Waves and ripples in water, air, and æther

by Sir J. A. Fleming

APPENDIX.

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Note A ([see p. 21]).

The distinction between the individual wave-velocity and a wave-group velocity, to which, as stated in the text, attention was first called by Sir G. G. Stokes in an Examination question set at Cambridge in 1876, is closely connected with the phenomena of beats in music.

If two infinitely long sets of deep-sea waves, having slightly different wave-lengths, and therefore slightly different velocities, are superimposed, we obtain a resultant wave-train which exhibits a variation in wave-amplitude along its course periodically. If we were to look along the train, we should see the wave-amplitude at intervals waxing to a maximum and then waning again to nothing. These points of maximum amplitude regularly arranged in space constitute, as it were, waves on waves. They are spaced at equal distances, and separated by intervals of more or less waveless or smooth water. These maximum points move forward with a uniform velocity, which we may call the velocity of the wave-train, and the distance between maximum and maximum surface-disturbances may be called the wave-train length.

Let v and v′ be the velocities, and n and n′ the frequencies, of the two constituent wave-motions. Let λ and λ′ be the corresponding wave-lengths. Let V be the wave-train velocity, N the wave-train frequency, and L the wave-train length. Then N is the number of times per second which a place of maximum wave-amplitude passes a given fixed point.

Then we have the following obvious relations:⁠—

v  =  nλ,  v′  =  n′λ′, N  =  n – n′  =  ^v / _λ – ^v′ / _λ′

Also a little consideration will show that⁠—

^L / _λ′  =  ^λ / _{λ – λ′}