Harmonic Wave Equation Calculator

A harmonic wave is a wave whose displacement follows a sinusoidal pattern over both space and time. It is characterized by a single frequency and wavelength, meaning all points along the wave oscillate in simple harmonic motion. Examples include sound waves in air (at low amplitudes) and electromagnetic waves in a vacuum.

The harmonic wave equation is y(x,t) = A·sin((2π/λ)·(x − vt) + φ), where A is the amplitude, λ is the wavelength, v is the wave velocity, t is time, x is the position along the wave, and φ is the initial phase angle. It describes the displacement of any point on the wave at any given time.

Simple harmonic motion is described by x(t) = A·cos(ωt + φ), where A is the amplitude, ω is the angular frequency (ω = 2πf), t is time, and φ is the initial phase. It represents the oscillation of a single particle, whereas the harmonic wave equation extends this to describe displacement across an entire medium.

If the wave equation is written in the form y = A·sin((2π/λ)·(x − vt) + φ), the wavelength λ is the spatial period — the distance over which the wave pattern repeats. For example, in y = 0.07·sin((2π/0.4)·(x − 320t) + π/6), the coefficient inside the parentheses gives 2π/λ = 2π/0.4, so λ = 0.4 m.

Common examples include sound waves produced by musical instruments (each note is a harmonic wave at a specific frequency), light waves (electromagnetic radiation with a defined wavelength), water ripples at low amplitudes, and waves on a taut string. Radio waves and microwaves are also harmonic waves operating at different frequency ranges.

The initial phase φ shifts the wave pattern left or right along the x-axis (or equivalently, forward or backward in time). A phase of 0 means the wave starts at zero displacement and moves in the positive direction. Changing φ allows you to describe waves that are offset relative to each other, which is important in wave interference and superposition problems.

Frequency f, wavelength λ, and wave velocity v are related by the equation v = f·λ, which rearranges to f = v/λ. A higher velocity at the same wavelength means higher frequency, while a longer wavelength at constant velocity means lower frequency. The angular frequency ω = 2πf = 2πv/λ is also directly derivable from these values.

To write the harmonic wave equation, identify the amplitude A, wavelength λ, wave velocity v, and initial phase φ. Then substitute into y(x,t) = A·sin((2π/λ)·(x − vt) + φ). For example, a wave with A = 0.05 m, λ = 2 m, v = 10 m/s, and φ = π/4 rad gives y(x,t) = 0.05·sin(π·(x − 10t) + π/4).