Oscillations and Waves

y - elongation

A - amplitude

$$\phi_0=\arcsin\frac{x_0}{A}$$ - initial phase

$$y=A*\sin(\omega*t+\phi_0)$$

$$v=-\omega*A*\cos(\omega*t+\phi_0)$$

$$a=-\omega^2*A*\sin(\omega*t+\phi_0)$$

$$k=m\omega^2$$

Elastic pendulum $$T=2*\pi*\sqrt{\frac{m}{k}}$$

Mathematical pendulum $$T=2*\pi*\sqrt{\frac{l}{g}}$$

$$\frac{kA^2}{2}=\frac{ky^2}{2}+\frac{mv^2}{2}$$

Same frequency oscillation composition

$$x_1=A_1\sin(\omega t+\phi_1)$$

$$x_2=A_2\sin(\omega t+\phi_2)$$

$$x=x_1+x_2=A\sin(\omega t+\phi)$$

$$A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(\phi_1-\phi_2)}$$

$$\tan \phi=\frac{A_1\cos \phi_1+A_2\cos \phi_2}{A_1\sin \phi_1+A_2\sin \phi_2}$$

Different frequency oscillation composition

$$x_1=A\sin(\omega_1 t+\phi_1)$$

$$x_2=A\sin(\omega_2 t+\phi_2)$$

$$x=2A \cos(\frac{\omega_1-\omega_2}{2}t+\frac{\phi_1-\phi_2}{2})\sin(\frac{\omega_1+\omega_2}{2}t+\frac{\phi_1+\phi_2}{2})$$

Perpendicular oscillation composition

$$x_1=A_1\sin(\omega t+\phi_1)$$

$$x_2=A_2\sin(\omega t+\phi_2)$$

$$x_2=\frac{A_2}{A_1}(x_1 \cos(\phi_2-\phi_1)\pm \sqrt{A_1^2-x_1^2}\sin(\phi_2-\phi_1))$$

Fluid friction

$$x=A_0e^{-bt}\sin(\omega t+\phi_0)$$

$$b=\frac{c}{2m}$$

c - drag coefficient, depending on the shape and size of the material

Dry friction

$$D=\frac{2\mu mg}{k}$$

$$\lambda=v*T=\frac{v}{\nu}=v\frac{2\pi}{\omega}$$

Waves

$$y=A\sin2\pi(\frac{t}{T}-{x}{\lambda})$$

Longitudinal waves $$v=\sqrt{\frac{E}{\rho}}$$

Transversal waves $$v=\sqrt{\frac{Tl}{m}}$$

T - tension

Refraction $$\frac{\sin i}{\sin r}=\frac{v_1}{v_2}$$

Doppler effect $$\nu'=\nu\frac{v_w+v_o}{v_w-v_s}$$

$$v_w$$ - the speed of the wave in the medium

$$v_o$$ - the speed of the observer relative to the medium

$$v_s$$ - the speed of the source relative to the medium

Acoustic intensity $$I=\frac{dW}{dt*dS}$$

Acoustic power $$p=\frac{dW}{dS}$$

Sound intensity level $$N=10\lg\frac{I}{I_0}$$

Hearing threshold $$I_0=10^{-16}W/cm$$