Question

Two waves, travelling along the same direction, are given by
y1(x, t) = asin(w1t − k1x)
and
y2 (x, t) = asin(w2t − k2x)
Suppose that w1and k1 are respectively slightly greater than w2 and k2. i) Obtain an
expression for the resultant wave arising due to their superposition and ii) explain the
formation of wave packets.

Answer 1

Answer on Question#54675, Physics/Mechanics | Kinematics | Dynamics

I) Using formula for sum of two sine functions, obtain:

$$y _ {1} + y _ {2} = a \sin (w _ {1} t - k _ {1} x) + a \sin (w _ {2} t - k _ {2} x) = 2 a \sin \left((w _ {1} + w _ {2}) \frac {t}{2} - (k _ {1} + k _ {2}) \frac {x}{2}\right) \cos \left((w _ {1} - w _ {2}) \frac {t}{2} - (k _ {1} - k _ {2}) \frac {x}{2}\right)$$

II) The resulting wave is a sine wave with frequency $\frac{(w_1 + w_2)}{2}$ with enveloping cosine function of frequency $\frac{w_1 - w_2}{2}$ . Because the frequencies of the initial two waves are not equal but almost the same, the enveloping cosine function has a small frequency (big period). This phenomenon is called beats. For equal frequencies of incident waves, the beats do not occur.