Wave Equation Calculator: v = f·λ, ω, k, Period, Phase

A traveling sinusoidal wave is fully specified by four numbers: amplitude A, wavelength λ, frequency f (or equivalently period T = 1/f), and an initial phase φ. The geometry is what those numbers describe. Amplitude is the peak displacement from equilibrium. Wavelength is the spatial repeat distance, the gap between two crests at a single instant. Frequency is the temporal repeat rate, the number of crests passing a fixed point per second. Phase is where in its cycle the wave sits at the chosen origin of x and t.

Two waves with the same A, λ, and f but different φ are the same wave shape shifted along the axis. That phase difference is what controls interference. When two crests align (Δφ = 0), they add to a 2A peak. When a crest meets a trough (Δφ = π), they cancel. Half the questions in this corner of physics reduce to "what's the phase difference between these two contributions," and the answer is almost always either a path-length difference times k, a time delay times ω, or a reflection that flipped a sign.

Snapshots and time histories are different views of the same y(x, t) surface. A snapshot fixes t and varies x. A time history fixes x and varies t. The ratio of how fast crests move spatially (1/λ) to how fast they move temporally (1/T) is the wave speed. That's all v = fλ is saying: spatial repeat times temporal rate equals propagation speed.

When a wave crosses from one medium into another, frequency is conserved and wavelength isn't. That's not a convention; it's a boundary condition. The source is still oscillating at the same rate, so the same number of crests per second has to enter the new medium as left the old one. If v changes, λ has to change to compensate. Light going from air to glass: f stays, v drops by a factor of n, so λ_glass = λ_air / n. Sound going from air to water: f stays, v jumps from 343 m/s to about 1480 m/s, so λ jumps by the same factor.

When wave speed v is constant inside one medium, λ and f move in lockstep through λ = v/f. Double f, halve λ. Each 10× increase in frequency causes a 10× decrease in wavelength. That's why low bass needs large speakers (long wavelengths to push air a long way) and tweeters can be tiny. At f = 100 Hz in air, λ = 3.43 m. At f = 1 kHz, λ = 0.343 m. At f = 10 kHz, λ = 0.034 m.

Sound speed in air depends on temperature because warmer molecules move faster and transfer pressure quicker. v ≈ 331 + 0.6T (T in Celsius) is the linearised approximation good across normal room conditions. At 0°C v = 331 m/s; at 20°C v = 343 m/s; at 30°C v = 349 m/s. Wind-instrument tuning is sensitive to this. An orchestra that warms up in a 25°C green room and walks onto a 18°C stage drops about 4 m/s in c, which shifts wavelengths in the pipes by roughly 1.2% and goes audibly flat. String tension, by contrast, is barely affected by air temperature, so violins go sharp relative to the woodwinds when the room cools. Conductors learn to retune the section after the first long break.

Common error: multiplying when you should divide. If your wavelength comes out in kilometres for an audible sound wave, you wrote λ = v × f instead of λ = v / f. Sanity check: audible-sound λ in air sits between roughly 1.7 cm (20 kHz) and 17 m (20 Hz). Visible-light λ sits between roughly 380 nm and 750 nm. Anything wildly outside those ranges means you applied the wrong operator or mixed units (kHz vs Hz, cm vs m, nm vs m).

v = fλ holds for all of them, but where v itself comes from is what tells the three apart. Sound is a longitudinal compression wave in a fluid or solid; the speed depends on the bulk modulus K and density ρ through v = √(K/ρ) (or √(γRT/M) for an ideal gas, which gives the 343 m/s in 20°C air). Light is a transverse electromagnetic wave; its speed in vacuum c = 299,792,458 m/s is exact by SI definition since 1983, and in a medium it's c/n where n is the refractive index. A wave on a string is a transverse mechanical wave; v = √(T/μ) with T the tension and μ the linear mass density. Same v = fλ on top, three different microscopic mechanisms underneath.

On a string, raising tension raises wave speed, which raises every standing-wave frequency on a string of fixed length, which is exactly what tuning a guitar does. Going from T = 75 N to T = 85 N increases v by about 6.4%, roughly a semitone. Bass strings are wound thick to push μ up so that v stays manageable; treble strings are thin so v stays high without absurd tension. The string-speed formula is also where the standing-wave frequencies f_n = nv/(2L) come from for a string fixed at both ends. The first harmonic has λ = 2L; the second has λ = L; the nth has λ_n = 2L/n.

For electromagnetic waves the spread of frequencies and wavelengths is enormous, so scientific-notation arithmetic gets unforgiving. FM radio at 100 MHz has λ = 3 m, which is roughly the length of a half-wave dipole antenna folded once. WiFi at 2.4 GHz has λ ≈ 12.5 cm; that's why router antennas are short stubs and why 2.4 GHz interferes with the rotational mode of liquid water (microwave ovens use the same band). Red light at 4.3×10¹⁴ Hz has λ ≈ 700 nm; blue at 6.7×10¹⁴ Hz has λ ≈ 450 nm. X-rays at 3×10¹⁷ Hz have λ ≈ 1 nm, which is why they can resolve atomic-spacing crystal lattices and softer wavelengths can't.

The formula v = ω/k is equivalent to v = fλ. In quantum mechanics photon energy is E = ℏω with ℏ = h/2π, and the dispersion relation for a relativistic free particle is ω² = c²k² + (mc²/ℏ)². For non-dispersive waves (sound in air at audio frequencies, light in vacuum), v is independent of f, so a pulse made of many frequencies travels intact. For dispersive waves (light in glass, deep-water gravity waves, plasma waves), v depends on f, and a pulse spreads as it propagates. Crawford's "Waves" treats both regimes carefully.

Wave problems with more than one source come up everywhere: two speakers in a room, two slits in a Young's experiment, an antenna array, the strings on a piano. Linearity is the rule that lets you add them up. If y_1(x, t) and y_2(x, t) are solutions to the wave equation, so is y_1 + y_2. The combined disturbance is the sum of the individual ones at every point in space and time. That's superposition, and it's the entire content of "interference."

For two sources of equal amplitude and frequency, the algebra collapses cleanly. Let y_1 = A sin(ωt) and y_2 = A sin(ωt + Δφ). Trig identity gives y_1 + y_2 = 2A cos(Δφ/2) sin(ωt + Δφ/2). The combined amplitude is |2A cos(Δφ/2)|. When Δφ = 0 you get 2A (constructive). When Δφ = π you get 0 (destructive). Everywhere in between scales smoothly. That single envelope cos(Δφ/2) is what makes diffraction patterns, room nulls, and the fringe pattern in a Michelson interferometer all look like the same picture viewed from different angles.

Standing waves are interference between a wave and its reflection. A string fixed at both ends supports modes where exactly an integer number of half-wavelengths fits the length: λ_n = 2L/n, f_n = nv/(2L). The fundamental has one antinode in the middle. The second harmonic has a node in the middle. For an organ pipe open at both ends the formula is the same (open ends are pressure nodes). For a pipe closed at one end, only odd harmonics survive: f_n = nv/(4L) with n = 1, 3, 5, .... That's why a clarinet (effectively closed at the reed) sounds an octave lower than its tube length suggests, while a flute (open at both ends) sounds at the open-pipe pitch.

Multi-source problems with N identical emitters in a line (an antenna array, a row of speakers, a diffraction grating with N slits) have constructive maxima where the path difference between adjacent elements is nλ. The pattern sharpens with N: the principal-maximum width drops as 1/N, which is why 100-element diffraction gratings can separate sodium-D doublet lines that two-slit setups can't. Same superposition rule, just applied to many terms instead of two.

When two sources have slightly different frequencies, you get beats. Set y_1 = A sin(ω_1 t) and y_2 = A sin(ω_2 t). The sum is 2A cos((ω_1 − ω_2)t / 2) sin((ω_1 + ω_2)t / 2): a high-frequency carrier at the average ω modulated by a slow envelope at half the difference. Audibly, you hear a single tone at the average pitch swelling and fading at the beat frequency f_beat = |f_1 − f_2|. Two tuning forks at 440 and 442 Hz give a clean 2 Hz beat, which is what piano tuners listen for: when the beat slows to nothing, the strings are in unison.

Phase-shift diagnostics show up wherever you can't measure phase directly but can measure amplitude or position. Two-slit interference: the fringe spacing on the screen tells you λ once you know the slit separation, because Δy = λL/d where L is the screen distance and d the slit gap. Put a thin slab of unknown index in front of one slit and the fringe pattern shifts laterally; count the shifted fringes and you get the optical path-length change. That's how Michelson and Morley measured to 10⁻⁹ in 1887, with no electronics, by counting fringes in a sodium lamp.

Reflections at fixed ends flip phase by π; reflections at free ends don't. A pulse on a string approaching a wall comes back inverted. The same pulse approaching a join with a much lighter string comes back upright (and the rest is transmitted). For sound, a closed end of a pipe is a velocity node and pressure antinode (hard reflection, no phase flip in pressure). An open end is the opposite: pressure node, velocity antinode. That's why open-open and closed-closed pipes have the same harmonic series, and why open-closed pipes (clarinets) skip even harmonics.

When the calculator returns something you didn't expect, the diagnostic order is: check units (Hz vs kHz, m vs cm, nm vs m); check that v matches the medium; check that frequency was held constant if you crossed media; check that you used the right boundary (fixed-fixed string vs open-open pipe vs closed-open pipe). Unit errors and boundary-condition errors account for nearly every wrong-by-an-octave or wrong-by-2π answer.