DAY 18

Summary of Topics Covered in Today’s Lecture

The Doppler Effect

See the Doppler Effect Demo program used in class (click here).

Also, for a nice Doppler Effect video clip, click here.

Shock Waves

See the Doppler Effect Demo program used in class (click here).

Also, for a nice video of a plane traveling faster than the speed of sound, click here. Note that you can both see the shock wave (where it intersects the water) and hear the shock wave (the sonic boom).

This video (click here) has a nice discussion of some interesting effects caused by shock waves in air. However, not all shock wave phenomena are so exotic. A boat wake is also a shock wave.

Example Problem #1

A source at rest gives off waves with wavelength l. The waves travel with speed v.

a) If the source moves with speed v_s, what will be the wavelength of waves ahead of the source? Derive an equation for this wavelength, which we will call l_ahead.

b) If the source moves at 25% of the wave speed, how much will the waves ahead of the source be foreshortened?

Solution:

Let’s say the source is moving to the right at speed v_s. When at A, the source emits a wave crest. The wave moves outward from A at speed v. The source moves to the right at v_s, and by the time it gives off the next wave crest it is at B.

By the time the source is at B, the wave it gave off when it was at A has expanded out to a circle of radius d_wave. d_wave is the distance the wave moved. The distance the source moved is d_s, the distance from A to B.

The time between wave crests is the period of the wave (T). This is the time that has elapsed. Since

distance = speed x time

the distance the wave traveled is

d_wave = v T

and the distance the source traveled is

d_s = v_s T

The wavelength ahead of the source is just the distance from B, where the 2^nd wave crest is being emitted, to C, where the first wave crest is located. We’ll call this wavelength l_ahead.

l_ahead = distance BC

= distance AC – distance AB

= d_wave - d_s

= v T – v_s T

l_ahead = T (v – v_s)

However, since

v = f l

= (l/T) l

then T = l/v. So now the yellow hilited equation above becomes

l_ahead = (l/v)(v – v_s)

l_ahead = l(1 – v_s/v)

That takes care of part a. Now let’s do part b.

v_s = 25% v

or

v_s = .25 v

l_ahead = l (1 – v_s/v)

= l (1 – 0.25 v/v)

= l (1 – 0.25) = .75 l

So the waves ahead of the source will be only 75% of the length of the waves that the source gives off at rest.