Point Source Radiating in a Moving Medium
Doppler shift for the general case of a transmitter and a receiver each moving with arbitrary velocities relative to the medium, is analyzed in this section. This is closely related to the Michelson-Morley experiment, which is a cornerstone of the theory of relativity. Yet, in all books I have seen, the behavior of a wave in a moving medium is essentially simply postulated. The behavior is intuitively reasonable, but is it really valid? Here we consider the equivalent experiment for sound, and show that the basic equations (17) and (19) of fluid mechanics are in fact satisfied far from the source (of course light, if ether existed, might obey a different set of equations).
The derivation for the general case is considerably more complex than the simple case where everything moves along one coordinate axis. The approach taken below is the least messy formulation I have tried. Vectors are shown in boldface. For some browsers this is hard to distinguish, so arrows are placed above vectors, and carets above unit vectors, in the numbered equations (I don't know how to do this within the body of the text).
Postulated Behavior of Sound in a Moving Medium
The medium is assumed to be homogenous, isotropic, and stationary with respect to some coordinate system. However rather than moving the source through the medium, the origin of the primary coordinate system is defined to be located at the source. In this system the source is stationary, and the system can be oriented, without loss of generality, such that the medium moves with velocity vo in the positive z-direction.
There are two postulates for the behavior of sound wave propagation: (1) any equiphase surface is spherical, with radius re, which expands at a constant velocity c; and (2) the entire surface also moves in the positive z-direction with velocity vo (i.e. it flows with the medium).
Let the receiver be located at
Where ax, ay, az, are Cartesian unit vectors, and r, θ, ɸ are the standard spherical coordinates. The source launches a wave surface of zero radius at t=0, and at time t the surface reaches the receiver. We define a secondary coordinate system whose origin is at the center of the surface, moving with the medium; in this system the receiver is at a radius re= ct. The geometry is shown here [24kb]. The primary coordinate system is used globally for all equiphase surfaces, and for the receiver velocity, whereas the secondary system is used for one surface launched at a particular instant in time. The solution for the equivalent coordinates is then found to be xe=x, ye=y, and
The equivalent radius is
At the point r the equiphase surface is moving with velocity
Equation (M4) shows that each point on the equiphase surface propagates along straight "rays" emanating from r=0, but with different velocities along each ray. A notable difference between the case of a stationary and moving medium is that the rays are always perpendicular to equiphase surfaces in the former case, but not the later case. This is evident in a plot of the phase contours and rays [11.3 kb] for the case of vo=c/2. In this global view, a given point on a surface moves, at a speed greater or less than c, along a "ray." When the point arrives at the receiver, the launch-point has moved along the z-axis, and it is also true that the point acts as if it came along the radius re with velocity c. The first viewpoint is that of an observer in the primary coordinate system, and is valid for all equiphase surfaces. The second viewpoint is only valid for one specific surface, since the origin for re varies from surface to surface.
The dot product between a unit vector in the ray direction and the normal to the equiphase surface is
The value of the dot product is 1.0 along the z-axis, or everywhere if vo=0.
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Michelson-Morley Experiment for Sound
The classic Michelson-Morley experiment compares the phase of two waves: (1) a wave that travels from the origin a distance d along the z-axis and then back to the origin, and (2) a wave that travels from the origin a distance d along the x-axis and then back to the origin. By the postulated behavior, phase in radians is given by
Where ω is the source frequency in radians per second. Along the z-axis
The plus and minus signs applying to propagation in the plus and minus z-directions respectively. Along the x-axis
Combining equations (M6-M8), after a round trip in the two perpendicular directions, when the phase of the x-leg is subtracted from the z-leg the difference is
Where the approximate value is accurate as long as vo << c. Putting in some numbers, for the real Michelson-Morley experiment (2nd try in 1887), d=11 meters (using on the order of 10 reflections), ω=3.24x1015, and vo=3x104. The resulting phase shift is 1.2 radians, or 68.7 degrees. This amount of phase shift would be easily detectable, if present, which of course it wasn't.
An equivalent experiment for sound at a frequency of 1000 Hz would use the equivalent distance d in wavelengths, and the same ratio of vo to the propagation velocity. This equates to a distance d = 6,509 kilometers, and vo =.0344 meters per second (0.12 kilometers per hour) - a very slow speed, and a very long distance.
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The frequency heard by an observer is equal to the dot product of the relative velocity of the equiphase surface towards the observer, and the gradient of the phase. The latter equals ω/c times the gradient of re, which is
For a stationary observer, the relative velocity is equal to the equiphase surface velocity. The dot product of equations (M4) and (M10) equals c, so a stationary observer hears the source frequency ω, unchanged. Since the relative velocity between the source and observer is zero, this must be true, and this simply verifies the equations for this case.
For an observer moving with velocity vd, the dot product of and the phase gradient produces a change in frequency given by
Along the z-axis this simplifies to the equation commonly given in textbooks
Where the upper and lower signs apply to the positive and negative parts of the z-axis, respectively. Along the x-axis
Where again the upper and lower signs apply to the positive and negative parts of the x-axis, respectively. I have seen one physics textbook where the equation for Doppler shift is incorrect, in that along the x-axis it always shows no shift for an observer moving in the z-direction. The error was in effect due to incorrectly assuming that the normal to the equiphase surface was equal to ar rather than are. The unit vectors are the same if vo=0, but not otherwise.
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Are the Equations of Fluid Mechanics Satisfied?
When the acoustic equations were derived from equations (17) and (19) for a stationary medium, the last terms on the right-hand side of equations (17) and (19) were completely neglected. For the present case, the constant velocity voaz of the medium makes these previously neglected terms significant. However we will still make the same approximation as before, and neglect the part related to the variable velocity term. In other words, if we write u=us+voaz, us will be neglected whenever it appears in addition to voaz. In all derivatives, of course, voaz disappears, and us is not neglected. The approximate forms of equations (17) and (19) are then
Case I: Plane Wave
As an initial check, consider a plane wave propagating in a medium moving with velocity voaz. Consistent with the postulated behavior, consider a plane wave
It is fairly easy to verify that this wave does satisfy equations (17) and (19), with the assumed approximations.
Case II: Point Source
A point source radiates a spherical wave in a stationary medium, given by equations (45) and (46), and a reasonable guess is that replacing r with re in these equations might represent the equivalent wave in a moving medium. Here we have another situation where a seemingly simple change significantly complicates the math. The proof of validity is pretty messy. If my algebra is correct - a big if - these equations do not in fact satisfy equations (M14) in general. However the primary interest here is in regard to the Michelson-Morley experiment. Each leg of the experiment has a path length on the order of millions of wavelengths. Therefore if the behavior is correct for large values of r/λ, that should suffice. So we consider, for r>>λ,
The required derivatives are found to be
When these values are plugged in, equations (M14) are satisfied. So at least for sound, the postulated behavior of a wave in a moving medium is valid at large distances from the source.
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Note: this section was originally posted September 1999, and was revised April 2006. The equations were not changed, but the text was hopefully clarified, and the geometry figure was added. I also programmed the Doppler equations in Matlab, and checked them against a different formulation of the problem based on the derivative of the transit time along a ray with respect to time. I will send the program to anyone who wants it.