Phase velocity

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

Let us examine in more detail a form of the third of the above sinusoidal wave functions,

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\left(x,t\right)=A{\rm \ sin\ }\left(kx-\omega t\right) . \end{align}

(21)

The argument of the sine function is known as the phase 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/":): \varphi of the wave 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/":): \varphi\ \ {\rm = }kx-\omega t . At 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{\rm =}x{\rm =0} , we have 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\left(0,0\right){\rm =0} , which is certainly a special case.

Instead of the above equation, we can write a sinusoidal wave in the more general form

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\left(x,t\right)&{\rm =}A{\rm \ sin\ }\left(kx-\omega t{\rm +}\varepsilon \right) , \end{align}

(22)

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/":): \varepsilon is the initial phase. The phase of this more general sinusoidal wave 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} \varphi \left(x,t\right)\ \ {=\ }kx-\omega t{\ +\ }\varepsilon \end{align}

(23)

and is evidently a function of both x and t. The partial derivative 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/":): \varphi with respect to t, holding x constant, is the rate of change of phase with respect to time and is equal to the negative angular frequency

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} \frac{\partial\varphi }{\partial t}{\ =\ }-\omega . \end{align}

(24)

Similarly, the rate of change of phase with distance x, holding t constant, is the wavenumber

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} \frac{\partial\varphi }{\partial x}=k. \end{align}

(25)

The condition of constant phase is expressed as

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} \varphi \left(x,t\right)&{\rm =}kx-\omega t{\rm +}\varepsilon {\rm =} \mathrm{constant}. \end{align}

(26)

Taking differentials, we have

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} d\varphi &{\rm =}kdx-\omega dt{\rm =0}. \end{align}

(27)

Solving this equation, we obtain (for 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/":): \varphi = constant)

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} {\left[\frac{\partial x}{\partial t}\right]}_{\varphi }&{\rm =}\frac{\omega }{k}. \end{align}

(28)

However,

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} \frac{\partial\varphi }{\partial t} {\rm =}-\omega \ \mathrm{and} \ \frac{\partial {\rm (}\varphi }{\partial x}{\rm =}k, \end{align}

(29)

so we have

$ {\begin{aligned}{\left[{\frac {\partial x}{\partial t}}\right]}_{\varphi }=-{\frac {\partial {\varphi }{/}\partial t}{\partial \varphi {/}\partial x}}={\frac {\omega }{k}}.\end{aligned}} $

(30)

The term on the left represents the velocity of propagation, subject to the condition of constant phase. Choose any point on the wave profile - for example, the crest of the wave. As the wave moves through space, the displacement u of the crest remains constant. Because the only variable in the sinusoidal wave function is the phase, it too must be constant. That is, the phase is fixed at a value producing the constant displacement u at the chosen point. The point moves along with the profile at velocity v, and so does the constant phase condition.

Because velocity is equal to frequency times wavelength, that 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/":): v{\rm =}f\lambda , and because 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 {\rm =2}\pi f 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/":): k{\rm =2}\pi {\rm /}\lambda , it follows 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{\rm =}\omega /k . Therefore, equation 30 also 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/":): \begin{align} {\left[\frac{\partial x}{\partial t}\right]}_{\varphi }&{\rm =}\frac{\omega }{k}{\rm =}\ v. \end{align}

(31)

Thus, the speed at which the profile moves is the wave velocity v or, more specifically, the phase velocity. The phase velocity carries a positive sign when the wave moves in the direction of increasing x, and it carries a negative sign when the wave moves in the direction of decreasing x.

Let us consider further the concept of propagation at constant phase and examine how this concept governs the propagation of any one of the sinusoidal waves, say, 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 {\rm =}A{\rm \ sin\ }\left(k\left(x\pm vt\right)\right) ). Assume that the quantity v in this equation is positive, and choose the negative sign. Then the condition of constant phase is 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/":): \varphi {\rm = }k\left(x - vt\right)\mathrm{=constant} . This equation says that as t increases, x also must increase. Even when 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{\rm < 0} , 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/":): \varphi {\rm < 0} , the quantity x must increase - that is, it will become less negative. Thus, the constant-phase condition implies propagation in the increasing x-direction. Let us once more assume that v is positive, but now we choose the negative sign in the constant-phase condition. We get

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} \varphi {\rm =}k\left(x{\rm +}vt\right)&{\rm =} \mathrm{constant}. \end{align}

(32)

This equation now tells us that as t increases, the quantity x can be positive and decreasing or negative and becoming more negative. Here, the constant-phase condition implies propagation in the decreasing x-direction.

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Table of Contents (book)

Also in this chapter

• Introduction
• Wavefronts and raypaths
• d’Alembert’s solution
• One-dimensional waves
• Sinusoidal waves
• 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

External links

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