Example 1. A car moving at a velocity 50 mi/hr~ 22 m/sec is passing a stationary sound source that emits
a frequency 500 Hz (Figure 8.4). The closest distance between source and the car is 20 m. Find the
apparent frequency heard by the driver as a function of the distance x. Assume cs = 340 m/sec.
Figure 8.4. Example 1. Observer moving along a line not intersecting the source.
In this case, the velocity of the car is not directed directly toward the sound source, and we have
to find the component of the velocity vector that is directed toward the source (Figure 8.5). It is given by
Figure 8.5. The velocity complement toward (or away from) the source is responsible for the Doppler
shift.
u0 cos   u0
x
x  20
2
2
 22
x
x  202
2
m sec,
where we have converted the velocity of 50 mi/hr into 22 m/sec. Then the apparent (Doppler-shifted)
frequency is
cs  u0 cos
cs
  22 

x
 500  1  
 2
 Hz.
2
  340  x  20 
v  x   v0
You should plot this as a function of x for 100  x  100 m. At x = 0, the car is moving purely
perpendicular to the wave and at the instant when the car passes this point, the driver hears the true
frequency, 500 Hz.