Apparent
change in the wave frequency (or in the period of periodic events) due
to motion of wave source and/or observer is known as Doppler effect. In
the 17th century (well before Doppler effect was predicted), Roemer (Danish
astronomer) estimated the speed of light from the change in the revolution
period of one of Jupiter's moons. When the earth is approaching Jupiter,
the period is shortened, while receding, the period is elongated. The first
animation shows wave patterns (surfaces of constant phase) emitted by a
moving wave source. Wavelengths in front of the moving source are squeezed
(shortened) according to

and
those behind are elongated,

where
*c _{w}*
is the wave (sound) velocity and

Similar effect occurs when the observer is moving. For sound waves, the velocities of source and observer are well defined relative to air, the wave medium. For stationary sound source and moving observer, the sound speed appears to change (the wavelength remains the same). Therefore, the Doppler shifted frequency for a stationary sound source heard by a moving observer is given by

where *V _{o}* is the velocity
of the observer relative to air, + sign for the case of approaching and
– sign for the case of receding.

For electromagnetic waves in vacuum**, the
velocity with respect to the wave medium (vacuum) is ill defined**. In
Doppler effect of electromagnetic waves,
**only the relative velocity
between wave source and observer matters**,

where

is the relative velocity between the wave
source and observer normalized by the velocity of light *c*. The factor
in the numerator,

is due to intrinsic Doppler shift caused by time dilation effect.

Some
objects (e.g., supersonic planes) can move faster than sound waves. Animation
below shows wave fronts emitted by a supersonic object when the Mach number
is 2. No waves can exist outside the cone created by the object. When the
cone surface hits an observer, sudden arrival of a sharp pulse is detected
as shock wave. In vacuum, nothing can travel faster than light. However,
in material media (glass, plastic, etc.), electromagnetic waves propagate
in general at a speed slower than c. Therefore, if a relativistic charged
particle with a velocity close to *c *enters, say, a plastic block,
electromagnetic shock wave is emitted. This phenomenon is called Cerenkov
radiation and is in wide use for detecting energetic particles.

**Doppler
effect. Volocity of wave source = one half of the wave velocity. Wavelengths
in front of the source are squeezed while those behind are elongated. **

> **animate({[u*sin(t),u*cos(t),t=-Pi..Pi],
[(u-1)*sin(t)*Heaviside(u-1)+.5,(u-1)*cos(t)*Heaviside(u-1),t=-Pi..Pi],**
**[(u-2)*sin(t)*Heaviside(u-2)+1.,(u-2)*cos(t)*Heaviside(u-2),t=-Pi..Pi],**
**[(u-3)*sin(t)*Heaviside(u-3)+1.5,(u-3)*cos(t)*Heaviside(u-3),t=-Pi..Pi],**
**[(u-4)*sin(t)*Heaviside(u-4)+2.,(u-4)*cos(t)*Heaviside(u-4),t=-Pi..Pi],**
**[(u-5)*sin(t)*Heaviside(u-5)+2.5,(u-5)*cos(t)*Heaviside(u-5),t=-Pi..Pi],**
**[(u-6)*sin(t)*Heaviside(u-6)+3.,(u-6)*cos(t)*Heaviside(u-6),t=-Pi..Pi],**
**[(u-7)*sin(t)*Heaviside(u-7)+3.5,(u-7)*cos(t)*Heaviside(u-7),t=-Pi..Pi]},**
**u=0..8,view=[-8..8,-8..8],frames=40,color=red);**

**Velocity
of wave source = twice of the wave velocity. Formation of shock wave.**

> **animate({[u*sin(t),u*cos(t),t=-Pi..Pi],
[(u-1)*sin(t)*Heaviside(u-1)+2,(u-1)*cos(t)*Heaviside(u-1),t=-Pi..Pi],**
**[(u-2)*sin(t)*Heaviside(u-2)+4,(u-2)*cos(t)*Heaviside(u-2),t=-Pi..Pi],**
**[(u-3)*sin(t)*Heaviside(u-3)+6,(u-3)*cos(t)*Heaviside(u-3),t=-Pi..Pi],**
**[(u-4)*sin(t)*Heaviside(u-4)+8,(u-4)*cos(t)*Heaviside(u-4),t=-Pi..Pi],**
**[(u-5)*sin(t)*Heaviside(u-5)+10,(u-5)*cos(t)*Heaviside(u-5),t=-Pi..Pi],**
**[(u-6)*sin(t)*Heaviside(u-6)+12,(u-6)*cos(t)*Heaviside(u-6),t=-Pi..Pi],**
**[(u-7)*sin(t)*Heaviside(u-7)+14,(u-7)*cos(t)*Heaviside(u-7),t=-Pi..Pi]},**
**u=0..8,view=[-8..16,-8..8],frames=40,color=red);**

>

> **interface(plotdevice=inline):**
**with(plots):**
**implicitplot({x^2+y^2=4.5^2,(x-2)^2+y^2=3.5^2,(x-4)^2+y^2=2.5^2,**
**(x-6)^2+y^2=1.5^2,(x-8)^2+y^2=0.5^2,x+1.732*y=9,x-1.732*y=9},x=-5..9,y=-5..5,numpoints=1000);**

>