Finding f_L as Source Speed Approaches Speed of Sound

In summary, the conversation discusses the value that f_L approaches as the source's speed approaches the speed of sound moving to the right, with the listener being stationary. The equation f_L=f_s\left ( \frac{v+v_L}{v+v_s} \right ) is used, where v is the speed of sound, v_L is the velocity of the listener, and v_s is the velocity of the source. By taking the limit of v_s approaching infinity, it is determined that f_L approaches \frac{1}{2}f_s. The misunderstanding of the meaning of "speed of sound" is clarified.
  • #1
JJBladester
Gold Member
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2

Homework Statement



Imagine that the source is to the right of the listener, so that the positive reference direction (from the listener to the source) is in the [itex]+\hat{x}[/itex] direction. If the listener is stationary, what value does [itex]f_L[/itex] approach as the source's speed approaches the speed of sound moving to the right?


Homework Equations



[tex]f_L=f_s\left ( \frac{v+v_L}{v+v_s} \right )[/tex]

Where [itex]v[/itex] is the speed of the sound in the medium,
[itex]v_L[/itex] is the velocity of the listener, and
[itex]v_s[/itex] is the velocity of the source.

The Attempt at a Solution



I'm assuming that the speed of sound, being so large can be seen as [itex]v_s[/itex] approaching infinity.

I think the way to solve this problem is to take the following limit:

[tex]\lim_{v_s \to \infty }f_s\left ( \frac{v+v_L}{v+v_s} \right )[/tex]

The answer given is that [itex]f_L[/itex] approaches [itex]\frac{1}{2}f_s[/itex].

I'm not sure how to evaluate the limit to get the answer.
 
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  • #2
Got it... [itex]v_s \to v[/itex] so the limit is just a simple substitution and the answer makes sense. I misunderstood what "the speed of sound" meant.
 

1. What is f_L and how does it relate to the speed of sound?

f_L refers to the frequency of a sound wave as it approaches the speed of sound. As the source speed increases towards the speed of sound, the frequency of the wave also increases.

2. What is the significance of finding f_L?

Finding f_L is important because it helps us understand the behavior of sound waves as they approach the speed of sound. It also allows us to calculate the speed of the source based on the frequency of the wave.

3. How is f_L calculated?

f_L can be calculated using the formula f_L = f/(1-V/V_s), where f is the original frequency, V is the speed of the source, and V_s is the speed of sound.

4. What factors can affect the accuracy of finding f_L?

The accuracy of finding f_L can be affected by factors such as the temperature and humidity of the surrounding air, the shape and size of the source, and any obstacles or reflections in the environment.

5. Why is it important to find f_L in certain scientific experiments?

Finding f_L can be crucial in certain experiments, such as those involving supersonic or ultrasonic speeds, as it allows for precise measurements and calculations related to the speed of sound. It can also help in understanding the behavior of sound waves in different environments and conditions.

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