Wave Summary

• Wave Equation
  

  $$\frac{d^2y}{dt^2} - v^2 \frac{d^2y}{dx^2} = 0$$ (93)

  
  
  where for waves on a string:
  

  $$v = \pm \sqrt{\frac{T}{\mu}}$$ (94)

  
  

• Superposition Principle
  

  $$y(x,t) = A y_1(x,t) + B y_2(x,t)$$ (95)

  
  
  (sum of solutions is solution). Leads to interference, standing waves.

• Travelling Wave Pulse
  

  $$y(x,t) = f(x \pm vt)$$ (96)

  
  
  where $f(u)$ is an arbitrary functional shape or pulse

• Harmonic Travelling Waves
  

  $$y(x,t) = y_0 \sin( kx \pm \omega t).$$ (97)

  
  
  where frequency $f$, wavelength $\lambda$, wave number $k = 2\pi/\lambda$ and angular frequency $\omega$ are related to $v$ by:
  

  $$v = f \lambda = \frac{\omega}{k}$$ (98)

  
  

• Stationary Harmonic Waves
  

  $$y(x,t) = y_0 \cos(kx + \delta) \cos(\omega t + \phi)$$ (99)

  
  
  where one can select $k$ and $\omega$ so that waves on a string of length $L$ satisfy fixed or free boundary conditions.

• Energy (of wave on string)
  

  $$E_{\rm tot} = \frac{1}{2} \mu \omega^2 A^2 \lambda$$ (100)

  
  
  is the total energy in a wavelength of a travelling harmonic wave. The wave transports the power
  

  $$P = \frac{E}{T} = \frac{1}{2} \mu \omega^2 A^2 \lambda f = \frac{1}{2} \mu \omega^2 A^2 v$$ (101)

  
  
  past any point on the string.

• Reflection/Transmission of Waves
  1. Light string (medium) to heavy string (medium): Transmitted pulse right side up, reflected pulse inverted. (A fixed string boundary is the limit of attaching to an ``infinitely heavy string'').
  2. Heavy string to light string: Transmitted pulse right side up, reflected pulse right side up. (a free string boundary is the limit of attaching to a ``massless string'').