Teaching Doppler Effect with a passing noise source Ivan F. Costa and Alexandra Mocellin
[email protected] University of Brasilia, 70919-970, Brasilia, DF, Brazil Abstract. The noise pitch variation of a passing noise source allows a low cost experimental approach to calculate speed and, for the first time, distance. We adjusted the recorded noise pitch variation to the Doppler shift equation for sound. We did this by taking into account the frequency delay due to the sound source displacement and performing a Fast Fourier Transform (FFT) of the noise signal using free software. This experimental method was successfully applied to aircraft and automobiles. Keywords: Doppler Effect, low cost experiments, teaching physics. PACS: 07.05.-t and 42.60.+d
INTRODUCTION There are three variations in the noise of a machine when it passes us: the intensity increase and decrease in time, the binaural (or stereo) hearing effect due to the difference in sound arrival time in each ear, and pitch range [1]. The pitch, or frequency, of sound observed is not the same as that emitted by a moving source. The difference observed when a wave source and an observer are moving in relation to each other is called the Doppler Effect. Traditionally, the Doppler Effect for sound is introduced in high school and college physics courses. Students calculate the detected frequency for several scenarios, in which the source or the detector are moving. Of course, these scenarios are not always easy for students to grasp, so a daily experimental approach can help. The work described in this paper illustrates a new and efficient technique for studying the Doppler Effect quantitatively. This low cost experiment challenges the students to understand technological apparatus, like medical ultrasound imaging [2,3].
FIGURE 1. Diagram of the distances traveled by the airplane and its noise, denoted by v.(t-τ) and u.τ respectively. The vectors are the source velocity and its projection in the observer’s direction.
projection vcosθ of this velocity in the observer’s direction. So, when the receiver is at rest [5]
f =
CALCULATIONS Figure 1 shows the diagram of the distances involved in the problem. The minimum distance between the source and the observer is r. The sound takes a time τ from the source to the observer due to its speed u (345 m/s for air temperature of 23 oC [4]). Considering v the velocity of the source, the equation for the Doppler shift can be directly written by the
fo f ou = 1 + v cos θ / u u + v cos θ
(1)
where fo is the frequency emitted by the source. The total time interval, t, is defined as the measured time minus the time at the origin, which is the moment that the source is passing by the detector. From the figure we can write
cosθ =
v.(t − τ ) u.τ
(2)
The sound file can then be read by software [9] which performs spectrograms of time versus frequency using Fast Fourier Transform (FFT), as can be seen in Fig. 2 for the noise of an airplane.
then,
f =
f ou 2τ . (3) (u 2 − v 2 )τ + v 2t
Again from Figure 1
6000 5000
Hence:
(v 2 − u 2 )τ 2 − (2v 2t )τ + (r 2 + v 2t 2 ) = 0 . (5) For speed of the source less than the sound’s speed (v
