Which of the following statements are correct ?

Options

A. The decrease in the speed of sound at high altitudes is due to a fall in pressure .
B. The standing wave on a string under a tension , fixed at its ends , does not have well - defined nodes .
C. The phenomenon of beats is not observable in the case of visible light waves.
D. The apparent frequency is `f_(1)` when a source of sound approached a stationary observer with a speed `u` and is `f_(2)` when the observer approaches the same stationary source with the same speed . Then `f_(2) lt f_(1) , if u lt v`, where `v` is the speed of sound.

Answer

B, C, D

Explanation

Statement `(a)` is incorrect .
A change in pressure has no effect on the speed of sound . The decrease in the speed at high altitudes is due to fall in temperature .
Statement `(b)` is correct .
Standing waves are produced due to superposition of the incident waves and the waves reflected from the fixed ends of the string . Since , the ends are never perfectly rigidly fixed , the amplitude of the reflected wave is always less than that of the incident wave. Consequently , the resultant amplitudes at nodes is not exactly zero . Thus , the nodes are not well defined .
Statement `( c)` is also correct .
To observe beats , the difference between the two interfering frequencies must be less than about `10 - 16 Hz`. Since , visible light waves have very high frequencies , beats are not observed due to persistence of vision.
Statement `(d)` is also correct . We know that
`f_(1) = (f)/( 1 - (u)/(v))` `(i)`
And `f_(2) = f( 1 + (u)/(v))` `(ii)`
Expression Eq. `(i)` may be written as
`f_(1) = f ( 1 - (u)/(v))^(-1)`
Extanding binomically and retaining terms up to order `u^(2)//v^(2)`, we have
`f_(1) = v( 1 + (u)/(v) + (u^(2))/(v^(2)))` `(iii)`
Comparing Eqs. `(ii)` and `(iii)` , we find that `f_(1) gt f_(2)`.