"A source of sound of frequency f0 moves horizontally at constant speed u in the x direction at a distance h above the ground. An observer is situated on the ground at the point x=0 (the source passes over this point at t=0).
Show that the signal received at any time Tr at the ground was emitted by the source at an early time Ts , such that:
[1-(u/v)2]*Ts=Tr-(1/v)√(h2*[1-(u/v)2]+u2*Tr2)
where v is the speed of the sound."
To solve this problem, we need to analyze the scenario and understand the nature of the signal received by the observer on the ground.
Let's break down the problem step by step:
1. Determine the time it takes for the sound to reach the observer:
We know that the speed of sound is v, and the signal travels a distance h from the source to the ground. So, it takes time t1 = h/v for the sound to reach the ground.
2. Determine the time it takes for the source to pass over the observer:
Since the source moves at a constant speed u, we can calculate the time it takes for the source to pass over the observer. The distance it needs to cover is h, so the time it takes is t2 = h/u.
3. Determine the time delay between the source passing over the observer and the observer receiving the signal:
The observer receives the signal at time Tr. So, the time delay is Tr - t2.
4. Find the time at which the signal was emitted by the source:
We need to find the time Ts at which the signal was emitted by the source, considering the time delay.
Now, let's represent the given equation in terms of the variables we have defined:
[1 - (u/v)^2] * Ts = Tr - (1/v) * √(h^2 * [1 - (u/v)^2] + u^2 * Tr^2)
Rearrange the equation to solve for Ts:
Ts = [Tr - (1/v) * √(h^2 * [1 - (u/v)^2] + u^2 * Tr^2)] / [1 - (u/v)^2]
This equation allows us to calculate Ts, the time at which the signal was emitted by the source, based on the given information about the frequency of the source, the speed of the source, and the height of the source above the ground.
Note: The derivation of the equation involves the use of the Doppler effect formula for sound, the distance formula, and some algebraic manipulation.