Doppler Shift Formula

A Doppler shift is a phenomenon of a change in frequency based on the observers point of view. The most common analogy of this is standing on the side of the road and listen to a passing car. As the car approaches, there is a definitive sound. As the car passes, the sound changes to a lower frequency. This is called a Doppler Shift. There are two types of Doppler shifts:

• Red-Shift or a shift of frequency to a lower wavelength (away from the observer)
• Blue-Shift or a shift of frequency to a higher wavelength (toward the observer)

Anything that emits wavelengths – light, radio, gamma rays, and the rest of the E-M Band – and changes frequency due to movement, a Doppler shift can be measured. This is the equation:

$\frac{\Delta&space;\lambda&space;}{\lambda&space;_{0}}=\frac{v}{c}$

$\Delta&space;\lambda&space;=wave\,length\,&space;shift$
$\lambda&space;_{0}=Wavelength\,&space;of\,&space;source\,&space;not\,&space;moving$
$v=velocity\,&space;of\,&space;source-line\,&space;of\,&space;site$
$C=Speed\,&space;of\,&space;light$

Question 1: A source and listener are moving towards each other with the speed of 54 km/hr. If the true frequency of sound emitted by the source is 500 Hz, calculate the observed frequency when both source and listener are moving towards each other.

Velocity of sound in air = 330 ms-1.

Solution:

Given: True frequency, f = 500 Hz,

Velocity of sound, vs = 54 km/hr = 54×1000360054×10003600 = 15 ms-1,

Source and observer moving towards each other

∴∴ Apparent frequency is given by f’ = v+vov−vsv+vov−vs f

= 330+15330−15330+15330−15 ×× 500

= 547.62 Hz.

Question 2: A fixed source emits sound of frequency 1000 Hz. What is the frequency as heard by a observer

(i) at rest

(ii) Moving towards the source at a constant speed of 20 ms-1 and

(iii) Moving away from the source at the same rate.

Solution:

Velocity of sound in air, v = 340 ms-1,

True frequency, f = 1000 Hz,

Velocity of observer, vo = 20 ms-1

(i) When both the source and observer are at rest, apparent frequency is same as true frequency

∴∴ Frequency as heard by listener at rest = f = 1000 Hz

(ii) Observer is moving towards the source

∴∴ Apparent frequency f1 = v+vovv+vov f

= 340+20340340+20340 ×× 1000

= 1059 Hz

(iii) Observer is moving away from the source

∴∴ Apparent frequency f2 = v−vovv−vov f

= 340−20340340−20340 ×× 1000

= 941 Hz.