PHYS212 : Lectures

Subsections

• 3.2.1 The Method
• 3.2.2 Limit of Validity
• 3.2.3 Standing Waves on an Inhomogeneous String

3.2 The WKB Approximation

3.2.1 The Method

• In cases in which the inhomogeneity of the medium varies gradually relative to the wavelength of waves propagating in it, we can generate approximate solutions to the wave equation consisting of plane waves with gradually varying amplitude and wavelength. This is known as the WKB approximation ^3.2.

• We assume solutions
  

  $$y(x,t) = Re\left\{ A(x) e^{i[\Phi(x) - \omega t + \phi]} \right\}$$ (3.15)

  
  
  where the amplitude and phase functions $A(x)$ and $\Phi(x)$ vary gradually.

• The wave equation (Eq. 1.69) with the solutions of Eq. 3.15 give
  

  $$A'' + 2iA'\Phi' - A \Phi'^2 + i A \Phi'' = - A k^2$$ (3.16)

  
  
  where we note that the wavenumber $k$ is now also a gradually varying function of position.

• The real and imaginary parts of Eq. 3.16 must satisfy the equation separately, yielding the equations,
  

  $$A'' + A (k^2 - \Phi'^2 ) = 0$$ (3.17)

  
  
  and
  

  $$2A'\Phi' + A \Phi'' = 0$$ (3.18)

  
  

• We assume that for gradually varying inhomogeneities, $\frac{A''}{A}$ is much smaller than the difference $(k^2 - \Phi'^2 )$ in Eq. 3.17 and can be neglected, which leaves
  
  $$\Phi'(x) = k(x)$$
  
  
  which we can simply integrate to find the WKB phase function
  

  $$\Phi(x) = \int_{x_o}^x k(x) dx$$ (3.19)

  
  

• Eq. 3.18 can be written
  
  $$2\frac{A'(x)}{A(x)} = - \frac{k'(x)}{k(x)}$$
  
  
  and integrated over $x$ to yield the WKB amplitude function
  

  $$A(x) = A_o \sqrt{\frac{k_o}{k(x)}}$$ (3.20)

  
  

• Eq.'s 3.19 and 3.20 require that we know the amplitude $A_o$ and wavenumber $k_o$ at some position $x_o$.

3.2.2 Limit of Validity

• We can quantify what we mean by ``gradual variation'' in order to find the limits of validity of the WKB solutions.

• The requirement that changes in wavelength must be small over a single local wavelength can be expressed
  

  $$\vert\lambda'(x)\vert \lambda(x) « \lambda(x)$$ (3.21)

  
  
  or simply as
  

  $$\vert\lambda'(x)\vert « 1$$ (3.22)

  
  

• In terms of the wavenumber function $k(x) = \frac{2\pi}{\lambda}$,
  

  $$\frac{\vert k'(x)\vert}{k^2(x)} « \frac{1}{2\pi}$$ (3.23)

  
  

• The left side of Eq. 3.23 is likely to be a function of $x$. Some effort may be required to find its maximum value on the position interval of interest.

3.2.3 Standing Waves on an Inhomogeneous String

• Consider a stretched string of length $L$ and tension $T_o$ with a linear mass density $\mu$ which increases linearly with position,
  

  $$\mu = \mu_o (1 + \epsilon x)$$ (3.24)

  
  

• We can use the expression for the phase velocity of waves on a homogeneous string (Eq. 1.68)
  
  $$\frac{\omega}{k} = \sqrt{\frac{T_o}{\mu}} = \sqrt{\frac{T_o}{\mu_o}} \frac{1}{\sqrt{1 + \epsilon x}}$$
  
  
  to find the wavenumber function of the inhomogeneous string
  

  $$k(x) = \omega \sqrt{\frac{\mu_o}{T_o}} \sqrt{1 + \epsilon x}$$ (3.25)

  
  

• Eq. 3.19 with Eq. 3.25 gives
  

  $$\Phi(x) = \omega \sqrt{\frac{\mu_o}{T_o}} \int_0^x \sqrt{1 + \epsilon x} dx$$ (3.26)

  $$= \omega \sqrt{\frac{\mu_o}{T_o}} \frac{2}{3 \epsilon} \left( (1 + \epsilon x)^\frac{3}{2} - 1 \right)$$ (3.27)

  
  

• Assembling the WKB solution, so far we have
  
  $$y(x,t) = \frac{A_o}{(1 + \epsilon x)^\frac{1}{4}} \sin \le... ...silon x)^\frac{3}{2} - 1 \right) \right) \cos(\omega t - \phi)$$
  
  
  where we have chosen the sine solution, because standing wave solutions must satisfy $y(0,t) = 0$. In addition, $y(L,t) = 0$ means
  
  $$\omega \sqrt{\frac{\mu_o}{T_o}} \frac{2}{3 \epsilon} \left( (1 + \epsilon L)^\frac{3}{2} - 1 \right) = n \pi$$
  
  
  leading to standing wave frequencies
  

  $$\omega_n = \sqrt{\frac{T_o}{\mu_o}} \frac{3 \epsilon n \pi}{2 \left( (1 + \epsilon L)^\frac{3}{2} - 1 \right)}$$ (3.28)

  
  

  Figure 3.3: The solid curve is the WKB approximation of the shape of the tenth standing wave mode of a stretched string with a mass density increasing linearly with position (Eq. 3.29 with $A_o = 1$, $L=1$, and $\epsilon = 0.25$). The dashed curve is the corresponding mode of a string with uniform density for comparison.

  

• With the frequencies of Eq. 3.28, we find that standing wave modes of oscillation of our inhomogeneous string are described by
  

  $$y(x,t) = \frac{A_o}{(1 + \epsilon x)^\frac{1}{4}} \sin \le... ...on L)^{\frac{3}{2}} - 1} n \pi \right) \cos(\omega_n t - \phi)$$ (3.29)

  
  
  A plot of the shape of the tenth mode with $A_o = 1$, $L=1$, and $\epsilon = 0.25$ is shown in Fig. 3.3.

• The homogeneous string has standing wave frequencies (Eq.'s 1.70 and 1.71)
  

  $$\omega_{on} = \sqrt{\frac{T_o}{\mu_o}} \frac{n \pi}{L}$$ (3.30)

  
  
  which compared with Eq. 3.28 gives a relative shift in standing wave frequency due to the inhomogeneity of
  
  $$\frac{\omega_{on} - \omega_n}{\omega_{on}} = \frac{\frac{1}... ...left( (1 + \epsilon L)^\frac{3}{2} - 1 \right)} }{\frac{1}{L}}$$
  
  
  which it turns out is independent of $n$ and reduces to
  

  $$\frac{\Delta \omega}{\omega_o} = 1 - \frac{3}{2} \frac{\epsilon L}{\left( (1 + \epsilon L)^\frac{3}{2} - 1 \right)}$$ (3.31)

  
  
  The example shown in Fig. 3.3 involves a 25% change in linear mass density ( $\epsilon = 0.25$) over the length of the string corresponding to a 6% shift in $\omega$.

• It can be shown that the limit of validity of the WKB approximation (Eq. 3.23) for this system places an upper limit on $\epsilon$,
  

  $$\epsilon « \frac{(\frac{3}{2} n + 1)^\frac{2}{3} - 1}{L}$$ (3.32)

  
  
  In the example shown in Fig. 3.3, $\epsilon$ is at about 5% of the limit.

Copyright © 2003-2009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006