Question

Consider a composite wave formed by two plane waves with slightly different frequencies of:

w1 = 3.1 x 10^12 rad/s , w2 = 3.2 x 10^12 rad/s

and their respective wavelengths are:

L1= 13.0 nm , L2= 14.0 nm

Calculate the wavelength of the envelope wave and give your results in units of meters with 1 digit precision, rounding off to 1 decimal place.

Answer 1

Amplitudes of the wave are not given. We assume amplitudes are same.

Two waves are given by

when these two waves are super imposed, we get

By using identity , sin(A) + sin(B) = 2 sin [ (A+B)/2 ] cos [ (A-B)/2 ]

if we substitute = 13 nm , = 14 nm , ₁ = 3.1 10¹² rad/s and ₂ = 3.2 10¹² rad/s

In the above expression, resultant wave is dominated by higher frequency component i.e. 3.15 10¹² rad/s and other lower frequency term 0.0510¹² rad/s behaves as envelope wave

As seen from above expression envelope wave has wavelength 364 nm = 364 10^-9 m