequency, wavelength, and velocity
- Human hearing spans 20 Hz to 20,000 Hz, with sensitivity peaking around 2-5 kHz
- Higher frequency = higher energy for electromagnetic radiation (E = hf)
- Frequency remains constant when waves change medium; only velocity and wavelength change
What Is Frequency? A Complete Physics Explanation
Frequency is one of the most fundamental concepts in physics, describing how often a repeating event occurs within a specific time interval. Whether discussing sound waves, electromagnetic radiation, mechanical vibrations, or alternating electrical currents, frequency provides the crucial measure of how rapidly these phenomena oscillate. Measured in hertz (Hz), frequency tells us the number of complete cycles that occur per second.
The concept of frequency connects intimately to our everyday experience. The pitch of a musical note directly corresponds to sound wave frequency - a violin string vibrating at 440 Hz produces the note A above middle C. The color of light we perceive depends on electromagnetic frequency - red light oscillates at about 430 trillion Hz, while violet light reaches 750 trillion Hz. Radio stations broadcast at specific frequencies, and our cell phones operate within allocated frequency bands. Even our hearts beat at a measurable frequency, and brain activity is characterized by frequency bands like alpha (8-12 Hz), beta (12-30 Hz), and theta (4-8 Hz) waves.
Essential Frequency Formulas and Equations
f = 1 / T
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, each vibration takes only 1/440 of a second, or about 2.27 milliseconds.
omega = 2 times pi times f = 2 times pi / T
Angular frequency (omega) measures the rate of rotation in radians per second. The factor of 2 times pi arises because one complete cycle corresponds to 2 times pi radians. Angular frequency appears naturally in simple harmonic motion equations like x = A cos(omega times t + phi) and in AC circuit impedance calculations for capacitors and inductors.
v = f times lambda
For any traveling wave, the wave equation connects frequency to wavelength through velocity. This can be rearranged to solve for any variable: f = v/lambda or lambda = v/f. For electromagnetic waves in vacuum, v equals the speed of light (approximately 3 times 10 to the 8th m/s).
Step-by-Step Guide: Calculating Frequency Values
Identify What You Know and Need
Determine which values you have (period, frequency, velocity, wavelength, or angular frequency) and which value you need to calculate. Select the appropriate formula.
Convert Units to Standard Form
Ensure all values are in SI units: frequency in Hz, period in seconds, wavelength in meters, velocity in m/s. Convert kHz to Hz by multiplying by 1000, MHz by 1,000,000, etc.
Apply the Appropriate Formula
For frequency from period: f = 1/T. For wavelength: lambda = v/f. For angular frequency: omega = 2 times pi times f. Substitute your values and calculate.
Convert Result to Appropriate Units
Express your answer in convenient units. Very high frequencies use kHz, MHz, GHz, or THz. Very small periods use ms, microseconds, or ns. Very small wavelengths use mm, micrometers, or nm.
Verify Your Answer
Check that your answer makes physical sense. Higher frequencies mean shorter periods and shorter wavelengths. Use the inverse relationship to verify: f times T should equal 1, and f times lambda should equal v.
Practical Examples: Real-World Frequency Calculations
Example 1: FM Radio Station Wavelength
lambda = (3 times 10^8) / (98.7 times 10^6) = 3.04 meters. This is why FM antennas are typically about 75 cm (quarter-wavelength).
Example 2: Green Light Frequency
f = (3 times 10^8) / (550 times 10^-9) = 5.45 times 10^14 Hz = 545 THz. Visible light oscillates incredibly fast!
Example 3: US Power Grid Angular Frequency
omega = 2 times pi times 60 = 377 rad/s. This value appears constantly in AC circuit analysis (Z = omega times L for inductors).
Units of Frequency: From Millihertz to Petahertz
| Unit | Symbol | Value in Hz | Typical Applications |
|---|---|---|---|
| Millihertz | mHz | 10^-3 Hz | Earth tides, seismic waves, heart rate variability |
| Hertz | Hz | 1 Hz | Human heartbeat, low bass notes, AC power |
| Kilohertz | kHz | 10^3 Hz | Audio frequencies, AM radio, ultrasound |
| Megahertz | MHz | 10^6 Hz | FM radio, TV broadcast, early processors |
| Gigahertz | GHz | 10^9 Hz | WiFi, microwave ovens, modern CPUs |
| Terahertz | THz | 10^12 Hz | Far infrared, molecular spectroscopy, security scanning |
| Petahertz | PHz | 10^15 Hz | Ultraviolet light, X-ray transitions |
The Electromagnetic Spectrum: Frequency Ranges
Electromagnetic radiation spans an enormous range of frequencies, from extremely low frequency (ELF) radio waves below 3 Hz to gamma rays exceeding 10^19 Hz. Each region of the spectrum has distinctive properties and applications.
| EM Region | Frequency Range | Wavelength Range | Common Uses |
|---|---|---|---|
| Radio Waves | 3 Hz - 300 GHz | 1 mm - 100,000 km | Broadcasting, communication, radar |
| Microwaves | 300 MHz - 300 GHz | 1 mm - 1 m | Cooking, satellite, 5G networks |
| Infrared | 300 GHz - 400 THz | 750 nm - 1 mm | Heat sensing, remote controls, fiber optics |
| Visible Light | 400 - 750 THz | 400 - 750 nm | Human vision, photography, displays |
| Ultraviolet | 750 THz - 30 PHz | 10 - 400 nm | Sterilization, black lights, tanning |
| X-rays | 30 PHz - 30 EHz | 0.01 - 10 nm | Medical imaging, security, crystallography |
| Gamma Rays | > 30 EHz | < 0.01 nm | Cancer treatment, sterilization, astronomy |
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 (elephants communicate using infrasound), while frequencies above 20 kHz constitute ultrasound (used by bats and dolphins, and in medical imaging).
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 (called A440 or concert pitch). 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).
Pro Tip: Musical Frequency Relationships
To find the frequency of any note, start from A440 and multiply or divide by 2 for each octave. For semitones, multiply by 1.0595. Middle C is approximately 261.6 Hz, and the lowest note on a piano (A0) is about 27.5 Hz while the highest (C8) reaches 4186 Hz.
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, MRI machines) and dangerous consequences (bridge collapse, building damage from earthquakes, wine glass shattering).
For a simple mass-spring system, the natural frequency is f = (1/2 times pi) times square root of (k/m), where k is the spring constant and m is the mass. For a simple pendulum of length L in gravitational field g, the natural frequency is f = (1/2 times pi) times square root of (g/L). Notice that a pendulum's frequency is independent of its mass - Galileo reportedly discovered this by timing the swings of a chandelier in the Pisa Cathedral.
Common Mistakes to Avoid
- Confusing frequency and period: Remember f = 1/T, not f = T. They are reciprocals, not equals.
- Forgetting unit conversions: 5 MHz is 5,000,000 Hz, not 5000 Hz. Always convert to base units before calculating.
- Using wrong velocity: Sound waves use sound speed (~343 m/s in air), light uses c (~3 times 10^8 m/s). Never mix them up!
- Ignoring medium effects: Sound travels faster in water (1480 m/s) and steel (5960 m/s) than in air.
- Misunderstanding angular frequency: omega is NOT the same as f. They differ by a factor of 2 times pi.
- Assuming frequency changes in new medium: Only wavelength and velocity change; frequency stays constant.
The Doppler Effect: Frequency Shifts in Motion
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 ambulance siren - higher pitch approaching, lower pitch receding.
For sound waves, the observed frequency is f' = f times (v plus/minus v_observer) / (v minus/plus v_source), where the upper signs apply for approaching and lower for receding. The Doppler effect also applies to electromagnetic waves, though relativistic corrections become necessary at high velocities. Doppler radar uses this principle to measure wind speeds and vehicle velocities, while astronomers use the redshift of spectral lines to measure the recession velocity of distant galaxies - the foundation of our understanding of the expanding universe.
Beat Frequency: When Waves Interfere
When two waves of slightly different frequencies interfere, they produce beats - periodic variations in amplitude at the beat frequency: f_beat = |f1 - f2|. 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.
Pro Tip: Using Beats for Precision Tuning
Professional piano tuners aim for about 1-2 beats per second when comparing octaves, due to the inharmonicity of piano strings. String and wind players typically tune to zero beats against a reference pitch. If you hear 3 beats per second between your A string and a 440 Hz tuner, your string is at either 437 Hz or 443 Hz.
Frequency in Electronics and Computing
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 = 1/(j times omega times C)), while an inductor's impedance increases (Z = j times omega times L). This frequency dependence enables the design of filters, tuned circuits, and frequency-selective networks that form the basis of radio receivers, audio equalizers, and signal processing systems.
Frequently Asked Questions
Frequency (f) counts complete cycles per second, measured in hertz (Hz). Angular frequency (omega) measures the rate of phase change in radians per second. They are related by omega = 2 times pi times f, meaning angular frequency is always about 6.28 times larger than regular frequency. Angular frequency is preferred in mathematical physics because wave equations like x = A cos(omega times t) are simpler without the extra 2 times pi factor.
Frequency is the reciprocal of period: f = 1/T. If something completes one cycle every 0.5 seconds (T = 0.5 s), its frequency is f = 1/0.5 = 2 Hz. Conversely, if the frequency is 100 Hz, the period is T = 1/100 = 0.01 seconds = 10 milliseconds. This inverse relationship means doubling the frequency halves the period, and vice versa.
Frequency and wavelength are inversely related through the wave equation: v = f times lambda. For a given wave speed, higher frequency means shorter wavelength, and lower frequency means longer wavelength. For electromagnetic waves in vacuum, using c = 3 times 10^8 m/s, a 1 GHz signal has wavelength 30 cm, while visible light at 500 THz has wavelength 600 nm.
Frequency is determined by the source and represents how many oscillations per second the source produces. When a wave enters a different medium, the particles at the boundary must oscillate at the same rate on both sides to maintain continuity. Therefore, frequency stays constant while wave speed and wavelength adjust. This is why a red laser remains red underwater, even though its wavelength changes.
Healthy young humans can typically hear frequencies from about 20 Hz to 20,000 Hz (20 kHz). This range narrows with age, particularly at high frequencies - most adults lose sensitivity above 15 kHz, and older adults may not hear above 8-10 kHz. Human hearing is most sensitive between 2-5 kHz, which is why emergency sirens and baby cries fall in this range. Sound below 20 Hz (infrasound) can be felt as vibration, and frequencies above 20 kHz (ultrasound) are used in medical imaging and pest deterrents.
In quantum mechanics, the energy of a photon is directly proportional to its frequency: E = h times f, where h is Planck's constant (6.626 times 10^-34 J s). This explains why gamma rays and X-rays (high frequency) are dangerous and can cause ionization, while radio waves (low frequency) pass harmlessly through our bodies. A single photon of blue light carries about 1.7 times more energy than a red photon, which is why UV light can cause sunburn while infrared just feels warm.
The fundamental frequency of a vibrating string depends on three factors: length (L), tension (T), and linear mass density (mu): f = (1/2L) times square root of (T/mu). Shorter strings produce higher frequencies (that's why pressing a fret raises the pitch), higher tension raises frequency (tuning by tightening), and heavier strings produce lower frequencies (bass strings are thicker). Guitar and violin strings demonstrate all three principles.
The Doppler effect is the change in observed frequency when the source and observer are moving relative to each other. When approaching, wavelengths are compressed and frequency increases (higher pitch for sound, blue shift for light). When receding, wavelengths stretch and frequency decreases (lower pitch, red shift). This principle enables police radar guns, weather radar, medical ultrasound blood flow measurement, and astronomers' measurements of stellar velocities and the expansion of the universe.