Eric Gisse says...

On Mar 23, 10:11=A0pm, Koobee Wublee wrote:

[...]

You've just confessed that you don't understand where Doppler shift
comes from, even in sound waves, despite your floundering attempts to
understand it using either Galilean or Lorentz transforms. This utter
incompetence on YOUR part you blame on others.

More cheap shots.

The bottom line is that using the same method the Galilean transform
predicts no Doppler shift even for sound waves.

Yes, double down on your stupidity.

He's wrong, of course.

Relativistic case: For simplicity, consider an electromagnetic
wave traveling in the +x direction.

Frame F:
E = A cos(kx - wt) e_y
B = A/c cos(kx - wt) e_z

where c = w/k

Under a Lorentz transform, in this special case,
x -- gamma (x' + v t')
t -- gamma (t' + v/c^2 x')
E_y -- gamma (E_y - v B_z)
B_z -- gamma (B_x - v E_y)

So in frame F':
E = gamma (A cos(kx - wt) - v/c A cos(kx - wt))
= gamma (1-v/c) A cos(kx-wt)
= gamma (1-v/c) A cos(k gamma (x'+vt') - w gamma (t'+v/c^2 x'))
= gamma (1-v/c) A cos(gamma(k-vw/c^2)x' - gamma (w - kv)t')

This has the same form as in frame F if we interpret:
A' = gamma (1-v/c) A = square-root((1-v/c)/(1+v/c)) A
k' = gamma (k-vw/c^2) = gamma (k-v/ck) = square-root((1-v/c)/(1+v/c)) k
w' = gamma (w-kv) = gamma (w-w/c v) = square-root((1-v/c)/(1+v/c))

So the relativistic Doppler shift for frequency is:

w'/w = square-root((1-v/c)/(1+v/c))

Non-relativistic case (sound): Again, for simplicity, consider
a sound wave traveling in the +x direction. Sound is a pressure
wave of the form:

P = A cos(kx - wt)

where in this case, w/k is the speed of sound.

Under a Galilean transform,
x -- x' + vt'
t -- t'
P -- P

(Pressure, being a scalar, doesn't change value under
a Galilean transform)

So in frame F':
P' = A cos(kx - wt)
= A cos(k(x'+vt') - wt')
= A cos(kx' - (w-kv) t')

This has the same form as in frame F if we interpret:
A' = A
k' = k
w' = w - kv = w - w/c v = w (1-v/c)

So the non-relativistic Doppler shift for frequency is:
w'/w = 1-v/c

--
Daryl McCullough
Ithaca, NY