PX154 - G1b - transfer of power by the wave

[YF 15.5]

• we know that waves transfer energy
• more useful to consider the power transfer, ie: rate of energy transfer
  • leads to
  $$P=-T\frac{\partial y}{\partial x}\frac{\partial y}{\partial t}$$
  • if we take $y(x,t)=A\cos (kx-\omega t)$:
  $$P=T{A}^{2}k\omega {\sin }^2(kx-\omega t)$$
  • 
  • power oscillates at twice the wave's frequency:
  $$f_P=2\omega$$
  • let's simplify this. take $v=\frac{\omega }{k}$ and $v^2=\frac{T}{\mu }$:
  $$⟹k=\omega \sqrt{\frac{\mu }{T}}$$$$\therefore P(x,t)={\omega }^2{A}^2\sqrt{\mu T}{\sin }^2(kx-\omega t)$$where, $z=\sqrt{\mu T}$, often called 'impedance'
  • the average power transmitted over a wave cycle:
  $$P_{av}=\frac{1}{2}{\omega }^2{A}^{2}z$$
  • $P$ is proportional to $z$, and the squares of $A$ and $\omega$ of the wave
    • this is true for mechanical waves
    • however, for EM waves, we will see later