Sum of waves of different frequencies and group velocity

From SEG Wiki
Jump to navigation Jump to search
ADVERTISEMENT

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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \omega _{1} =\omega _{0} +\Delta \omega, \omega_2= \omega_0 -\Delta \omega, k_{0} =2\pi /\lambda _{0} =\omega _{0} /V ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Show that the composite wave is given approximately by the expression


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} B\cos (\kappa_{0} x-\omega _{0} t), \end{align}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V (the velocity with which a given frequency travels), the pulse changes shape as it moves along. The group velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

we get for the composite wave


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Problem 2.7b

Why do we regard Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B as the amplitude? Show that the envelope of the pulse is the graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B plus its reflection in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis.

Solution

The solution in (a) can be written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi =B\psi _{0} . We regard Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B as the amplitude for two reasons: (1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B repeats every time that $ x $ increases by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi /{\mathrm{\Delta}} \kappa or each time that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t increase by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi /{\mathrm{\Delta}} \omega . But Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\Delta}} \kappa\ll \kappa_{0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\Delta}} \omega \ll \omega _{0} \} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B must repeat more slowly than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi _{0} . (2) Each time that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi _{0} attains its limiting values, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm 1 , $ \psi $ has the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm B and therefore never exceeds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left|B\right| ; thus the curves of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): +\left|B\right| and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\left|B\right| pass through the maxima and minima of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \psi _{0} and therefore constitute the envelope.

Problem 2.7c

Show that the envelope moves with the group velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U where


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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} (2.7a)

(see Figure 2.7a).

Solution

If we consider the quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B as a wave superimposed on the primary wavelet $ \cos \left(\kappa _{0}x-\omega _{0}t\right) $, comparison with the basic wave type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(x-Vt) of problem 2.5a shows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left[x-\left({\mathrm{\Delta}} \omega /{\mathrm{\Delta}} \kappa\right)t\right] takes the place of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (x-Vt) , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left({\mathrm{\Delta}} \omega /{\mathrm{\Delta}} \kappa\right) is the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U with which the envelope travels. In the limit, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U is given by


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Figure 2-7a  Illustrating group and phase velocity.

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f is the frequency. The result is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} U=-{\mathrm{\lambda}} ^{2} \frac{\mathrm{d}f}{\mathrm{d}\lambda}. \end{align}

We introduce the phase velocity $ V $ by noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V={\mathrm{\lambda}} f , so


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Thus,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

To replace the derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{d}V/\mathrm{d}{\mathrm{\lambda}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathrm{d}V/\mathrm{d}f , we find the relation between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathrm{\lambda}}/\mathrm{d}{\mathrm{\lambda}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f/\mathrm{d}f ; to do this we let $ V $ be constant so that


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V={\mathrm{\lambda}} f,\; \mathrm{d}V=0={\mathrm{\lambda}} \mathrm{d}f+f\mathrm{d}\lambda, \end{align}

hence,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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}

Continue reading

Previous section Next section
Wave equation in cylindrical and spherical coordinates Magnitudes of seismic wave parameters
Previous chapter Next chapter
Introduction Partitioning at an interface

Table of Contents (book)

Also in this chapter

External links

find literature about
Sum of waves of different frequencies and group velocity
Retrieved from "http://wiki.seg.org/w/index.php?title=Sum_of_waves_of_different_frequencies_and_group_velocity&oldid=119889"