# Phase velocity

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

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

 {\displaystyle {\begin{aligned}u\left(x,t\right)=A{\rm {\ sin\ }}\left(kx-\omega t\right).\end{aligned}}} (21)

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

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

 {\displaystyle {\begin{aligned}u\left(x,t\right)&{\rm {=}}A{\rm {\ sin\ }}\left(kx-\omega t{\rm {+}}\varepsilon \right),\end{aligned}}} (22)

where ${\displaystyle \varepsilon }$ is the initial phase. The phase of this more general sinusoidal wave is

 {\displaystyle {\begin{aligned}\varphi \left(x,t\right)\ \ {=\ }kx-\omega t{\ +\ }\varepsilon \end{aligned}}} (23)

and is evidently a function of both x and t. The partial derivative of ${\displaystyle \varphi }$ 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

 {\displaystyle {\begin{aligned}{\frac {\partial \varphi }{\partial t}}{\ =\ }-\omega .\end{aligned}}} (24)

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

 {\displaystyle {\begin{aligned}{\frac {\partial \varphi }{\partial x}}=k.\end{aligned}}} (25)

The condition of constant phase is expressed as

 {\displaystyle {\begin{aligned}\varphi \left(x,t\right)&{\rm {=}}kx-\omega t{\rm {+}}\varepsilon {\rm {=}}\mathrm {constant} .\end{aligned}}} (26)

Taking differentials, we have

 {\displaystyle {\begin{aligned}d\varphi &{\rm {=}}kdx-\omega dt{\rm {=0}}.\end{aligned}}} (27)

Solving this equation, we obtain (for ${\displaystyle \varphi }$ = constant)

 {\displaystyle {\begin{aligned}{\left[{\frac {\partial x}{\partial t}}\right]}_{\varphi }&{\rm {=}}{\frac {\omega }{k}}.\end{aligned}}} (28)

However,

 {\displaystyle {\begin{aligned}{\frac {\partial \varphi }{\partial t}}{\rm {=}}-\omega \ \mathrm {and} \ {\frac {\partial {\rm {(}}\varphi }{\partial x}}{\rm {=}}k,\end{aligned}}} (29)

so we have

 {\displaystyle {\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, ${\displaystyle v{\rm {=}}f\lambda }$, and because ${\displaystyle \omega {\rm {=2}}\pi f}$ and ${\displaystyle k{\rm {=2}}\pi {\rm {/}}\lambda }$, it follows that ${\displaystyle v{\rm {=}}\omega /k}$. Therefore, equation 30 also can be written

 {\displaystyle {\begin{aligned}{\left[{\frac {\partial x}{\partial t}}\right]}_{\varphi }&{\rm {=}}{\frac {\omega }{k}}{\rm {=}}\ v.\end{aligned}}} (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, ${\displaystyle u{\rm {=}}A{\rm {\ sin\ }}\left(k\left(x\pm vt\right)\right)}$). Assume that the quantity v in this equation is positive, and choose the negative sign. Then the condition of constant phase is that ${\displaystyle \varphi {\rm {=}}k\left(x-vt\right)\mathrm {=constant} }$. This equation says that as t increases, x also must increase. Even when ${\displaystyle x{\rm {<0}}}$, so that ${\displaystyle \varphi {\rm {<0}}}$, 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

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