Question

Find c, yenvelope(x,t), and ycarrier(x,t). express your answer in terms of a, k1, k2, x, t, ω1, and ω2. separate the three parts of your response with commas. recall that yenvelope (the second term) varies slowly whereas ycarrier (the third term) varies quickly. both yenvelope and ycarrier should be trigonometric functions of unit amplitude.

Answer 1

`C,Y_(envelope)(x,t), Y_(carrier)(x,t)=2A, \cos ((k_1-k_2)x/2-(\omega_1 -\omega_2)t / 2 ) , \sin ((k_1+k_2)x / 2 - (\omega_1 +\omega_2)t / 2 )`

Explanation:

Given

`Y_1(x,t)=A \sin(K_1x- \omega _1 t)\\\\Y_2(x,t)=A \sin(K_2x- \omega _2 t)`

using a trigonometrical identity

sin p + sin q = 2 sin ( p+q/2) cos ( p-q/2)

and here the condition is

the choice is in between sinax and cosax

where a > b

so we get using above equation

`C,Y_(envelope)(x,t), Y_(carrier)(x,t)=2A, \cos ((k_1-k_2)x/2-(\omega_1 -\omega_2)t / 2 ) , \sin ((k_1+k_2)x / 2 - (\omega_1 +\omega_2)t / 2 )`