Sinusoidal waves

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	1
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

Until now, we have not given the wave function $u\left(x,t\right)$ an explicit functional dependence; that is, we have not specified its shape. Let us now examine the simplest waveform for which the profile is a sine or cosine curve. Such waves are known as sinusoidal waves or, in older terminology, as simple harmonic waves. Because, by Fourier’s theorem, any periodic function can be synthesized by a superposition of sinusoidal waves, it is important that we know their properties.

Choose as the profile the simple function

${\begin{aligned}u\left(x,t\right){\rm {|}}_{\rm {t=0}}&{\rm {=}}u\left(x,0\right){\rm {=}}Asinkx=f\left(x\right),\end{aligned}}$

(10)

where k is a positive constant known as the wavenumber, or propagation number, and the product kx is in radians. The sine varies in magnitude from ${\rm {+l}}$ to $-{\rm {1}}$ so that the maximum value of $u\left(x,0\right)$ is A, which is known as the amplitude of the wave. To transform equation 10 into a progressive wave traveling at speed v in the positive x-direction, we need to replace x by $x-vt$, in which case

${\begin{aligned}u\left(x,t\right)&{\rm {=}}Asink\left(x-vt\right){\rm {=}}f\left(x-vt\right).\end{aligned}}$

(11)

This is a solution of the wave equation. Holding either x or t fixed results in a sinusoidal disturbance, so the wave is periodic in both space and time.

Let us now examine the spatial period. The spatial period is known as the wavelength and is denoted by $\lambda$. An increase or decrease in x by the amount $\lambda$ leaves u unaltered; that is,

${\begin{aligned}u\left(x,t\right)&{\rm {=}}u\left(x\pm \lambda ,t\right).\end{aligned}}$

(12)

Thus, an increase or decrease in x by the amount $\lambda$ leaves the sine representation unaltered; that is,

${\begin{aligned}{\rm {\ sin\ }}\left(k\left(x-vt\right)\right)&{\rm {=\ sin\ }}\left(k\left(x\pm \lambda -vt\right)\right){\rm {=\ sin\ }}\left(k\left(x-vt\right)\pm k\lambda \right).\end{aligned}}$

(13)

We also know that an increase or decrease in the argument by the amount ${\rm {2}}\pi$ leaves the sine representation unaltered; that is,

${\begin{aligned}{\rm {\ sin\ }}\left(k\left(x-vt\right)\right)&{\rm {=\ sin\ }}\left(k\left(x-vt\right)\pm {\rm {2}}\pi \right).\end{aligned}}$

(14)

From the above two equations, we see that ${\rm {|}}k\lambda {\rm {|}}{\rm {=2}}\pi$. Because both k and $\lambda$ are taken as positive numbers, we obtain the fundamental equation $k{\rm {=2}}\pi {\rm {/}}\lambda$. The wavelength $\lambda$ is the distance between successive crests of the spatial sinusoidal wave and commonly is expressed in meters.

In a completely analogous manner, we can examine the temporal period T, which usually is called the period. It is the amount of time one complete oscillation takes to pass a stationary observer. In this case, the repetitive behavior of the wave in time is of interest, so $u\left(x,t\right){\rm {=}}u\left(x,t\pm T\right)$ and

${\begin{aligned}{\rm {\ sin\ }}\left(k\left(x-vt\right)\pm {\rm {2}}\pi \right)&{\rm {\ =\ sin\ }}\left(k\left(x-vt\right)\right){\rm {=\ sin\ }}\left(k\left(x-v\left(t\pm T\right)\right)\right)\end{aligned}}$

${\begin{aligned}{\rm {=\ sin\ }}&\left(k\left(x-vt\right)\mp kvT\right).\end{aligned}}$

(15)

Therefore, ${\rm {|}}kyT{\rm {|}}{\rm {=2}}\pi$. But these are all positive quantities, so $kyT{\rm {=2}}\pi$. As we have just seen, $k{\rm {=2}}\pi {\rm {/}}\lambda$, so ${\rm {2}}\pi vT/\lambda {\rm {=2}}\pi$ from which it follows that $T{\rm {=}}\lambda /v$. The period T is the number of units of time per cycle. The period commonly is expressed in seconds. Thus, the velocity is $v{\rm {=}}\lambda /T$.

To describe the period, we first choose a fixed point in space. We then measure the time interval between a wave’s successive peaks (or troughs) to obtain the period. To describe the wavelength, we first fix time by taking a photograph of the wave. We then measure the distance between successive peaks (or troughs) on the photograph to obtain the wavelength (Figure 4).

Figure 4.  A sinusoidal wave. Top diagram: When the wave is plotted as a function of time at a given distance, the period is the interval of time between two adjacent crests. Bottom diagram: When the wave is plotted as a function of distance at a given time, the wavelength is the interval of distance between two adjacent crests. The velocity of the wave is equal to the quotient of wavelength divided by the period.

The reciprocal of the period is the frequency f, or the number of cycles per unit of time. The frequency f commonly is expressed in Hertz (Hz), or cycles per second. The equation $T{\rm {=}}\lambda /v$ becomes $v{\rm {=}}\lambda f$. The velocity v commonly is expressed in meters per second. The two quantities that often are used in the literature of wave motion are the angular frequency

${\begin{aligned}\omega &{\rm {=}}{\frac {{\rm {2}}\pi }{T}},\end{aligned}}$

(16)

which commonly is expressed in radians per second, and the angular wavenumber

${\begin{aligned}k&{\rm {=}}{\frac {{\rm {2}}\pi }{\lambda }},\end{aligned}}$

(17)

which commonly is expressed in radians per meter. We see that the angular wavenumber is the same quantity we previously called the propagation number. The wavelength, wavenumber, period, and frequency all describe aspects of the repetitive nature of a wave in space and time. These concepts can be applied to waves that are not sinusoidal, as long as each wave is made up of a regularly repeating pattern (i.e., of periodic waves, examples of which are shown in Figure 5).

Figure 5.  Three examples of periodic waves.

Let us now look at the equation $v{\rm {=}}\lambda f$. If we divide the wave velocity v (expressed in meters per second) by the wavelength $\lambda$ (expressed in meters), the length term (meters) cancels out, and the result must be expressed as something per second. As we know, we call this result frequency f because it states how frequently the new wave crests pass a given point. Frequency f is expressed in cycles per second because it tells us how many wave troughs and crests (i.e., how many cycles) pass the given point in one second. More specifically, we can say that wavelength $\lambda$ is expressed in meters per cycle. Then we have the equation

${\begin{aligned}{\frac {\rm {meters\ \ per\ \ second}}{{\rm {meters\ \ per}}{\rm {\ \ cycle}}}}=\mathrm {cycles\ \ per\ \ second,or{\frac {v}{\lambda }}=f} .\end{aligned}}$

(18)

Thus, we can write

${\begin{aligned}\mathrm {frequency\times wavelength{\rm {=}}velocity,or\ f\ \lambda {\rm {\ =}}\ v} .\end{aligned}}$

(19)

In a medium of constant velocity, we see that a wave with a short wavelength has a high frequency, and one with a long wavelength has a low frequency.

For example, suppose that we have a machine that generates the same pulse shape, one after the other, at equal time intervals T. In doing this, the wave generator repeats its motion once every interval T, the period of the motion. If the motion repeats every 0.01 s, then the frequency is 100 Hz (i.e., 100 cycles/s).

Let us now concentrate on some spatial point. The pulses produced by the generator move toward this point, and they pass the point with the same frequency as the one at which they leave the source. The frequency of the wave motion is therefore also 100 Hz, and the time between passages of successive pulses is also 0.01 s. Furthermore, as the waves move, the spatial distance between any two adjacent pulses is always the same and is the wavelength $\lambda$. Because the pulses are separated by a distance $\lambda$ and because each pulse moves over this distance in a time T, it follows that the velocity of propagation is $v{\rm {=}}\lambda /T$. Using the relation $f{\rm {=}}1/T$, we again find that $v{\rm {=}}f\lambda$ or that the velocity of propagation of a periodic wave is the product of the frequency and the wavelength. This is an important relationship; in particular, it holds for sinusoidal waves.

Now we come to an application of the formula $v{\rm {=}}f\lambda$. Instead of watching a periodic wave continuously, we look at it through a shutter that is closed most of the time and opens periodically for short time intervals. Such an instrument is the stroboscope. The first time the shutter opens, we see the wave pattern in a certain position. When the shutter is closed, all the pulses move a distance equal to their velocity multiplied by that time duration. As we look through the shutter while it periodically opens and closes, the pattern usually appears to move. However, if the period of the shutter is the same as the period of the wave motion, then while the shutter is closed, each pulse moves up to the position of the pulse just ahead of it. Consequently, we see the same pattern each time the shutter opens. In other words, we see a stationary pattern from which it is easy to measure the wavelength. In addition, as we stated earlier, the period of the shutter is equal to the period of the wave, which we obtain by simply counting the number of times the shutter is opened each second - that is, by measuring the frequency of the shutter. Now we have both f and $\lambda$ for the wave, so we can use the formula $v{\rm {=}}1\lambda$ to determine the velocity of the wave.

Thus far, we have defined several quantities that characterize various aspects of wave motion. Accordingly, several equivalent formulations of the progressive sinusoidal wave exist. Some of the more common are

${\begin{aligned}u{\rm {=}}A{\rm {\ sin\ }}k\left(x\pm vt\right)\end{aligned}}$

${\begin{aligned}u{\rm {=}}A{\rm {\ sin\ 2}}\pi \left({\frac {x}{\lambda }}\pm {\frac {t}{T}}\right)\end{aligned}}$

${\begin{aligned}u=A{\rm {\ sin\ }}\left(kx\pm \omega t\right)\end{aligned}}$

${\begin{aligned}u{\rm {=}}A{\rm {\ sin\ 2}}\pi f\left({\frac {x}{v}}\pm t\right).\end{aligned}}$

(20)

Note that these waves are all of infinite extent. That is, for any fixed value of time t, the distance x varies from $-\infty$ to $\infty$. Each wave has a single constant frequency and therefore is said to be monochromatic.

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Table of Contents (book)

Also in this chapter

• Introduction
• Wavefronts and raypaths
• d’Alembert’s solution
• One-dimensional waves
• Phase velocity
• Wave pulses
• Geometric seismology
• The speed of light
• Huygens’ principle
• Reflection and refraction
• Ray theory
• Fermat’s principle
• Fermat’s principle and reflection and refraction
• Diffraction
• Analogy
• Appendix A: Exercise

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