Harmonic Wave Equation Calculator

A harmonic wave is a repeating disturbance whose displacement varies sinusoidally with position and time. The standard traveling-wave expression used here is y(x,t) = A sin(kx - omega t + phase). Amplitude A is the greatest displacement from equilibrium, x is position, t is elapsed time, k is angular wave number, omega is angular frequency, and phase sets the wave's starting point in its cycle. Frequency tells how many complete cycles pass a fixed point each second and is measured in hertz. Angular frequency equals two pi times frequency and is measured in radians per second. Wavelength is the distance between corresponding points on adjacent cycles. Wave number equals two pi divided by wavelength and describes phase change per meter. Their ratio gives wave speed, equivalently frequency multiplied by wavelength. The minus sign before the time term describes propagation in the positive x direction. Replacing it with a plus sign would describe a wave moving in the negative x direction under this convention. The phase is entered in degrees for convenience and converted internally to radians before evaluating the sine. Phase values outside the usual zero-to-360-degree interval are mathematically valid because a sinusoid repeats every full cycle. Units must be consistent. With amplitude and wavelength in meters, position in meters, frequency in hertz, and time in seconds, displacement is in meters and wave speed is in meters per second. Other consistent length units can be used, but amplitude, wavelength, and position should then share that length unit. The sine argument itself must be dimensionless because each spatial and temporal contribution represents an angle in radians. Harmonic wave models appear in acoustics, optics, vibrating strings, water-wave approximations, and electromagnetic theory. A single ideal sinusoid extends indefinitely and has constant amplitude, unlike a finite pulse or damped oscillation. Real waves may disperse, attenuate, reflect, interfere, or contain many frequencies. This calculator evaluates one one-dimensional ideal component, making it useful for homework checks and physical intuition while keeping the sign and phase convention explicit.