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Some Theorems in Group Velocity
Published online by Cambridge University Press: 20 January 2009
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We consider a medium in which the equation satisfied by a disturbance ø(x,t) is capable of solutions of the form sin(vx—wt), with v and ω real but not necessarily of the same sign. If the phase velocity U = ω/v is not of constant magnitude the medium is said to be dispersive, and the group velocity V may conveniently be defined as dω/v, an expression easily re-written in other familiar forms. From this definition the two physical interpretations of V may easily be seen. In one interpretation we consider a superposition of two harmonic waves with slightly different v (and ω); the velocity of advance of the group form, being the velocity of advance of a point at which the two waves have a constant phase difference, is dω/v.
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- Copyright © Edinburgh Mathematical Society 1956
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1 The point and dash mean ∂/∂t and ∂/∂x; and the range of all the integrals in the paper is from —∞ to ∞, except where otherwise obvious.
1 It will be found that there are also a large number of such expressions containing higher differential coefficients.
1 Similarly, of course, in media where there are relations of the form 
and 
the quantity 
is independent of time, and the expression for it as E multiplied by an average of V 2 is easily writtan down. It is also evident that, owing to the ultimate sorting out of the frequencies in space, there will be a large number of X m's whose m-th time differential coefficients, though not constant, will tend with increasing time to constant values given by simple expressions.
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