Stationary waves are produced in a length of wire fixed between two points . Compare the amplitudes at an antinode for the fundamental and the first overtone . Assume that the total energy of the initial waves is, on an average, equally divided between the two modes.

Options

A. `3/sqrt2`
B. `1/sqrt2`
C. `1/2`
D. `1/(2sqrt2)`

Answer

B

Explanation

Let ` a` be the amplitude of each component wave . Then total energy of waves
` 2 xx (1)/(2) rho a^(2) omega^(2) c = rho a^(2) omega^(2) c`
where `rho =` density and omega = cyclic frequency of each wave
Let `A_(0)` be the amplitude of he fundamental mode .
`:. rho a^(2) omega^(2) c = (1)/(2) rho A_(0)^(2) omega^(2) c or A_(0) = a sqrt(2)`
Let `A_(1)` be the amplitude of the first overtone . There are two loops in the first overtone .
`:. rho a^(2) omega^(2) c = 2 xx (1)/(2) rho A_(1)^(2) omega^(2) c rArr A_(1) = a`
`:. (A_(1))/(A_(0)) = (a)/( a sqrt(2)) or (A_(1))/(A_(0)) = (1) /(sqrt(2))`