Sum of waves of different frequencies and group velocity

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	2
Pages	7 - 46
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 2.7a

A pulse composed of two frequencies, $\omega _{0}\pm {\mathrm {\Delta } }\omega$, 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

${\begin{aligned}A\cos \left(\kappa _{1}x-\omega _{1}t\right),{\quad }A\cos \left(\kappa _{2}x-\omega _{2}t\right),\end{aligned}}$

where $\omega _{1}=\omega _{0}+\Delta \omega ,\omega _{2}=\omega _{0}-\Delta \omega ,k_{0}=2\pi /\lambda _{0}=\omega _{0}/V$,

${\begin{aligned}\kappa _{1}&\approx \kappa _{0}+\Delta \kappa \approx \left(\omega _{0}+\Delta \omega \right)/V,\ \mathrm {and} \\\kappa _{2}&\approx \kappa _{0}-\Delta \kappa \approx \left(\omega _{0}-\Delta \omega \right)/V.\end{aligned}}$

Show that the composite wave is given approximately by the expression

${\begin{aligned}B\cos(\kappa _{0}x-\omega _{0}t),\end{aligned}}$

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

Background

When different frequency components in a pulse have different phase velocities $V$ (the velocity with which a given frequency travels), the pulse changes shape as it moves along. The group velocity $U$ 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

${\begin{aligned}\cos \theta +\cos \phi =2\cos \left[{\frac {1}{2}}\left(\theta +\phi \right)\right]\cos \left[{\frac {1}{2}}\left(\theta -\phi \right)\right],\end{aligned}}$

we get for the composite wave

${\begin{aligned}2A\cos \left\{{\mathrm {\Delta } }\kappa \left[x-\left({\mathrm {\Delta } }\omega /{\mathrm {\Delta } }\kappa \right)t\right]\right\}\cos \left(\kappa _{0}x-\omega _{0}t\right)=B\cos \left(\kappa _{0}x-\omega _{0}t\right).\end{aligned}}$

Problem 2.7b

Why do we regard $B$ as the amplitude? Show that the envelope of the pulse is the graph of $B$ plus its reflection in the $x$-axis.

Solution

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

Problem 2.7c

Show that the envelope moves with the group velocity $U$ where

${\begin{aligned}U={\frac {{\mathrm {\Delta } }\omega }{{\mathrm {\Delta } }\kappa }}\approx {\frac {\mathrm {d} \omega }{\mathrm {d} \kappa }}\approx V-{\mathrm {\lambda } }{\frac {\mathrm {d} V}{\mathrm {d} {\mathrm {\lambda } }}}=V+\omega {\frac {\mathrm {d} V}{\mathrm {d} \omega }}\end{aligned}}$

(2.7a)

(see Figure 2.7a).

Solution

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

${\begin{aligned}U=\mathrm {d} \omega /\mathrm {d} {\kappa }=\mathrm {d} f/\mathrm {d} \left(1/{\mathrm {\lambda } }\right)=\left(\mathrm {d} f/\mathrm {d} {\mathrm {\lambda } }\right)\left[\mathrm {d} {\mathrm {\lambda } }/\mathrm {d} \left(1/{\mathrm {\lambda } }\right)\right],\end{aligned}}$

Figure 2-7a  Illustrating group and phase velocity.

where $f$ is the frequency. The result is

${\begin{aligned}U=-{\mathrm {\lambda } }^{2}{\frac {\mathrm {d} f}{\mathrm {d} \lambda }}.\end{aligned}}$

We introduce the phase velocity $V$ by noting that $V={\mathrm {\lambda } }f$, so

${\begin{aligned}\mathrm {d} V={\mathrm {\lambda } }\mathrm {d} f+f\mathrm {d} \lambda ,\;{\mathrm {\lambda } }{\frac {\mathrm {d} f}{\mathrm {d} {\mathrm {\lambda } }}}={\frac {\mathrm {d} V}{\mathrm {d} {\mathrm {\lambda } }}}-f.\end{aligned}}$

Thus,

${\begin{aligned}U=-{\mathrm {\lambda } }\left({\mathrm {\lambda } }{\frac {\mathrm {d} V}{\mathrm {d} {\mathrm {\lambda } }}}\right)={\mathrm {\lambda } }f-{\mathrm {\lambda } }{\frac {\mathrm {d} V}{\mathrm {d} {\mathrm {\lambda } }}}=V-{\mathrm {\lambda } }{\frac {\mathrm {d} V}{\mathrm {d} \lambda }}.\end{aligned}}$

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

${\begin{aligned}V={\mathrm {\lambda } }f,\;\mathrm {d} V=0={\mathrm {\lambda } }\mathrm {d} f+f\mathrm {d} \lambda ,\end{aligned}}$

hence,

${\begin{aligned}{\mathrm {\lambda } }/\mathrm {d} {\mathrm {\lambda } }=-f/\mathrm {d} f,\ \mathrm {and} \ U=V+f{\frac {\mathrm {d} V}{\mathrm {d} f}}=V+\omega {\frac {\mathrm {d} V}{\mathrm {d} \omega }}.\end{aligned}}$

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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

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