Siren frequency heard by observer when an ambulance passes

[songoku]

Homework Statement

An ambulance siren emits a sound with a single frequency f. The ambulance travels towards, passes close to, and then travels away from a stationary observer. Which statement describes the frequency of the sound detected by the observer as the ambulance passes the observer?
A. equal to f and decreasing
B. equal to f and increasing
C. greater than f and constant
D. less than f and constant

Relevant Equations

Doppler Effect

I am not sure I understand the question.

Based on "The ambulance travels towards, passes close to, and then travels away from a stationary observer", I would answer greater than f then less than f.

If based on "as the ambulance passes the observer", I would say less than f and constant (option D)

But the answer is (A).

What is my mistake?

Thanks

 

[kuruman]

What is my mistake?

Your mistake is that you probably did this in your head. Write down the relevant equation instead of just "Doppler effect", which is not even an equation, and apply it to the given situation. I am sure you will find your mistake.

 

[Orodruin]

I would rather say it is an issue of the problem being possible to interpret in two different ways:

1. The ambulance passing “close to” the observer means that we can use the approximation that the ambulance passes through the same location as the observer to good accuracy. The frequency will have to be given infinitesimally after passage to be well defined.

2. The approximation cannot be made and you have to take the angle between direction of motion and direction to the ambulance into account.

The listed answer is compatible with 2 whereas your response is compatible with 1.

 

[kuruman]

1. The ambulance passing “close to” the observer means that we can use the approximation that the ambulance passes through the same location as the observer to good accuracy.

My interpretation of "close to" is "close but not in such a manner as to run over the observer." There is a distance of closest approach. We are to answer the question when the ambulance is exactly at that distance.

 

[Orodruin]

My interpretation of "close to" is "close but not in such a manner as to run over the observer." There is a distance of closest approach. We are to answer the question when the ambulance is exactly at that distance.

Which is:

2. The approximation cannot be made and you have to take the angle between direction of motion and direction to the ambulance into account.

1 is not necessarily about running the observer over, but about the passage being so fast that the approximation of a step function is a good approximation to the frequency change during the passage. What is a "good approximation" is of course always up for discussion and a matter of observational accuracy.

 

[kuruman]

I think the questions are "at the point of closest approach, (a) what is the value of the function and (b) what is the sign of the slope?"

I have observed what is described with a train blowing its siren while I being stopped at a railroad crossing. It's one continuous sound with changing frequency and intensity. The longitudinal component of the train's velocity relative to me is a continuous function of the siren's radial distance from me and goes through zero at the point of closest approach.

 

[pbuk]

Based on "The ambulance travels towards, passes close to, and then travels away from a stationary observer", I would answer greater than f then less than f.

But the question is not "Which statement describes the frequency of the sound detected by the observer as the ambulance travels towards, passes close to, and then travels away from a stationary observer" so that is the not an answer to the question that is asked.

If based on "as the ambulance passes the observer", I would say less than f and constant (option D)

Then you would be wrong. Before the ambulance passes the observer it is travelling towards them and so the observed frequency is greater than f. After the ambulance passes the observer it is travelling away from them and so the observed frequency is less than f. At the instant that the ambulance passes the observer it is neither travelling towards them nor away from them and so the observed frequency is equal to f and decreasing.

 

[pbuk]

1 is not necessarily about running the observer over, but about the passage being so fast that the approximation of a step function is a good approximation to the frequency change during the passage.

Maybe but both the value and the gradient of an unqualified step function are undefined at the point of discontinuity and "undefined" is not among the answers.

 

[Orodruin]

Maybe but both the value and the gradient of an unqualified step function are undefined at the point of discontinuity and "undefined" is not among the answers.

Hence why I said:

The frequency will have to be given infinitesimally after passage to be well defined.

I am not arguing that it is the intended interpretation. I am arguing that it is a possible interpretation given the usual fuzziness with implicit assumptions of many physics teachers.

 

[haruspex]

1 is not necessarily about running the observer over, but about the passage being so fast that the approximation of a step function is a good approximation to the frequency change during the passage.

As ever, the right way to treat an idealisation (zero mass, instantaneous impact, inextensible string…) is as the limit of realistic scenarios. The idealisation is valid if the limit (which may involve several parameters varying independently) always exists and is constant.
In the present case, we have the proximity, s, the velocity, v, and the raw frequency, f.
For any non-infinite/infinitesimal combo of those, the answer is A.