Waves

The required relation can be given as Given, Then,

The equation of travelling wave

Compare with the standard wave equation

we get
Amplitude,
Angular frequency, and
Angular wave number,
Velocity of the wave

Frequency of the wave

Wavelength of the wave

There is positive sign between and terms, the given wave is moving in the negative -direction.

and

Velocity of transverse wave is given by

Tension in string
Mass per unit length



Now, time taken by wave to travel full length of the rope

Given that fundamental frequency of open organ pipe
(i)
Frequency of second harmonic-of open organ pipe
and frequency of third harmonic of closed organ pipe
Now according to question
(ii)
Form E (i), we get


Now from E (ii), we get

Let the frequency of the fork be .
In the first case, the length of the wire is .
Therefore, (i)
On decreasing the length of the wire, its frequency will increase.
Hence, in the second case when the length is ,
then (ii)
Dividing Eqs. (i) by (ii), we get

Given, speed of rocket,
Frequency emitted by the rocket,
Velocity of sound,

Apply Doppler's effect in sound's equation

Putting values,


This frequency is reflected back to the rocket

Apply Doppler's effect in sound's equation

[from E (i)]

Thus, the most appropriate option is (d)