Vibration wave some equations

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)