Doppler effect

An aircraft is north-east of you, flying north with a steady course and speed, half the speed of sound. It is emitting a loud sound with a pitch of 400 Hz. What pitch do you hear?

Obvious (?) answer: show

The answer below is tedious, uninstructive, and possibly wrong. I recommend skipping it.

To the right is a diagram for the problem.
    O is the observer.
    C is the current position of the aeroplane.
    B is its position when it emitted the sound we are now hearing.
    A is the point on its flight path nearest to us.
    S is the speed of sound

B is x units (think of them as miles if you like) north of A.
While the aeroplane has travelled from B to C, the sound it emitted at B has travelled, at twice the speed of the aeroplane, from b to O. So we apply Pythagoras to triangle ABC:
    sqrt( 1² + x² ) = 2(1-x)
    x² + 1 = 4x² - 8x = 4
    3x² -8x = 3 = 0
    x = (4 ± sqrt(7) ) / 3
    x = 0.4514
    so angle AOB is arctan( 0.4514 )
so angle AOB is 24.3°
so when the aeroplane was at B, the resolved part of its velocity away from the observer was S/2 sin(24.3°)
= 0.2057 S
So the pitch heard is 400 * (1/1.2057) Hz
=332 Hz

Alternative interpretation of question: show

My physics teacher claimed that the method used above was too difficult for the examiners to expect us to apply it. Instead, we should interpret the question as

An aircraft is north-east of you, flying north with a steady course and speed, half the speed of sound. It is emitting a loud sound with a pitch of 400 HZ. What pitch ~~do you hear~~ will you hear when the sound it is now making reaches you?

This is of course much easier:

The resolved part of its velocity away from the observer was S/2 sin(45°)
= 0.3536 S
So the pitch heard is 400 * (1/1.3536) Hz
=296 Hz

Doppler effect

Problem statement

Obvious (?) answer: show

Alternative interpretation of question: show

Comments: show