The one-dimensional wave equation describes how a disturbance \(y(x,t)\) travels through space and time. In many ideal settings
(a taut string with small transverse displacements, or sound waves in a uniform medium), the displacement satisfies
\[
\frac{\partial^2 y}{\partial t^2}=v^2\frac{\partial^2 y}{\partial x^2},
\]
where \(v\) is the wave speed set by the medium. A large class of solutions are sinusoidal waves, which are convenient because they
have a single wavelength and frequency and can be superposed to build more complicated signals.
Traveling waves. A sinusoidal traveling wave moving in the \(+x\) direction can be written as
\[
y(x,t)=A\sin(kx-\omega t+\phi)\quad \text{(or cosine)}.
\]
Here \(A\) is the amplitude (maximum displacement), \(\phi\) is a phase shift, and the parameters
\[
k=\frac{2\pi}{\lambda},\qquad \omega=2\pi f
\]
are the wavenumber and angular frequency. The argument \(kx-\omega t+\phi\) is the phase of the wave. If you hold the phase
constant and solve for how a crest moves, you find that it travels at the phase speed
\[
v=\frac{\omega}{k}=\lambda f.
\]
A wave moving in the \(-x\) direction uses a plus sign, \(kx+\omega t+\phi\). This calculator calls these progressive (\(+x\)) and
regressive (\(-x\)) directions.
Standing waves. A standing wave forms from the superposition of two equal traveling waves moving in opposite directions. A common form is
\[
y(x,t)=A\sin(kx+\phi)\cos(\omega t),
\]
(or \(A\cos(kx+\phi)\cos(\omega t)\) depending on boundary conditions). Unlike a traveling wave, a standing wave does not transport the pattern
through space; instead, the shape “breathes” in time. Points where the displacement is always zero are nodes, and points that reach the
maximum amplitude are antinodes. Standing waves are central to resonance on strings and in air columns.
Reading y(x,t). Once \(A\), \(\lambda\), \(f\), and \(\phi\) are chosen, the wave function gives a unique displacement at any
position \(x\) and time \(t\). Because the sine/cosine argument is in radians, the phase in degrees must be converted:
\[
\phi_{\text{rad}}=\phi_{\text{deg}}\cdot\frac{\pi}{180}.
\]
The calculator computes \(k\) and \(\omega\), selects the appropriate formula (traveling or standing, sine or cosine, direction if traveling),
and evaluates \(y(x,t)\) step-by-step.
Beyond the sinusoid. University-level wave theory extends to general solutions like \(y(x,t)=F(x-vt)+G(x+vt)\), dispersion
(where \(v\) depends on wavelength), and attenuation/damping. In dispersive media, the phase speed \(v_p=\omega/k\) differs from the
group speed \(v_g=d\omega/dk\). This tool focuses on ideal sinusoidal waves but is a useful starting point for those more advanced topics.
