constructing the wave equation

  • consider a series of masses on a string
    • at \( \ldots, j-1, j, j+1, \ldots \)
    • spaced by \( \Delta z \)
  • the pieces of strings connecting the masses have tension \(T\)
  • the masses are at different heights from a reference horizontal
    • this also leads to a vertical force on the masses
    • because of the angle between the tensions in different pieces of string

Masses on a String

  • forces
    • \( T \sin \theta_j \) pulls mass \(j\) upwards
    • \( T \sin \theta_{j-1} \) pulls mass \(j\) downwards
    • the net upwards force on mass \(j\) is
      • \( F_j = T (\sin \theta_j - \sin \theta_{j-1}) \)
  • for small angles
    • \( \sin \theta_j \approx \frac{ y_{j+1} - y_j }{\Delta z} \)
    • \( \sin \theta_{j-1} \approx \frac{ y_j - y_{j-1} }{\Delta z} \)
  • using the small angles assumption in the force equation
    • \( F_j = T (\sin \theta_j - \sin \theta_{j-1}) \)
    • \( F_j \approx T \left[ \frac{ y_{j+1} - y_j }{ \Delta z } - \left( \frac{ y_j - y_{j-1} }{\Delta z } \right) \right] \)
    • \( F_j = T \left[ \frac{ y_{j+1} - 2y + y_{j-1} }{\Delta z} \right] \)
      • \( \text{ as } \Delta z \text{ reaches infinitesimal limit } \)
    • \( F = T \left[ \frac{ y_{j+1} - 2y_j - y_{j-1} }{\Delta z} \right] \)
    • \( F = T \Delta z \left[ \frac{y_{j+1} - 2y_j + y_{j-1}}{ (\Delta z)^2 } \right] \)
    • \( F = T \Delta z \frac{\partial^2 y }{\partial z^2} \)
  • the resulting equations implies that force \(F \) is proportional to the curvature of the string of masses
    • if all the masses are in a line, then there is no net force perpendicular to the line, either vertical or horizontal

linear density

  • consider the mass string as a linear density
    • the amount of mass per unit length in the \(z\) direction is \(\rho\)

  • mass, \(m\) is given by \( m = \rho \Delta z \)

  • to get the force in terms of density, apply newton’s II law:
    • \( F = m\frac{\partial^2 y}{\partial t^2} = \rho \Delta z \frac{\partial^2 y}{\partial t^2} \)

putting the force from curvature and linear density force equations together

  • \( \rho \Delta z \frac{\partial^2 y}{\partial t^2} = T \Delta z \frac{\partial^2 y }{\partial z^2} \)

  • gives: \( \rho \Delta z \frac{\partial^2 y}{\partial t^2} = T \Delta z \frac{\partial^2 y }{\partial z^2} \)

  • rearranging:
    • \( \frac{\partial^2 y}{\partial z^2} = \frac{\rho}{T}\frac{\partial^2 y}{\partial t^2 } \)
  • consider identity:
    • \( v^2 = \frac{T}{\rho} \)
  • this gives
  • \( \frac{\partial^2 y}{\partial z^2} - \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2} = 0 \)
  • this is a wave equation for a wave with velocity \( v = \sqrt{\frac{T}{\rho}} \)
    • any function of the form \( f(z-ct) \) is a solution of the wave equation
      • and is a wave moving to the right with velocity \(c\)
    • any function of the form \( g(z+ct) \) is a solution of the wave equation
      • and is a wave moving to the left with velocity \(c\)

monochromatic waves

  • waves oscillating at one specific (angular) frequency \( \omega \)

  • temporal behavior of the form
    • \( T(t) = exp(i\omega t), exp(-i\omega t), cos(\omega t), sin(\omega t) \)
    • complex and real
    • or any combination of these, provided they comply with boundary conditions
  • consider a monochromatic wave written as a product of variation in \(z\) and a variation in \(t\)
    • \( \phi(z,t) = Z(z)T(t) \)
    • double partial derivative w.r.t time gives
      • \( \frac{\partial^2 \phi}{\partial t^2} = -\omega^2 \phi \)

helmholtz wave equation

  • plugging this into the simple harmonic equation
    • \( \frac{ d^2 Z(z) }{ d z^2 } + k^2 Z(z) = 0 \)
      • where \( k^2 = \frac{\omega^2}{c^2} \)
      • \(c\): wave - velocity
    • this is called the helmholtz wave equation
      • simplest wave equation
      • used in CM for monochromatic wave

standing waves

  • particularly important in QM

  • needs matter that functions as a pair of walls

  • standing waves are an equal combination of forward and backward waves
    • \( \phi(z,t) = sin(kz - \omega t) + \sin (kz + \omega t) \)
  • applying trigonometric identity
    • \( \phi(z,t) = 2 \cos(\omega t) \sin(kz) \), where \( k = \frac{\omega}{c} \)
  • analogous to forces being in equilibrium
    • and net force begin equal to zero
  • every point along the wave oscillates in time
    • but the spatial shape remains the same

example

  • for a rope tied two two walls at distance L apart
    • \( k = 2\pi \)
    • \( \omega = \frac{2\pi c}{L} \)

Standing Wave