## Tag Archives: wave equation

### Revamping Wave Mechanics

The Doppler ‘Mess’

Wave Motion – some basics

Demystifying Wave Motion

### The Doppler ‘Mess’

Most educated people would have heard about Doppler Effect as this is part of the science curriculum from secondary school level. A change in the observed frequency of a wave due to relative motion between the source and observer is called Doppler Effect.

We all experience this phenomenon in our everyday life. For example when a police siren approaches us, we hear a high pitched (high frequency) sound and as the siren passes away from us, we hear a low pitched sound even though the siren produces the sound at the same frequency throughout. Even more familiar is the scenario of an approaching train. Also one experiences the same effect by moving towards or away from a stationary sound source. This is true with any type of waves including water waves and light waves.

So different observers note a different frequency for the same wave depending upon their relative motion with respect to the source.

Imagine that a projectile is moving at 10meters/sec with respect to its source. An observer moving at 2meters/sec with respect to the source in the same direction, will measure the velocity of the projectile as only 8meters/sec. So while the ‘original’ velocity of projectile is 10meters/sec, the observer measures that as 8meters/sec.

Shall we call this velocity shift or change in the observed velocity as Doppler Effect or Einstein Effect? Obviously the velocity of any particle or projectile will be different to different observers depending upon their reference frames and we know that all reference frames are equally valid and there is nothing special about the reference frame of the source. Scientific folk knows this very well.

But what is not realised by the scientific folk is that like a particle’s velocity, a wave’s frequency is relative and is dependent upon the reference frame of the observer. Frequency by definition is the number of waves crossing a reference point per second or the number of waves received by a detector per second. So it is what an observer measures of a wave.

We don’t consider the velocity of a projectile with respect to its source as its actual velocity. Similarly if a source releases a wave at some particular frequency, that frequency doesn’t become its ‘actual’ frequency –the said frequency is only true in the reference frame of the source. The medium’s particles will experience a different frequency for the same wave if the source and medium are in relative motion. Some other observer may experience an even different frequency depending upon his relative motion. Just like the case with velocity, no one’s measurement of a wave’s frequency can be sworn upon as absolute to call others’ as apparent.

Not only are frequency and velocity frame dependent, same is also the case with energy because energy is dependent upon frequency and velocity.

### Disentangling the Doppler ‘Mess’

Like with so many things in physics, physicists have messed up with Doppler Effect to such an extent that disentangling the entire mess is not an easy task and demands a separate volume for itself.

We know that it is the relative motion between the source and the observer that results in Doppler shift. And we also know that motion is a relative thing, so whether it is the source or the observer who is moving, things should remain the same for the observer in both the scenarios. In other words the observer will not be able to tell whether he is moving towards the source or the source is moving towards him because from his perspective both scenarios are identical. Why I am stressing this point is that the implications are enormous- in case if the observer is able to tell who is actually moving by looking at a particular wave or projectile released by the source, then that obviously goes against the basic premise that motion is relative.

Imagine that a stationary source fires a projectile towards a stationary observer at 10meters/sec. The observer here obviously sees the projectile approaching him at velocity 10meters/sec. Now imagine the following two scenarios-

1) the source moving at 10meters/sec towards the observer as it fires the projectile

2) the observer moving towards the source at 10meters/sec as the source fires the projectile

From the observer’s perspective he is always stationary and hence he notes no difference in the above scenarios- according to him, the source is approaching him at 10meters/sec and the projectile at 20meters/sec. He sees no difference in what he experiences and he will not be able to tell who is actually moving.

But how come an observer experiences a wave differently depending upon who is moving? Physicists preach that when the observer moves towards the stationary source, he receives waves with same wavelength but at a faster velocity. But when it is the source that moves towards the stationary observer, he receives waves of shorter wavelength travelling at the same velocity as before. Moreover, apparently the observer experiences a different frequency shift in both the scenarios. So the same wave will be noted to have a different frequency, wavelength and velocity depending upon whether the observer himself moves towards the source or the source moves towards the observer.

And our physicists have devised different formulae to calculate the observed frequency, wavelength and velocity of a wave depending upon who is ‘actually’ moving.

http://hyperphysics.phy-astr.gsu.edu/hbase/sound/dopp.html

http://www.phys.uconn.edu/~gibson/Notes/Section6_3/Sec6_3.htm

http://physics.bu.edu/~redner/211-sp06/class19/class19_doppler.html

So motion is not relative! What an awful science we have been religiously listening to and carrying on in our minds. On one hand physicists preach that motion is relative and on the other hand they preach weird formulae which imply that motion is not relative.

Of course Nature is neither stupid nor weird (unlike our physicists’ understanding of Her) to contradict Her own basic rules. The reason for the apparent discrepancy is that our physicists have not been careful about the various reference frames while preaching the above- they mess up with frequency, wavelength and velocity of a wave under study by viewing each of them from different reference frames i.e. they talk about frequency from the source/observer’s frame, wavelength from medium’s perspective and velocity from which ever frame that makes them comfortable. Often they don’t specify from whose perspective something is being discussed and instead they toggle between source to medium to observer even without realising.

For example in the scenario of an observer moving towards a stationary source, when physicists say frequency increases but wavelength remains the same, they are talking about frequency in the observer’s frame but wavelength from the medium’s frame. The truth is that when an observer approaches a stationary source, not only does the frequency increase but also the wavelength decreases if one measures the same from the observer’s frame. But how to measure the wavelength from the observer’s frame?

Let us analyse that. The observer hears an increase in the sound frequency because his tympanic membrane or ear drum vibrates more rapidly as he receives more waves per second while he approaches the sound source. These vibrations from the ear drum get conducted across the middle ear via three tiny interconnected bony structures called ossicles and reach the inner ear and from there signals go to the brain in the form of nerve impulses.

Of course one need not bother too much about the anatomic details of middle ear or on how the signals get transmitted from the inner ear to the brain. The point of interest is that the vibrations from the ear drum also get transmitted to the air column in the middle ear. The sound waves that get transmitted in this air column vibrate at the same frequency as the ear drum (which is obviously higher than the frequency of source) and will have a shorter wavelength compared to the waves that reach the ear drum via the external air medium.

So in the reference frame of the observer (represented by the air column in his middle ear), not only does the frequency of the sound increase, but also its wavelength decreases. So obviously physicists are wrong when they say that wavelength doesn’t change in the scenario of a moving observer.

So as the observed frequency increases, the observed wavelength correspondingly decreases for the moving observer. That implies that the observed velocity must remain the same (because Velocity = Frequency x Wavelength). But then how come the observed velocity of the sound is more when the observer moves towards the sound source? Well, physicists have a totally wrong concept about wavelength so they can’t obviously be correct with the velocity formula that incorporates ‘their’ wavelength! (Much of our misunderstanding in wave motion comes from that of the wavelength. Things will surely become clear as we talk about the fundamentals of wave motion but until then we will imagine wavelength as a measure of spatial spread or dispersion of the energy pocket in the medium)

The truth is – the thing which should ‘sit’ by the side of the frequency in the above velocity formula should be amplitude and not wavelength. So it is the frequency and the amplitude which will decide the velocity of propagation of a wave. And as shown in the picture above, while there is an increase in the observed frequency, the amplitude remains the same (of course the particles will oscillate at a ‘higher level’ i.e. in an excited state and I will be discussing this in a separate section) and thus we can explain the ‘apparent’ increase in the observed velocity without making things contradict each other.

(Of course physicists almost never talk about amplitude when they preach Doppler Effect – http://www.physicsforums.com/showthread.php?t=509032. They only talk about change in frequency and change in wavelength and only occasionally about change in velocity).

To conclude for now, whether it is the source or the observer that moves, what the observer feels will remain the same i.e. the observed frequency, wavelength, velocity and amplitude will remain the same in either scenarios. And it makes sense because we believe that motion is relative. (But of course there is a catch here and I will discuss that elsewhere)

### Doppler Effect and EM waves

Physicists believe that in case of mechanical waves (sound waves and water waves), motion of the source doesn’t alter the observed velocity of the waves- that is whether a source is moving or stationary, the waves it produces reach the observer at the same velocity. And when it comes to the scenario of a moving observer and stationary source they teach that a wave’s velocity changes depending upon the observer’s motion. That means an observer sees the waves reaching him at a faster velocity when he moves towards the source and sees the same approaching him slower when he moves away from the source.

Not surprisingly, they don’t extend the same rules to the EM waves because doing so would destroy their wonderful but stupid theory of relativity which preaches that speed of light is constant irrespective of the motion of the source or the observer. I can understand their logic here- A major stupidity can’t be challenged or overthrown by a minor stupidity. Obviously physicists are only half stupid when they talk about mechanical waves but become totally stupid when they start talking about light and EM waves.

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