Two harmonic waves moving in the same direction superimpose to form x = a cos(1.5t) cos(50.5t), where t is in seconds. Find the beat period, rounded to the nearest integer.
Options

Question

Two harmonic waves moving in the same direction superimpose to form x = a cos(1.5t) cos(50.5t), where t is in seconds. Find the beat period, rounded to the nearest integer.

G1 Physics · Accepted Answer

Use the identity cosA cosB = ½[cos(A+B) + cos(A-B)]. Thus, x = (a/2)cos[(50.5+1.5)t] + (a/2)cos[(50.5-1.5)t] = (a/2)cos(52t) + (a/2)cos(49t). Therefore the angular frequencies are ω₁ = 52 rad/s, ω₂ = 49 rad/s. The beat angular-frequency difference is Δω = 52-49 = 3 rad/s. Hence the beat frequency is f b e a t f_beat f b ​ e a t = Δω/(2π) = 3/(2π) Hz. Beat period: T b e a t T_beat T b ​ e a t = 1/ f b e a t f_beat f b ​ e a t = 2π/3 s ≈ 2.09 s. Rounded to the nearest integer, T b e a t T_beat T…