Vibration wave some equations
1 min read ·
Some equations related to propagation of vibration as wave.
wave frequency
Wave frequency $f$ can be obtained from
$$\tag{1} f = v \lambda, $$
with speed of wave $v$ and wavelength $\lambda$.
wave equation
Wave, propagation of oscillation, is described by
$$\tag{2} \frac{\partial^2 \psi}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} = 0 $$
for 1-D case and
$$\tag{3} \nabla^2 \psi - \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} = 0 $$
for for 3-D case.
solution of wave equation
For (2) the solution is
$$\tag{4} \psi(x, t) = A \sin(kx - \omega t + \phi_0), $$
with amplitude $A$, wavenumber $k$, angular frequency $\omega$, and initial phase $\phi_0$, and
$$\tag{5} \psi(\vec{r}, t) = A \sin(\vec{k} \cdot \vec{r} - \omega t + \phi_0), $$
is the solution for (3)