Doppler Effect

Strings (SiPjAjk) = S7P4A44     Base Sequence = 12735     String Sequence = 12735 - 4 - 44

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Gaussian Lens Equation
Math (a) What is the relationship between the frequency of the sound from the siren of the futuristic ambulance illustrated above and the frequency heard by an observer if both ambulance and observer are approaching each other at velocities va and vo respectively?
(b) If ambulance and observer are moving in opposite directions at velocities va and vo respectively?

The strings: S7P4A44 (Motion - Oscillatory Motion).

The math:
Pj Problem of Interest is of type motion (oscilatory). When the source of sound and the receiver (hearer) of sound are in relative motion, the frequency of the sound heard by the receiver is different from the frequency of the sound emitted by the source. This change in freqency is referred to as the Doppler effect.
(a) The relationship between the frequencies when source of sound and receiver of sound are approaching each other is as follows:
fr = [(c + vr)/(c - vs)]fs
Where fr is frequency of receiver of sound
fs is frequency of sound emitted from sound source
c is velocity of sound (331.45 m/sec)
vs is velocity of sound source
vr is velocity of receiver of sound

So, for the ambulance and observer approaching each other we have:
fr = [(331.45 + vo)/(331.45 - va)]fs
Sound waves pile up, therefore shorter wavelengths and higher frequency sound reaches the receiver.

(b) The relationship between the frequencies when source of sound and receiver of sound are moving awat from each other:

fr = [(c - vr)/(c + vs)]fs
The signs of the respective velocities are negative.

So, for the ambulance and observer moving away from each other we have:
fr = [(331.45 - vo)/(331.45 + va)]fs
Sound waves spread out, therefore longer wavelengths and lower frequency sound reaches the receiver. The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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