PHYS212 : Lectures

Subsections

• 2.1.1 Beats
• 2.1.2 Fourier Series
• 2.1.3 A Square Wave

2.1 Periodic and Confined Packets

2.1.1 Beats

Figure 2.1: A snapshot of the superposition (Eq. 2.2) of two harmonic waves with slightly different wavelengths and frequencies at time $t = 0$.

• You have shown in homework that
  

  $$\cos(\omega_1 t) + \cos(\omega_2 t) = 2 \cos \left( \bar{\omega} t \right) \cos \left( \frac{\Delta \omega}{2} t \right)$$ (2.1)

  
  
  where $\bar{\omega}$ is the average of the two frequencies, and $\Delta \omega$ is the difference between them.

• Similarly, a superposition of two plane waves with equal amplitudes and phases but with different $k$ and $\omega$ has the two representations
  

  $$y(x,t) = A \cos(k_1 x - \omega_1 t) + A \cos(k_2 x - \omega_2 t)$$

  $$= 2A \cos \left( \bar{k} x - \bar{\omega} t \right) \cos \left( \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t \right)$$ (2.2)

  
  
  where $\bar{k}$ and $\bar{\omega}$ are averages and $\Delta k$ and $\Delta \omega$ are differences. A snapshot of such a superposition at $t = 0$ is shown in Fig. 2.1.

• Animation : ndBeats.py (see Appendix A on running animations)

• Velocities

  Two velocities can be calculated via Eq. 1.66 from the two wavenumber, frequency pairs of Eq. 2.2,

  • Phase velocity

    We recognize the velocity of the fast oscillations as the phase velocity
    

    $$v_p = \frac{\bar{\omega}}{\bar{k}}$$ (2.3)

    
    

  • Modulation velocity
    

    $$v_g = \frac{\Delta \omega}{\Delta k}$$ (2.4)

    
    
    as we consider superpositions of larger groups of waves, we will call this the group velocity. If we take a superposition of a continuous distribution of wavenumbers, the group velocity becomes
    

    $$v_g = \frac{d \omega}{d k}$$ (2.5)

    
    
    For waves on a string $\omega = v_p k$, and these two velocities are the same.

• Dispersion
  • The frequency written as a function of wavenumber $\omega(k)$ is called the dispersion relation for a wave.

  • The velocities of waves with nonlinear dispersion relations depend on wavenumber. They are called dispersive waves.

  • For example, quantum mechanical wavefunctions are dispersive. ^2.1 Solutions to the Schrödinger equation have the dispersion relation
    

    $$\hbar \omega = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2m}.$$ (2.6)

    
    
    (We will return to this topic later in the course.)

  • Animation : quantumBeats.py (see Appendix A on running animations)
    Note : In order for this animation to run, the file planeWave.py must be present in the same directory.

  • Waves can be non-dispersive in some media and dispersive in others. For example, rainbows arise from the dispersion of electromagnetic waves in water.

• Standing Beats
  • The superposition of two standing waves (Eq. 1.76) of different wavenumber gives a standing beat pattern.

  • Animation : standingBeats.py (see Appendix A on running animations)
    Note : In order for this animation to run, the file planeWave.py must be present in the same directory.

2.1.2 Fourier Series

• Inner products and orthogonality: first, a familiar example
  • The dot product that we use in the context of vectors in 3-space is an example of an inner product. Among other things, we use dot products to find the components $(a_x, a_y, a_z)$ of a vector
    

    $$\vec{a} = a_x \hat{\imath} + a_y \hat{\jmath} + a_z \hat{k}$$ (2.7)

    
    
    via
    

    $$a_x = \hat{\imath} \cdot \vec{a}$$

    $$a_y = \hat{\jmath} \cdot \vec{a}$$

    $$a_z = \hat{k} \cdot \vec{a}$$ (2.8)

    
    

  • The unit vectors $\hat{\imath}$, $\hat{\jmath}$, $\hat{k}$ are orthogonal to each other,
    

    $$\hat{\imath} \cdot \hat{\jmath} = 0, \hat{\jmath} \cdot \hat{k} = 0, \hat{k} \cdot \hat{\imath} = 0$$ (2.9)

    $$\hat{\imath} \cdot \hat{\imath} = 1, \hat{\jmath} \cdot \hat{\jmath} = 1, \hat{k} \cdot \hat{k} = 1$$ (2.10)

    
    

  • The unit vectors are also a complete set of orthogonal vectors. That is, any vector in 3-space can be represented as a linear combination of $\hat{\imath}$, $\hat{\jmath}$, and $\hat{k}$.

• Functions as vectors
  • Functions can be thought of as vectors in an infinite dimensional space. Just as we can find the components of a velocity vector in its three-dimensional space, we can find the components of functions in its infinite-dimensional space.

  • The definition of the inner product depends on the functions we plan to consider. In the context of Fourier series, in the function space of sines and cosines, we use
    

    $$\vec{f}(x) \cdot \vec{g}(x) = \int_0^\lambda f(x) g(x) dx$$ (2.11)

    
    
    where $\lambda$ is a wavelength we specify.

  • The orthogonality relationships, analogous to Eq.'s 2.10, sines and cosines are
    

    $$\int_0^{\lambda} \cos \left( \frac{2 \pi n x}{\lambda} \right) \cos \left( \frac{2 \pi m x}{\lambda} \right) dx = \frac{\lambda}{2} \delta_{nm}$$

    $$\int_0^{\lambda} \sin \left( \frac{2 \pi n x}{\lambda} \right) \sin \left( \frac{2 \pi m x}{\lambda} \right) dx = \frac{\lambda}{2} \delta_{nm}$$

    $$\int_0^{\lambda} \cos \left( \frac{2 \pi n x}{\lambda} \right) \sin \left( \frac{2 \pi m x}{\lambda} \right) dx = 0$$ (2.12)

    
    
    where the Kronecker delta $\delta_{nm}$ is
    

    $$\delta_{nm} = \left\{ \begin{array}{lc} 1 & (n = m)\\ 0 & (n \neq m) \end{array} \right.$$ (2.13)

    
    
    and $n$ and $m$ are integers.

• Fourier coefficients are components
  • The set of functions $\cos \left( \frac{2 \pi n x}{\lambda} \right)$ and $\sin \left( \frac{2 \pi n x}{\lambda} \right)$ are a complete set of ``vectors'' describing functions that are periodic on the interval $\lambda$.

  • This complete set can be used to describe a function $f(x)$ provided...
    • ...it is defined at the boundaries $x = 0$ and $x=\lambda$.
    • ...it is at least piecewise continuous.
    • ...it is square integrable. That is, the integral
      
      $$\int_0^L \vert f(x)\vert^2 dx$$
      
      
      must exist.

  • Any such function can be described by a Fourier series,
    

    $$f(x) = a_o + \sum_{n=1}^{\infty} a_n \cos \left( \frac{2 \p... ...=1}^{\infty} b_n \sin \left( \frac{2 \pi n x}{\lambda} \right)$$ (2.14)

    
    
    where the Fourier coefficients {$a_0$, $a_1$, $a_2$, ..., $b_1$, $b_2$, ...} are components, analogous to the Cartesian components {$a_x$, $a_y$, $a_z$} of a vector is 3-space.

  • In direct analogy to Eq.'s 2.8, the Fourier coefficients are found by taking inner products with unit vectors,
    

    $$a_o = \frac{1}{\lambda} \int_0^{\lambda} f(x) dx$$

    $$a_n = \frac{2}{\lambda} \int_0^{\lambda} f(x) \cos \left( \frac{2 \pi n x}{\lambda} \right) dx$$

    $$b_n = \frac{2}{\lambda} \int_0^{\lambda} f(x) \sin \left( \frac{2 \pi n x}{\lambda} \right) dx$$ (2.15)

    
    

• A word about intervals
  • In the above development of Fourier series and in the two examples we have considered we used the interval $0 \leq x \leq \lambda$. In general, any interval of length $\lambda$ will do.

  • Hence, a more general version of Eq.'s 2.15 is
    

    $$a_o = \frac{1}{\lambda} \int_{x_o}^{x_o + \lambda} f(x) dx$$

    $$a_n = \frac{2}{\lambda} \int_{x_o}^{x_o + \lambda} f(x) \cos \left( \frac{2 \pi n x}{\lambda} \right) dx$$

    $$b_n = \frac{2}{\lambda} \int_{x_o}^{x_o + \lambda} f(x) \sin \left( \frac{2 \pi n x}{\lambda} \right) dx$$ (2.16)

    
    

  • Often it is convenient to define a function in the interval $-\frac{\lambda}{2} \leq x \leq \frac{\lambda}{2}$. You will consider an example in which this is the case for homework.

• Fourier series occupy a small corner of a broad theory of orthogonal functions called Sturm-Liouville Theory. Different sets of orthogonal functions are best suited to different systems. For example, while waves on a horizontal stretched string are described by sine and cosine functions, waves on a hanging chain can be described by special functions called Bessel functions.

2.1.3 A Square Wave

• Traveling square waves

  Figure 2.2: Partial Fourier series representations (Eq. 2.22) of a square wave with $A = 1$ and $\lambda=2$ including 10 and 100 terms.

  
  • Consider a square wave described by
    

    $$f(x) = \left\{ \begin{array}{lc} A & (0 < x < \frac{\lambda... ...)\\ -A & (\frac{\lambda}{2} < x < \lambda) \end{array}\right.$$ (2.17)

    
    

  • Applying Eq.'s 2.15, we have
    

    $$a_o = A \frac{1}{\lambda} \int_0^{\frac{\lambda}{2}} dx - A \frac{1}{\lambda} \int_{\frac{\lambda}{2}}^{\lambda} dx$$

    $$a_n = \frac{2A}{\lambda} \int_0^{\frac{\lambda}{2}} \cos \left( \frac{2 \pi n x}{\lambda} \right)$$

    $$- \frac{2A}{\lambda} \int_{\frac{\lambda}{2}}^{\lambda} \cos \left( \frac{2 \pi n x}{\lambda} \right) dx$$

    $$b_n = \frac{2A}{\lambda} \int_0^{\frac{\lambda}{2}} \sin \left( \frac{2 \pi n x}{\lambda} \right)$$

    $$- \frac{2A}{\lambda} \int _{\frac{\lambda}{2}}^{\lambda} \sin \left( \frac{2 \pi n x}{\lambda} \right) dx$$ (2.18)

    
    
    which we can evaluate
    

    $$a_o = \frac{A}{2} - (A - \frac{A}{2})$$

    $$a_n = \frac{A}{n\pi} \left[ \sin \left( \frac{2 \pi n x}{\lambda} \right) \right\vert _0^{\frac{\lambda}{2}}$$

    $$-\frac{A}{n\pi} \left[ \sin \left( \frac{2 \pi n x}{\lambda} \right) \right\vert _{\frac{\lambda}{2}}^{\lambda}$$

    $$b_n = -\frac{A}{n\pi} \left[ \cos \left( \frac{2 \pi n x}{\lambda} \right) \right\vert _0^{\frac{\lambda}{2}}$$

    $$+ \frac{A}{n\pi} \left[ \cos \left( \frac{2 \pi n x}{\lambda} \right) \right\vert _{\frac{\lambda}{2}}^{\lambda}$$

    $$= \frac{2A}{n\pi} (1 - cos(n\pi))$$ (2.19)

    
    
    and simplify to obtain
    

    $$a_o = 0$$

    $$a_n = 0$$

    $$b_n = \frac{2A}{n\pi} (1 - (-1)^n).$$ (2.20)

    
    

  • We have identified the Fourier components of the square waveform. Now, we construct a Fourier series of plane waves, including time dependence,
    

    $$y(x,t) = a_o + \sum_{n=1}^{\infty} a_n\cos\left(\frac{2 \pi n}{\lambda}(x-v_pt)\right)$$

    $$+ \sum_{n=1}^{\infty} b_n\sin\left(\frac{2 \pi n}{\lambda}(x-v_pt)\right)$$ (2.21)

    
    
    where we have used $\omega = v_p k$. With Eq.'s 2.20, a traveling square wave can be represented by
    

    $$y(x,t) = \frac{2A}{\pi} \sum_{n=1}^{\infty} \frac{(1 - (-1)^n)}{n} \sin\left(\frac{2 \pi n}{\lambda}(x-v_pt)\right)$$ (2.22)

    
    

  • This series converges in a finite number of terms to an approximation of the square wave form. Fig. 2.2 shows partial series of 10 and 100 terms.

  • Animation : travelingSquare.mws [217 kB] (see Appendix A on running animations)

  • It is difficult to imagine actually making a string do this. Strings do not bend or stretch in this way. However, square electromagnetic wave forms can be realized.

• Standing square waves
  • The traveling square wave is made of plane waves. Hence, two square waves traveling in opposite directions with a relative phase of $\pi$ should give a standing square wave.

  • A standing square wave can also be obtained with a linear combination of standing plane waves (Eq. 1.76),
    

    $$y(x,t) = \frac{2A}{\pi} \sum_{n=1}^{\infty} \frac{(1 - (-1)^... ...right) \cos\left( \frac{2 \pi n v_p t}{\lambda} + \phi \right)$$ (2.23)

    
    

  • Recall that the longest wavelength standing wave has wavelength $\lambda = 2L$. We can rewrite our standing square wave solution in terms of the length $L$ of the string
    

    $$y(x,t) = \frac{2A}{\pi} \sum_{n=1}^{\infty} \frac{(1 - (-1)^... ... x}{L} \right) \cos\left( \frac{\pi n v_p t}{L} + \phi \right)$$ (2.24)

    
    

  • The phase of our square waves is such that the fixed end points of the string can fall on $x = 0$ and $x = L$ or on any other adjacent pair of nodes.

  • Animation : standingSquare.mws [232 kB] (see Appendix A on running animations)

Copyright © 2003-2009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006