Harmonic Wave Equation Calculator

Harmonic Wave Equation

A harmonic (sinusoidal) wave traveling along the x-axis is described by $y = A\sin(kx - \omega t)$, where $A$ is amplitude, $k$ is the wave number, and $\omega$ is angular frequency. The minus sign indicates a wave propagating in the positive x direction.

Key Relationships

$$v = \lambda f, \quad k = \frac{2\pi}{\lambda}, \quad \omega = 2\pi f$$

Wave speed $v$ equals wavelength $\lambda$ times frequency $f$. The wave number $k$ and angular frequency $\omega$ link spatial and temporal periodicity. Period $T = 1/f$.

Displacement at a Point

$$y(x, t) = A\sin(kx - \omega t)$$

Substitute position $x$ and time $t$ to find instantaneous displacement. At $kx - \omega t = 0$, displacement equals zero and rises with amplitude $A$ as the phase advances.

Related tools: Note Frequency Calculator and BPM Calculator.

Frequently Asked Questions

What is the difference between frequency and angular frequency?

Frequency $f$ is cycles per second (hertz). Angular frequency $\omega = 2\pi f$ is radians per second and appears naturally in the sine argument of the wave equation.

How is wave speed related to wavelength?

Wave speed $v = \lambda f$. A longer wavelength at the same frequency means a faster wave. For sound in air at room temperature, $v \approx 343$ m/s.

What does the wave number k represent?

The wave number $k = 2\pi/\lambda$ measures spatial frequency in radians per meter. It tells how quickly the wave oscillates in space.

Can the wave travel in the negative x direction?

Yes. A wave traveling left uses $y = A\sin(kx + \omega t)$. The sign of the $\omega t$ term sets propagation direction.

What units does amplitude use?

Amplitude has the same units as displacement $y$. For a string wave, meters are typical. For sound pressure, pascals apply.

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