Definition: Wave Equation

The wave equation is $$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2},$$ where $c$ is the speed of the wave in the medium and $\psi(x,t)$ is the wave, which is a function of position and time.

Theorem: Solutions to the Wave Equation

The most general solution to the wave equation is a linear combination of two functions $f(x-ct)$ and $g(x+ct)$.

Definition: Standing Waves

A normal mode / standing wave is a wavefunction of the form $$\psi (x,t) = f(x) \cos (\omega t).$$ It essentially oscillates in place. We can find them by imposing boundary conditions.

Idea: Reflection and Transmission

At any boundary, we must have continuity and any other physics satisfied. Some things that can create boundary conditions are:

• Massless ring — net vertical force should be 0
• Fixed point — vertical coordinate is always that fixed y

A continuity condition would look like: $$\psi_{\text{in}}(x_0,t) + \psi_r(x_0,t) = \psi_t(x_0,t).$$ Hard walls lead to $\pi$ phase shifts and soft walls lead to in phase reflections.

Theorem: Snell's Law

Snell's Law tells us that the angle of incidence and angle of refraction are related by $$n_1 \sin \theta_1 = n_2 \sin \theta_2.$$

Theorem: Doppler Effect

Moving sources and observers cause changes in frequency for sound. The formula is, $$f' = \frac{c+v_o}{c-v_s}f,$$ where $v_o$ and $v_s$ are the radial velocities towards each other. This means that there is no transverse doppler effect for sound.