bei48482_FM

It is not hard to find the velocity gwith which a wave group travels. Let us sup-

pose that the wave group arises from the combination of two waves that have the same

amplitude Abut differ by an amount in angular frequency and an amount kin

wave number. We may represent the original waves by the formulas

y 1  A cos (t kx)

y 2  A cos [( )t (k k)x]

The resultant displacement yat any time tand any position xis the sum of y 1 and y 2.

With the help of the identity

cos cos  2cos^12 ( )cos^12 ( )

and the relation

cos ( ) cos

we find that

y y 1 y 2

 2 A cos^12 [(2)t (2k k)x]cos^12 ( t kx)

Since and kare small compared with and krespectively,

2  2 

2 k k 2 k

and so

Beats y  2 A cos (t kx) cos t  x (3.10)

k



2





2

100 Chapter Three

Figure 3.4Beats are produced by the superposition of two waves with different frequencies.

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