Question

Discuss the following case-

Both are in motion

1. Source and Observer approach each other
2. Source and Observer resides from each other
3. Source chases Observer
4. Observer chases Source

Answer 1

(a) Source and observer approach each other:

Let v_s and v_o be the respective velocities of source and observer approaching each other as shown in Figure. In order to calculate the apparent frequency observed by the observer, let us have a dummy (behaving as observer or source) in between the source and observer. Since the dummy is at rest, the dummy (observer) observes the apparent frequency due to the approaching source as given in equation f ‘ = `"f"/((1 - "v"_"s"/"v"))`

`"f"_"d" = "f"/((1 - "v"_"s"/"v"))`  ....(1)

The true observer approaches the dummy from the other side at that instant of time. Since the source (true source) comes in a direction opposite to the true observer, the dummy (source) is treated as a stationary source for the true observer at that instant. Hence, apparent frequency when the true observer approaches the stationary source (dummy source), f’ = `"f"(1 + "v"_0/"v")`.

f’ = `"f"_"d"(1 + "v"_0/"v")`

`=> "f"_"d" = "f'"/((1 + "v"_0/"v"))`  ....(2)

Since this is true for any arbitrary time, therefore, comparing equation (1) and equation (2), we get

`"f"/((1 - "v"_"s"/"v")) = "f'"/((1 + "v"_0/"v"))`

`=> "vf'"/(("v + v"_0)) = "vf"/(("v - v"_s))`

Hence, the apparent frequency as seen by the observer is

`"f'" = (("v" + "v"_0)/("v" -"v"_"s"))"f"`   ....(3)

(b) Source and observer recede from each other:

It is noticed that the velocity of the source and the observer each point in opposite directions with respect to the case in (a) and hence, we substitute (v_s → – v_s) and (v₀ → – v₀) in equation (3), and therefore, the apparent frequency observed by the observer when the source and observer recede from each other is f’ = `(("v" - "v"_0)/("v" + "v"_"s"))"f"`

(c) Source chases the observer:

Only the observer’s velocity is oppositely directed when compared to case (a). Therefore, substituting (v₀ → – v₀) in equation (3), we get f’ = `(("v" - "v"_0)/("v" - "v"_"s"))"f"`

(d) Observer chases the source:

Only the source velocity is oppositely directed when compared to case (a). Therefore, substituting (v_s → – v_s) in equation (3), we get f’ = 

= `(("v" + "v"_0)/("v" + "v"_"s"))"f"`