View source for Sum of waves of different frequencies and group velocity

{{publication
 | name    = Problems in Exploration Seismology and their Solutions
 | image   = [[File:897.Def.Lz-cropped.png|frameless|180px|link=Problems in Exploration Seismology and their Solutions]]
 | width   = 120px
 | series  = Geophysical References Series
 | title   = [[Problems in Exploration Seismology and their Solutions]]
 | part    = 
 | chapter = 2
 | pages  = 7 - 46
 | author  = [[Lloyd P. Geldart]] and [[Robert E. Sheriff]]
 | doi     = 10.1190/1.9781560801733
 | pdf     = 
 | store   = https://shop.seg.org/shop/products/detail/897
 | isbn    = ISBN 9781560801153
}}
__TOC__
== Problem 2.7a ==
A pulse composed of two frequencies, <math>\omega _{0} \pm {\mathrm{\Delta}}  \omega </math>, can be represented by factors involving the sum and difference of the two frequencies. If the two components have the same amplitudes, we can write


<math>
\begin{align}
A \cos \left(\kappa_{1} x-\omega _{1} t\right),{\quad} A \cos \left(\kappa_{2} x- \omega_{2} t\right),
\end{align}
</math>

where <math>\omega _{1} =\omega _{0} +\Delta \omega, \omega_2= \omega_0 -\Delta \omega, k_{0} =2\pi /\lambda _{0} =\omega _{0} /V</math>,

<math>
\begin{align}
\kappa_{1} &\approx \kappa_{0} + \Delta \kappa\approx \left(\omega _{0} +\Delta \omega \right)/V,\ \mathrm{and} \\
\kappa_{2} &\approx \kappa_{0} -\Delta \kappa\approx \left(\omega _{0} -\Delta \omega \right)/V.
\end{align}
</math>

Show that the composite wave is given approximately by the expression


<math>
\begin{align}
B\cos (\kappa_{0} x-\omega _{0} t),
\end{align}
</math>

where <math>B=2A\cos \left\{{\mathrm{\Delta}}  \kappa\left[x-\left({\mathrm{\Delta}}  \omega /{\mathrm{\Delta}}  \kappa\right)t\right]\right\}</math>.

=== Background ===

When different frequency components in a pulse have different ''phase velocities'' <math>V</math> (the velocity with which a given frequency travels), the pulse changes shape as it moves along. The ''group velocity'' <math>U</math> is the velocity with which the envelope of the pulse travels.

The envelope of a pulse comprises two mirror-image curves that are tangent to the waveform at the peaks and troughs, and therefore define the general shape of the pulse.

=== Solution ===

Adding the two components and using the identity


<math>
\begin{align}
\cos \theta +\cos \phi =2\cos \left[\frac{1}{2} \left(\theta +\phi \right)\right]\cos \left[\frac{1}{2} \left(\theta -\phi \right)\right],
\end{align}
</math>

we get for the composite wave


<math>
\begin{align}
2A\cos \left\{{\mathrm{\Delta}}  \kappa\left[x-\left({\mathrm{\Delta}}  \omega /{\mathrm{\Delta}}  \kappa\right)t\right]\right\}\cos \left(\kappa_{0} x-\omega _{0} t\right)=B\cos \left(\kappa_{0} x-\omega _{0} t\right).
\end{align}
</math>

== Problem 2.7b ==
Why do we regard <math>B</math> as the amplitude? Show that the envelope of the pulse is the graph of <math>B</math> plus its reflection in the <math>x</math>-axis.

=== Solution ===

The solution in (a) can be written <math>\psi =B\psi _{0} </math>. We regard <math>B</math> as the amplitude for two reasons: (1) <math>B</math> repeats every time that <math>x</math> increases by <math>2\pi /{\mathrm{\Delta}}  \kappa</math> or each time that <math>t</math> increase by <math>2\pi /{\mathrm{\Delta}}  \omega</math>. But <math>{\mathrm{\Delta}}  \kappa\ll \kappa_{0}</math>, <math>{\mathrm{\Delta}}  \omega \ll \omega _{0} \}</math>, so <math>B</math> must repeat more slowly than <math>\psi _{0} </math>. (2) Each time that <math>\psi _{0} </math> attains its limiting values, <math>\pm 1</math>, <math>\psi </math> has the value <math>\pm B</math> and therefore never exceeds <math>\left|B\right|</math>; thus the curves of <math>+\left|B\right|</math> and <math>-\left|B\right|</math> pass through the maxima and minima of <math>\psi _{0} </math> and therefore constitute the envelope.

== Problem 2.7c ==
Show that the envelope moves with the group velocity <math>U</math> where


{{NumBlk|:|<math>
\begin{align}
U=\frac{{\mathrm{\Delta}}  \omega }{{\mathrm{\Delta}}  \kappa} \approx \frac{\mathrm{d}\omega }{\mathrm{d}\kappa} \approx V-{\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } =V+\omega \frac{\mathrm{d}V}{\mathrm{d}\omega }
\end{align}
</math>|{{EquationRef|2.7a}}}}

'''(see Figure 2.7a).'''

=== Solution ===

If we consider the quantity <math>B</math> as a wave superimposed on the primary wavelet <math>\cos \left(\kappa_{0} x-\omega _{0} t\right)</math>, comparison with the basic wave type <math>f(x-Vt)</math> of problem 2.5a  shows that <math>\left[x-\left({\mathrm{\Delta}}  \omega /{\mathrm{\Delta}}  \kappa\right)t\right]</math> takes the place of <math>(x-Vt)</math>, i.e., <math>\left({\mathrm{\Delta}}  \omega /{\mathrm{\Delta}}  \kappa\right)</math> is the velocity <math>U</math> with which the envelope travels. In the limit, <math>U</math> is given by


<math>
\begin{align}
U=\mathrm{d}\omega /\mathrm{d}{\kappa} =\mathrm{d}f/\mathrm{d}\left(1/{\mathrm{\lambda}} \right)=\left(\mathrm{d}f/\mathrm{d}{\mathrm{\lambda}} \right)\left[\mathrm{d}{\mathrm{\lambda}} /\mathrm{d}\left(1/{\mathrm{\lambda}} \right)\right],
\end{align}
</math>

[[file:ch02_fig2-7a.png|thumb|{{figure number|2-7a}} Illustrating group and phase velocity.]]

where <math>f</math> is the frequency. The result is


<math>
\begin{align}
U=-{\mathrm{\lambda}} ^{2} \frac{\mathrm{d}f}{\mathrm{d}\lambda}.
\end{align}
</math>

We introduce the phase velocity <math>V</math> by noting that <math>V={\mathrm{\lambda}} f</math>, so


<math>
\begin{align}
\mathrm{d}V={\mathrm{\lambda}} \mathrm{d}f+f\mathrm{d}\lambda, \; {\mathrm{\lambda}} \frac{\mathrm{d}f}{\mathrm{d}{\mathrm{\lambda}} } =\frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } -f.
\end{align}
</math>

Thus,


<math>
\begin{align}
U=-{\mathrm{\lambda}} \left({\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } \right)={\mathrm{\lambda}} f-{\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}{\mathrm{\lambda}} } =V-{\mathrm{\lambda}} \frac{\mathrm{d}V}{\mathrm{d}\lambda}.
\end{align}
</math>

To replace the derivative <math>\mathrm{d}V/\mathrm{d}{\mathrm{\lambda}} </math> with <math>\mathrm{d}V/\mathrm{d}f</math>, we find the relation between <math>{\mathrm{\lambda}}/\mathrm{d}{\mathrm{\lambda}} </math> and <math>f/\mathrm{d}f</math>; to do this we let <math>V</math> be constant so that


<math>
\begin{align}
V={\mathrm{\lambda}} f,\; \mathrm{d}V=0={\mathrm{\lambda}} \mathrm{d}f+f\mathrm{d}\lambda,
\end{align}
</math>

hence,


<math>
\begin{align}
{\mathrm{\lambda}} /\mathrm{d}{\mathrm{\lambda}} =-f/\mathrm{d}f,\ \mathrm{and}\ U=V+f\frac{\mathrm{d}V}{\mathrm{d}f} =V+\omega \frac{\mathrm{d}V}{\mathrm{d}\omega }.
\end{align}
</math>

== Continue reading ==
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| align="center" | [[:Category:Introduction to Problems_in_Exploration_Seismology_and_their_Solutions|Introduction]]
| align="center" | [[:Category: Partitioning at an interface| Partitioning at an interface]]
|-
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[[Problems_in_Exploration_Seismology_and_their_Solutions|Table of Contents (book)]]
|}

== Also in this chapter ==
* [[The basic elastic constants]]
* [[Interrelationships among elastic constants]]
* [[Magnitude of disturbance from a seismic source]]
* [[Magnitudes of elastic constants]]
* [[General solutions of the wave equation]]
* [[Wave equation in cylindrical and spherical coordinates]]
* [[Magnitudes of seismic wave parameters]]
* [[Potential functions used to solve wave equations]]
* [[Boundary conditions at different types of interfaces]]
* [[Boundary conditions in terms of potential functions]]
* [[Disturbance produced by a point source]]
* [[Far- and near-field effects for a point source]]
* [[Rayleigh-wave relationships]]
* [[Directional geophone responses to different waves]]
* [[Tube-wave relationships]]
* [[Relation between nepers and decibels]]
* [[Attenuation calculations]]
* [[Diffraction from a half-plane]]

== External links ==
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[[Category:Problems and Solutions]]
[[Category:Publications]]
[[Category: Theory of seismic waves]]