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This answer suggests that when there is zero radial velocity… there will be no Doppler shift but it's not likely to be intended as precise and absolute.

I remember reading about something like this in a post here but I can't find it. While to first order there is no frequency shift when moving perpendicular (we wouldn't expect there to be be due to symmetry i.e. which way? Up or down?) I think there is a second order effect which is a Doppler shift associated with the small change in angle due to astronomical aberration.

**Question:** Is this a thing? If so, what would be an expression for the frequency shift and what is it called? If the magnitude of each effect is $|v/c|$ is it always a red shift and simply

$$Delta f/f = -frac{v^2}{c^2}?$$

Since an object moving perpendicular to a given "line of sight" has a constantly changing **range** , there is a Doppler shift, blue when approaching and red when leaving.

The shift drops to zero at the point of crossing the line of sight because at that instant the radial speed is zero, as you suggested. So, the general magnitude is calculated using trigonometry: the range is effectively the hypotenuse and the cosine leg is the distance to the crossing point. The sine leg is the distance along the object's line of travel between the object and the crossing point. Differentiate to get the delta(range)/delta(time) .

Yes, there is a transverse relativistic Doppler shift. You can think of it as being caused by time dilation. https://en.m.wikipedia.org/wiki/Relativistic_Doppler_effect

There can be a redshift or a blueshift depending on when, where and who does the measurement.

e.g. a receiver with a source going around it in a circular orbit. The receiver sees a lower frequency (redshift), by a factor $gamma = (1 -v^2/c^2)^{-0.5}$. On the other hand, a receiver orbiting the source would receive a blueshifted signal by the same factor. When $v ll c$ then $$gamma simeq 1 + frac{v^2}{2c^2} ,$$ so whilst the standard (longitudinal) Doppler shift is of order $v/c$, the transverse Dopper shift is of order $v^2/c^2$.

NB: This scenario is chosen so that the relative motion of the source and receiver are perpendicular to the line between them. All other scenarios are complicated by the usual Doppler shift you would see because there is a component of the velocity along the line joining the source and receiver.

## Doppler shift

When a body that is emitting radiation has a non-zero radial velocity relative to an observer, the wavelength of the emission will be shortened or lengthened, depending upon whether the body is moving towards or away from an observer.

**Doppler Shift**

Red and blue shifts

Light from moving objects will appear to have different wavelengths depending on the relative motion of the source and the observer.

A **Doppler shift** is a phenomenon of a change in frequency based on the observers point of view. The most common analogy of this is standing on the side of the road and listen to a passing car. As the car approaches, there is a definitive sound. As the car passes, the sound changes to a lower frequency.

. This effect can occur with both sound and light, because both sound and light demonstrate wave-like behavior.

A shift in an object's spectrum due to a change in the wavelength of light that occurs when an object is moving toward or away from Earth.

E .

A change in the wavelength of radiation received from a source because of its motion along the line of sight.

The change in wavelength of a wave motion arising from the motion of the emitting object and/or The observer along the line of sight. The wavelength is increased when the relative motion is away, and decreased when the relative motion is towards each other.

DOUBLE STAR .

of hot coronal lines in a moss area of an active region A115

N. Dadashi, L. Teriaca, D. Tripathi, S. K. Solanki and T. Wiegelmann

DOI: .

(a) The blueshift or redshift produced by an object's motion toward or away from us. If a star moves toward us, its light waves get compressed and its spectrum is blueshifted if a star moves away from us, its light waves get stretched and its spectrum is redshifted.

(or Doppler Effect) is an increase or decrease in wavelength as the object emitting the wave moves relative to the observer.

This is used in Cosmology & Spectroscopy (see image below).

★ Double star Two stars that appear close together in the sky. Optical doubles are chance alignments of the stars as seen from Earth, while the stars in a binary or multiple system are actually linked by mutual gravity.

- This formula was originally introduced when light was discussed, but now we're looking at it again. Basically it shows how much light is effected by velocity and allows us (astronomers) to determine velocities based upon the measurable effects.

Formula: v = c x / where: .

-- UCLA

Cepheid Variable Stars and Distance Determination - Australia Telescope Outreach and Ed.

Measuring distances to stars

The Cosmic Distance Scale and Standard Candles - UTK

Cosmic Distance Ladder -- Wiki

Andromeda/Milky Way collision - Hayden Planetarium .

studies of galaxy clusters by Fritz Zwicky in 1937 found that most galaxies were moving much faster than seemed to be possible from what was known about the mass of the cluster.

from the Big Bang

Science Fiction scenario for a Flight to Mars

Motion of the Moon .

- (n.)

A change in frequency resulting from relative motion along the line between the transmitter and the receiver. If the source and the receiver are approaching each other, the frequency received is higher than the frequency transmitted by a factor, depending on the actual relative velocity.

24.6 Evidence for Black Holes, 27.1 Quasars

Drake30.4 The Search for Extraterrestrial Intelligence

Drake equation30.4 The Search for Extraterrestrial Intelligence .

in a star's light spectrum often indicates the presence of planets. Unlike previous equipment, which frequently missed some of that light, the new system uses a cluster of optical fibres to gather all the starlight, boosting efficiency and doubling the Doppler precision.

of spectral features toward longer wavelengths, indicating recession of the source.

redshift of galaxies

The shift toward longer wavelengths in light of distant galaxies, due to their recession from the solar system. It increases with galaxies' distances.

: This method also relies on the fact that the planet and star are both orbiting a shared center of mass. If the orbit is edge-on, the star will move towards us and then away from us in its tiny orbit.

s that they focused on were the ones most sensitive to magnetic activity.

gives us the velocity of a star toward or away from us, i.e., the radial motion. Motion perpendicular to our line of sight is known as the proper motion of a star. The proper motion can be determined by careful measurement of stellar positions over a long period of time.

Also known to most as the Doppler effect, the

explains the phenomenon of the change in frequency of a wave in relation to an observer. This can be observed when an ambulance drives past you and the volume of the siren doesn't quite match with the proximity of the ambulance to you.

ing of its light, Andromeda is speeding toward us at 68 miles per second (110 kilometers per second). Compare this to the light from Andromeda, which is moving toward us at 186,000 miles per second (300,000 km/s).

or Doppler effect: The change in wavelength due to the relative motion of source and receiver. Things moving toward you have their wavelengths shortened. Things moving away have their wavelengths lengthened.

. A change in the perceived frequency of a radiated signal caused by motion of the source relative to the observer. dose rate. The rate at which radiation energy is absorbed in living tissue, expressed in centisieverts per unit time. DSD. See dark surge on the disk. DSF.

Refers to the apparent shift in spectral lines If a galaxy is moving towards our Galaxy, or away from it, the light we see coming from that galaxy appears different from what it would be if the galaxies were 'standing still'.

: The change in frequency of a wave (light, sound, etc.) Due to the relative motion of source and receiver. Things moving toward you have their wavelengths shortened. Things moving away have their emitted wavelengths lengthened.

DORSUM: A ridge.

is most familiar from the sounds vehicles make. An approaching train, for example, makes a higher-pitched sound than one moving away from you. The sound waves are "pushed together" in the first case, and "stretched apart" in the second.

- the change in frequency (or wavelength) from a source due to the relative motion of the source and observer Galaxy - a gravitationally bound grouping of stars, gas, and dust that is physically isolated in space General relativity - a theory which argues that everything, .

On the other hand, if the two are moving apart, the observed wavelength appears longer that the wavelength that would be measured at rest. The

or Doppler Effect is a recognition that the change of wavelength Ø&lambda .

is a simple shift in frequency due to the radial velocity difference vr between the source and observer, given by Dl = (vr / c) l0, where l0 is the original wavelength.

. In the case of sound, or any other wave motion where a real medium of propagation exists (excepting, therefore, light and other electromagnetic radiations) one must distinguish two principal cases: If the source is in motion with speed v relative to a medium which propagates the waves in .

We know that Sigma Aql is double because two stars appear in the spectrum (each showing their

s) and because every 1.95026 days they eclipse each other, the main eclipse yielding a dip of about 0.2 magnitudes.

This was accomplished by measuring the

in the S-band tracking signal as it reaches Earth, which can be converted to spacecraft accelerations.

, we know that the lines from one of the stars will be blueshifted while the lines from the other star are redshifted. As the two stars orbit each other, each set of lines will appear to shift back and forth.

in the spectral lines depend on the relative speed? How does the direction of shift of the spectral lines depend on the direction of motion?

(Sec. 24.5) Up to now we have explained the redshift of galaxies as a

, a consequence of their motion relative to us. However, we have just argued that the galaxies are not in fact moving with respect to the universe, in which case the Doppler interpretation is incorrect.

) is an increase or decrease in the wavelength of the radiation emitted by an object, as observed from Earth, as the object moves relative to the observer.

In the late 1980s, a search using multiple, precise

measurements for weak gravitational perturbations by companion objects as small as 20 times Jupiter's mass that are located within 10 AUs of Ross 248 was negative (Butler and Benitz, 1989).

s, these variations imply orbital velocities on the order of 1/1000 the speed of light.

At some point the effects of special relativity come into play, and the light from a galaxy is

lowers the rate at which photons from a galaxy arrive at Earth.

"By measuring the changes in the

, we can use Chandra to pinpoint where the radiation is coming from on these stars and it turns out it's not where many scientists would have expected it," said SAO's Andrea Dupree.

is readily detected in astronomical bodies by the shifts in the wavelengths of spectrum lines. An approaching body has its spectrum lines shifted towards longer wavelengths, in the vernacular of astronomy 'towards the blue' (a 'blue shift', no matter what the lines' actual colours).

Supergranules are much larger versions of granules (about 35,000 km across) but are best seen in measurements of the "

" where light from material moving toward us is shifted to the blue while light from material moving away from us is shifted to the red.

of radio signals sent back to Earth will allow precise determination of changes in the orbit, which will allow for a model of the Mars gravity field. As the spacecraft passes over the poles on each orbit, radio signals pass through the Martian atmosphere on their way to Earth.

Galactic rotation curves plot a galaxy's circular velocity (which can be measured using the

of HI regions of the trailing and leading sides as viewed from the Earth) vs. the distance from the center of rotation.

At its distance of nearly 37 light years, this motion, when combined with its motion along our line of sight measured spectroscopically using the

, yields a space velocity of about 76 miles per second with respect to our Sun.

Vesto Slipher tried to measure the

of light from Venus, but found he could not detect any rotation. He surmised the planet must have a much longer rotation period than had previously been thought.[124] Later work in the 1950s showed the rotation was retrograde.

Pulsars make finding exoplanets quite easy, because even small orbiting bodies cause a detectable

in the pulses. The smallest object detected outside the solar system is a Moon-sized object orbiting a pulsar.

Neutron stars that are not pulsars are very hard to detect.

Black holes .

However, if that star is hurtling away from us, all those absorption lines undergo a

and move toward the red part of the rainbow. This is what we call a redshift.

Spectroscopic observations in the early 20th century also gave the first clues about the Venusian rotation. Vesto Slipher tried to measure the

of light from Venus. After finding that he could not detect any rotation, he surmised the planet must have a very long rotation period.

The star and its planet orbit around each other. The planet moves in a wide orbit, while the star just appears to wobble slightly. By measuring the

of the light coming from the star, scientists can detect the tiny motion caused by the planet. Most of the distant planets were discovered this way.

This is the measurement of the quantity of light from the galaxies that was stretched out due to their motion. Using the

he discovered that the galaxies that were around the Milky Way galaxy were traveling at very fast speed and moving away from us.

Doppler Weather Radar: Similar principle, bounce microwave radar signals of known wavelength off of clouds, measure the wavelength reflected back. The

and its sign (blue or red) gives the speed and direction of the clouds.

Spectroscopic Binary - A pair of stars whose binary nature can be detected by observing the periodic

s of their spectral lines as they move about one another

Spectroscopy - The recording and analysis of spectra

Spicule - A hot jet of gas moving outward through the Sun's chromosphere .

For example, at the limits of HARPS' sensitivity it can detect a

in the rotation of a star of just four kilometres per hour, which would be caused by the gravity of a two Earth-mass planet tugging on the star each orbit.

transverse motion Motion perpendicular to a particular line of sight, which does not result in

navigation satellite, artificial satellite designed expressly to aid the navigation of sea and air traffic. Early navigation satellites, from the Transit series launched in 1960 to the U.S. navy's Navigation Satellite System, relied on the

Zeta Leporis has the stellar classification of A2 IV-V(n). The (n) indicates that the absorption lines in the star's spectrum look nebulous because the star is a rapid spinner, which causes the absorption lines to broaden as a result of the

. The star has a rotational velocity of 245 km/s.

On 1/17/96 Geoffrey Marcy and Paul Butler announced the discovery of planets orbiting the stars 70 Virginis and 47 Ursae Majoris. 70 Vir is a G5V (main sequence) star about 78 light-years from Earth 47 UMa is a G0V star about 44 light-years away. These were discovered using the same

For instance, let's say you wanted to measure velocities of expansion of a planetary nebula, which are typically about 10 kilometers per second, using lines in the red part of the optical spectrum (about 6500 Angstroms). The equation for

s says you would want to make sure your spectrograph can make .

As the stars orbit each other, the light they emit shifts slightly blueward or redward depending on whether they're moving toward or away from us, respectively. These so-called

s are what astronomers picked up in 1890, suggesting that Spica A had a companion.

Binary stars produce shifting spectra because of their orbital motion, which is alternately redshifted and blueshifted. (Binary star motions are called

s.) Much of our knowledge of astronomy, for example, the masses of stars and distances to galaxies, .

The spectrum of fhis star shows the typical

of binaries. The two stars revolve each other each 5.6 days. This is of course not an amateur object. Some more information can be found in an article of the observatory of the Jagiellonian University in Krakow.

## Answers and Replies

I think Mathman is correct that stars are not generally spiralling inwards in a galaxy. However your post prompts another similar thought. We are situated at somewhere mid way between the centre of the Milky Way and the perimiter. Should there not be a detectable blue shift from stars further out towards the perimeter than us and a corresponding red shift from stars nearer the centre simply due to their location in the gravitational well of the galaxy as a whole? I imagine this effect would be very slight and difficult to detect due to the individual motions of stars in orbit around the galaxy centre.

Does anyone know if the long axis of the individual eliptical orbits around the galaxy are generally parallel or randomly orientated?

Spiraling toward the center was intended to be general. Each object will have an orbit that most probably would be ellipical but the gravititational fields that it would encounter from other near objects in non-parallel orbits would cause a "bumpy" ride.

For an object to be in orbit requires a velocity with a radial orientated vector that retains it in orbit. It's this radial component that I'm interested in and how it affects the shift. I imagined that it would be only slight but it would appear that a red shift would always be larger than a blue shift due to this radial velocity from any reference frame except parallel to a central axis of the galaxy. This appears to be what is happening and I've been unable to find a reference that considers this vector or attempts to measure it.

The red and blue shift in the 21cm HI radio line is routinely used to measure the rotation of our and other galaxies. So yes, this effect is known and utilised. However, the magnitude of this shift in frequency is relatively small, so for distant galaxies the cosmological redshift is much greater.

When taking the spectrum of a distant galaxy to determine its redshift, the light from the whole galaxy is considered. Indeed this does mean that some of that light is a little more redshifted and some a little more blueshifted due to the rotation of the galaxy. The effect of this, when the light is considered together, is to smear out each line in the spectrum a little bit, since the redshift is not all exactly the same. As I say though, this effect is relatively small and this smearing not a huge problem. It doesn't alter the centre of the peak in frequency and hence doesn't affect out measurement of redshift.

I understand the reason why we can't see the center of our galaxy. But the effect is not limited to our galaxy.

There simply does not appear to be anyone who has access to a large telescope that has scanned across a galaxy to verify experimentally what you say. If there is I haven't found a reference.

Related to my search there is a question that if space is expanding and galaxies are moving apart, how could they collide.

Have a look http://hubblesite.org/gallery/album/galaxy_collection/" for many beautiful pictures of galaxies taken with Hubble. As you will see, the centre is almost always the brightest part, except in cases where a disk galaxy is seen edge on.

Mapping the rotation curves of galaxies (i.e. scanning across a galaxy measuring the relative red and blueshift) has been done routinely for about 70 years or so. A quick search over at ADS yielded something over 80 thousand papers. http://adsabs.harvard.edu/cgi-bin/n. xt_wgt=YES&ttl_sco=YES&txt_sco=YES&version=1" is the search result.

Just for kicks, have a look at http://adsabs.harvard.edu/abs/1914LowOB. 2. 66S" paper from 1914 that describes mapping the rotation, using spectroscopy (i.e. to look for the doppler shift), of the "Virgo Nebula', now known to be a galaxy. When this measurement was made we didn't even know there were other galaxies than our own.

As for your last question, space isn't expanding in the literal sense of the words. It's perfectly acceptable to think of the expanding universe as galaxies flying apart, just remember that everything is moving away from everything else, not from any central point. Indeed, observations show that galaxy mergers were much more common in the past than today, since the Universe used to be a lot more crowded. In any case, the Universe is not completely smooth, and there are many regions, such as clusters of galaxies, where there is no expansion at all. The local overdensity of gravity has captured the galaxies such that they are now a bound system. Within these clusters, galaxies fly around roughly orbiting the centre of the cluster. Sometimes they collide with other galaxies in the same cluster.

## Is there still a red shift when moving perpendicular to the direction of incidence? - Astronomy

*I've been out of college and grad school for a number of years now, but one little question that popped into my head during freshman astronomy still vexes me. If we can use red shift to determine the rate and direction at which objects are moving away from Earth, couldn't we take a sampling of objects and extrapolate from their speed and movement the origin, or point in space, from which they are travelling, ie, the origination point of the Big Bang? I read the other question that explains how all objects are moving away from each other and how space is expanding, but that doesn't account for the fact that the big bang is always described as this tiny point of super condensed matter. I'm thinking that extrapolating redshifts could point us back to that pinpoint of matter. Thanks for your time.*

The Big Bang is often described as a tiny bit of matter, but that's an oversimplification. If the Big Bang occurred in a specific point in space, spewing galaxies in all directions, then we would expect our galaxy to be one of many galaxies sitting on an expanding shell of galaxies, with the center of that shell being the point of the "Bang." This, however, is not what we see, and not what the BB predicts.

If we were on a shell of galaxies, we would see many galaxies when we looked in directions along the shell, and few galaxies when we looked perpendicular to (up out of or down into) the shell. Moreover, distances and redshifts in such a scenario would depend on the direction we were looking. As we looked tangent to the shell, we would see many nearby galaxies with small redshifts. As we looked down into the shell, we would see more distant galaxies with higher redshifts. (Up out of the shell we would see only empty space.) This is not what we see. Galaxies, distant and nearby, are evenly distributed all around us. The number of galaxies and their redshifts are completely independant of which direction we look (we say that they are "homogeneous"), and that homogeneous distribution is also "isotropic," meaning that no matter where in the univerese you were, you would see exactly the same average distribution of galaxies and redshifts.

No, that little point of matter that was the Big Bang was not a little point of stuff inside an empty universe. It was, in fact, the entire observable universe. There was no "outside" of that point into which it could explode. In fact, the Big bang was not an explosion at all it was simply the very hot state of the early universe. Distances between objects were much shorter back then, but the universe was still homogeneous and isotropic. Wherever you were in the early universe, you would see a homogeneous, even, distribution of matter and energy around you. There was no empty "space" outside of this point of matter into which it could expand, for all of space was already there, in that little "point." The expansion of the universe is manifested only in the stretching of space itself, perpetually increasing distances between distant objects, not in some "empty space" gradually getting filled as matter streams into it. These distances expand in all directions equally, and so cannot be traced back to a single point. If you try to do this, you find that the single point is your telescope, no matter where in the universe you observe from. After all, the "point" in question was all there was of space: the entire observable universe. The Big Bang happened everywhere. It happened right where you are sitting, where the Andreomeda galaxy is now, and in the most distant reaches of the universe. It's just that the reaches of the universe were not quite as distant those many billions of years ago.

*This page was last updated June 27, 2015.*

### About the Author

#### Dave Kornreich

Dave was the founder of Ask an Astronomer. He got his PhD from Cornell in 2001 and is now an assistant professor in the Department of Physics and Physical Science at Humboldt State University in California. There he runs his own version of Ask the Astronomer. He also helps us out with the odd cosmology question.

## Length Contraction

The Lorentz transformation leads to a contraction of the apparent length of an object in a moving frame as seen from a fixed frame. The length of a ruler in its own frame of reference is called the *proper length*. Consider an accurately measured rod of known proper length (L_p = x^*given time in the stationary frame*, (t_1 = t_2), by taking a photograph of the moving rod. The corresponding locations in the moving frame are:

[x^

Since (t_2 = t_1), the measured lengths in the two frames are related by:

That is, the lengths are related by:

Note that the moving rod appears shorter in the direction of motion. As (v
ightarrow c) the apparent length shrinks to zero in the direction of motion while the dimensions perpendicular to the direction of motion are unchanged. This is called the *Lorentz contraction*. If you could ride your bicycle at close to the speed of light, you would observe that stationary cars, buildings, people, all would appear to be squeezed thin along the direction that you are travelling. Also objects that are further away down any side street would be distorted in the direction of travel. A photograph taken by a stationary observer would show the moving bicycle to be Lorentz contracted along the direction of travel and the stationary objects would be normal.

## Cosmology is Your Friend

And I will start by saying that this is a popular and very common misconception. The bang was not an explosion, and cannot properly be thought of as analogous to an explosion. It is an entirely different kind of thing. The "bang" *happened simultaneously everywhere in the universe, at all places and all times*. There is no center and there is no "edge". Or so it is if general relativity (GR) is a true representation of the physics of spacetime. If not, then scratch everything I said. All statements about the origin and early history of the universe are entirely dependent on assumptions based on a cosmological model. In this case, the model is provided by general relativity, but there are those who argue for other theories, on which to base other models.

A literal interpretation of GR requires that the universe began in a "singular" state, which is commonly misinterpreted to mean "infinitely small", but really means "undefined". The origin of the universe can be neither described nor explained in terms of classical GR. As a result, cosmology has generally not concerned itself with the question of origin, and has contented itself with describing the post-origin history of the universe.

Any theory intended to describe or explain the origin of the universe must, so far as we know, include quantum field theory. But GR, which has no quantum aspects, is the only viable theory of spacetime. So there is much work going on in attempting to "quantize" GR. String Theory & Loop Quantum Gravity are the best known, though there are other efforts underway. Only when such a theory becomes available, will anyone be able to meaningfully address the question of the origin of the universe.

*Short answer:* Nobody knows. *Long Answer:* There is no good physical answer to this question. There are several candidate mathematical answers. The most popular answer amongst cosmologists at the moment is the "cosmological constant" (CC). Originally invented by Einstein and inserted into his equations of general relativity, in order to maintain a static universe, the CC works just as well to force the universe to expand faster than it would if it were just coasting after the bang. A physical interpretation of the CC might be quantum mechanical energy embedded in spacetime. But there are other bits & pieces in Einstein's equations where an expanding force can hide. Quintessence is a cosmological idea that expresses the accelerated expansion in terms of a force embedded in the spacetime equations of general relativity. It has an advantage over the CC in that it is not required to be constant everywhere & everywhen. But it has a disadvantage, in that its functional form is entirely unknown (and maybe unknowable), so we can use quintessence to make universes that accelerate, but then suddenly screech to a halt & contract, for no particular physical reason, but just because that's how the functional form behaves. And there are several other, similar candidates for an acelerating force.

No, but again the answer may not come too easily. The real point of expanding universe ("big bang") cosmology is that the expansion is not a motion of the galaxies through spacetime, but rather a motion of spacetime which carries the galaxies along for the ride. Think of spacetime as a flat elastic sheet, which expands by simultaneously stretching outwards in all directions from every point. As the sheet expands outward, it will carry anything sitting on it. Things that are farther away have a longer distance of stretching sheet between it & us. If the sheet expands at a constant rate, *per unit length*, a longer distance will then expand faster, because it has a larger number of unit lengths in it. You can see that in the units used to express a numerical value for the Hubble Constant, say about 70 km/sec/Mpc. That's 70 kilometers per second *per megaparsec*, a velocity (km/sec) per unit distance (Mpc). So, the farther out you get, the greater the expansion velocity. That is the way the expansion of the universe appears to us.

Now, consider the difference between what the universe *is*, and what the universe *looks like*. If we understand things more or less correctly, then as we look out at the deep sky, we look farther away in distance, and farther back in time. If the universe is expanding, then as we look farther back in time, we look at a smaller universe. But from where we sit, the universe appears to surround us as ever more distant spherical shells, which look bigger the farther we go. So, as the universe actually gets smaller, the farther awy we look, the bigger it looks from here. So we see the universe as if through a strongly distorting lens, it looks much different than it really is. That's why cosmology is more dominated by mathematics & mathematical models, than are more conventional astronomy & astrophysics.

The universe "is" what it looks like from here. After all, how can we even use the word "is" to describe something that stretches over time, from yesterday, to 10 billion years ago, all at once? This, along with the distortion of distance, makes cosmology a trying thing to talk about!

While above answers are spot on, I would like to add that even if we would assume that the redshifts would arise from actual motion, the objects closest to us would still exhibit lowest motions (low redshifts) even in accelerating universe. One important thing is that you are talking about motion of distant objects, but you also have to consider our own motion, because it adds to the redshift too. I'll give you one dimensional illustration of this. Here I have one dimensional universe presented in 4 moments of time (A - D), and distance between each object is accelerating so that it doubles between each moment of time (objects: 1 - 4 are galaxies, E is Earth, points represent certain distance of empty space):

Here, object 1 is most distant object from Earth, and during time Earth along some other objects are moving away from it. Notice that object 1 appears to remain still. At moment of time A, distance from all objects to their closest neighbours is one unit of space, presented by one point between each object.

You see that distance between E and 4 increases from one unit of space to two units of space between A and B moments of time so we can say that velocity of E and 4 with reference to each other is one unit of space per one moment of time (at moment of time B).

Similarily, distance between E and 3 increases from two units of space to four units of space between A and B. That means that velocity of E and 3 with reference to each other is 4 - 2 = 2 units of space per one moment of time. So, object 3 has two times higher velocity from Earth's point of view than object 4, and object 3 is twice as far from Earth as object 4. You can repeat this exercise with other objects and other moments of time in my illustration, and you see that in all cases you get higher velocities for more distant objects.

So, my answer to your question would be that even if our closest objects have had more time to accelerate than far away objects, we are also accelerating with them, so there's not much velocity difference between us and our closest objects, but there is huge velocity difference between us and far away objects because we have had time to accelerate.

The illustration can also be presented like this:

Now Earth appears to remain still, but calculated velocities work as before. We can also try to illustrate what **Tim** says so that objects remain still and more space "grows" between them:

## Chapter 6 Acceleration and General Relativity

General relativity is Einstein&rsquos extension of special relativity to include gravity. An important aspect of general relativity is that spacetime is no longer necessarily flat, but in fact may be curved under the influence of mass. Understanding curved spacetime is an advanced topic which is not easily accessible at the level of this text. However, it turns out that some insight into general relativistic phenomena may be obtained by investigating the effects of acceleration in the flat (but non-Euclidean) space of special relativity.

The central assumption of general relativity is the equivalence principle, which states that gravity is a force which arises from being in an accelerated reference frame. To understand this we must first investigate the concept of acceleration. We then see how this leads to phenomena such as the gravitational red shift, event horizons, and black holes. We also introduce in a preliminary way the notions of force and mass.

### Acceleration

Imagine that you are in a powerful luxury car stopped at a stoplight. As you sit there, gravity pushes you into the comfortable leather seat. The light turns green and you &ldquofloor it&rdquo. The car accelerates and an additional force pushes you into the seat back. You round a curve, and yet another force pushes you toward the outside of the curve. (But the well designed seat and seat belt keep you from feeling discomfort!)

Figure 6.1: Example of linear motion.

Let us examine the idea of **acceleration** more closely. Considering first acceleration in one dimension, figure 6.1 shows the position of an object as a function of time, (x) ((t) ). The velocity is simply the time rate of change of the position:

The acceleration is the time rate of change of velocity:

[ dv(t) d2x-(t) a(t) = dt = dt2 . label<6.2>]

In figure 6.1, only the segment OA has zero velocity. Velocity is increasing in AB, and the acceleration is positive there. Velocity is constant in BC, which means that the acceleration is zero. Velocity is decreasing in CD, and the acceleration is negative. Finally, in DE, the velocity is negative and the acceleration is zero.

In two or three dimensions, position (x), velocity (v), and acceleration (a) are all vectors, so that the velocity is

while the acceleration is

Thus, over some short time interval &Delta(t), the changes in (x) and (v) can be written

These are vector equations, so the subtractions implied by the &ldquodelta&rdquo operations must be done vectorially. An example where the vector nature of these quantities is important is motion in a circle at constant speed, which is discussed in the next section.

### Circular Motion

Figure 6.2: Two different views of circular motion of an object. The left panel shows the view from the inertial reference frame at rest with the center of the circle. The tension in the string is the only force and it causes an acceleration toward the center of the circle. The right panel shows the view from an accelerated frame in which the object is at rest. In this frame the tension in the string balances the centrifugal force, which is the inertial force arising from being in an accelerated reference frame, leaving zero net force.Imagine an object constrained by an attached string to move in a circle at constant speed, as shown in the left panel of figure 6.2. We now demonstrate that the acceleration of the object is toward the center of the circle. The acceleration in this special case is called the **centripetal** **acceleration** .

Figure 6.3 shows the position of the object at two times spaced by the time interval &Delta(t). The position vector of the object relative to the center of the circle rotates through an angle &Delta(&theta ) during this interval, so the angular rate of revolution of the object about the center is (&omega ) = &Delta(&theta∕) &Delta(t). The magnitude of the velocity of the object is (v), so the object moves a distance (v) &Delta(t ) during the time interval. To the extent that this distance is small compared to the radius (r ) of the circle, the angle &Delta(&theta ) = (v) &Delta(t∕r). Solving for (v ) and using (&omega ) = &Delta(&theta∕) &Delta(t), we see that

[v = &omegar (circular motion ). label<6.6>]

The direction of the velocity vector changes over this interval, even though the magnitude (v ) stays the same. Figure 6.3 shows that this change in direction implies an acceleration (a ) which is directed toward the center of the circle, as noted above. The magnitude of the vectoral change in velocity in the time interval &Delta(t ) is (a) &Delta(t). Since the angle between the initial and final velocities is the same as the angle &Delta(&theta ) between the initial and final radius vectors, we see from the geometry of the triangle in figure 6.3 that (a) &Delta(t∕v ) = &Delta(&theta). Solving for (a ) results in

[a = &omegav (circular motion ). label<6.7>]

Figure 6.3: Definition sketch for computing centripetal acceleration.

Combining equations (6.6) and (6.7) yields the equation for centripetal acceleration:

[a = &omega2r = v2∕r (centripetal acceleration ). label<6.8>]

The second form is obtained by eliminating (&omega ) from the first form using equation 6.6>.

### Acceleration, Force, and Mass

We have a good intuitive feel for the concepts of force and mass because they are very much a part of our everyday experience. We think of force as how hard we push on something. Mass is the resistance of an object to acceleration if it is otherwise free to move. Thus, pushing on a bicycle on a smooth, level road causes it to accelerate more readily than pushing on a car. We say that the car has greater mass. We can summarize this relationship with Newton&rsquos second law

where (F ) is the total force on an object, (m ) is its mass, and (a ) is the acceleration resulting from the force.

Three provisos apply to equation 6.9>. First, it only makes sense in unmodified form when the velocity of the object is much less than the speed of light. For relativistic velocities it is best to write this equation in a slightly different form which we introduce later. Second, the force must be the total force, including all frictional and other incidental forces which might otherwise be neglected by an uncritical observer. Third, it only works in a reference frame which itself is unaccelerated, i.e., an **inertial reference frame** . We deal below with accelerated reference frames.

### Acceleration in Special Relativity

Figure 6.4: World line of the origin of an accelerated reference frame.As noted above, acceleration is just the time rate of change of velocity. We use the above results to determine how acceleration transforms from one reference frame to another. Figure 6.4 shows the world line of an **accelerated** **reference frame** , with a time-varying velocity (U) ((t) ) relative to the unprimed inertial rest frame. Defining &Delta(U ) = (U) ((T) ) - (U) (0) as the change in the velocity of the accelerated frame (relative to the unprimed frame) between events A and C, we can relate this to the change of velocity, &Delta(U) &prime , of the accelerated frame **relative to an inertial frame moving with the initial velocity** , (U) (0). Applying the equation for the relativistic addition of velocities, we find

We now note that the mean acceleration of the reference frame between events A and C in the rest frame is just (a ) = &Delta(U∕T), whereas the mean acceleration in the primed frame between the same two events is (a) &prime = &Delta(U) &prime (∕T) &prime . From equation 6.10> we find that

and the acceleration of the primed reference frame as it appears in the unprimed frame is

Since we are interested in the instantaneous rather than the average acceleration, we let (T ) become small. This has three consequences. First, &Delta(U) and &Delta(U) &prime become small, which means that the term (U) (0)&Delta(U) &prime (∕c) 2 in the denominator of equation 6.12> can be ignored compared to 1. This means that

with the approximation becoming exact as (T ) &rarr 0. Second, the &ldquotriangle&rdquo with the curved side in figure 6.4 becomes a true triangle, with the result that (T) &prime = (T) [1 - (U) (0) 2 (∕c) 2 ] 1 ∕ 2 . The acceleration of the primed frame with respect to an inertial frame moving at speed (U) (0) can therefore be written

Third, we can replace (U) (0) with (U), since the velocity of the accelerated frame doesn&rsquot change very much over a short time interval.

Dividing equation 6.13> by equation (6.14) results in a relationship between the two accelerations:

which shows that the acceleration of a rapidly moving object, (a), as observed from the rest frame, is less than its acceleration relative to an inertial reference frame in which the object is nearly stationary, (a) &prime , by the factor (1 - (U) 2 (∕c) 2 ) 3 ∕ 2 . We call this latter acceleration the **intrinsic acceleration** . This difference in observed acceleration between the two inertial reference frames is purely the result of the geometry of spacetime, but it has interesting consequences.

Identifying (a ) with (dU∕dt), we can integrate the acceleration equation assuming that the intrinsic acceleration (a) &prime is constant and that the velocity (U ) = 0 at time (t ) = 0. We get the following result (verify this by differentiating with respect to time):

which may be solved for (U∕c) :

This is plotted in figure 6.5. Classically, the velocity (U ) would reach the speed of light when (a) &prime (t∕c ) = 1. However, as figure 6.5 shows, the rate at which the velocity increases with time slows as the object moves faster, such that (U ) approaches (c) asymptotically, but never reaches it.

Figure 6.5: Velocity divided by the speed of light as a function of the product of the time and the (constant) acceleration divided by the speed of light.

The results for this section are valid only for acceleration components in the direction of motion. The components perpendicular to this direction behave differently and are treated in more advanced texts.

### Accelerated Reference Frames

Referring back to the forces being felt by the occupant of a car, it is clear that the forces associated with accelerations are directed opposite the accelerations and proportional to their magnitudes. For instance, when accelerating away from a stoplight, the acceleration is forward and the perceived force is backward. When turning a corner, the acceleration is toward the corner while the perceived force is away from the corner. Such forces are called **inertial** **forces** .

The origin of these forces can be understood by determining how acceleration changes when one observes it from a reference frame which is itself accelerated. Suppose that the primed reference frame is accelerating to the right with acceleration (A ) relative to the unprimed frame. The position (x) &prime in the primed frame can be related to the position (x ) in the unprimed frame by

where (X ) is the position of the origin of the primed frame in the unprimed frame. Taking the second time derivative, we see that

where (a ) = (d) 2 (x∕dt) 2 is the acceleration in the unprimed frame and (a) &prime = (d) 2 (x) &prime (∕dt) 2 is the acceleration according to an observer in the primed frame.

We now substitute this into equation 6.9> and move the term involving (A ) to the left side:

This shows that Newton&rsquos second law represented by equation 6.9> is not valid in an accelerated reference frame, because the total force (F ) and the acceleration (a) &prime in this frame don&rsquot balance as they do in the unaccelerated frame the additional term - (mA ) messes up this balance.

We can fix this problem by considering - (mA ) to be a type of force, in which case we can include it as a part of the total force (F). This is the inertial force which we mentioned above. Thus, to summarize, we can make Newton&rsquos second law work when objects are observed from accelerated reference frames if we include as part of the total force an inertial force which is equal to - (mA), (A ) being the acceleration of the reference frame of the observer and (m ) the mass of the object being observed.

The right panel of figure 6.2 shows the inertial force observed in the reference frame of an object moving in circular motion at constant speed. In the case of circular motion the inertial force is called the **centrifugal force** . It points away from the center of the circle and just balances the tension in the string. This makes the total force on the object zero in its own reference frame, which is necessary since the object cannot move (or accelerate) in this frame.

General relativity says that gravity is nothing more than an inertial force. This was called the **equivalence principle** by Einstein. Since the gravitational force on the Earth points downward, it follows that we must be constantly accelerating upward as we stand on the surface of the Earth! The obvious problem with this interpretation of gravity is that we don&rsquot appear to be moving away from the center of the Earth, which would seem to be a natural consequence of such an acceleration. However, relativity has a surprise in store for us here.

It follows from the above considerations that something can be learned about general relativity by examining the properties of accelerated reference frames. In particular, we can gain insight into the above apparent paradox. Equation (6.17) shows that the velocity of an object undergoing constant intrinsic acceleration (a) (note that we have dropped the &ldquoprime&rdquo from (a ) to simplify the notation) is

where (t ) is the time and (c ) is the speed of light. A function (x) ((t) ) which satisfies equation 6.21> is

(Verify this by differentiating it.) The interval OB in figure 6.6 is of length (x) (0) = (c) 2 (∕A).

Figure 6.6: Spacetime diagram showing the world line of the origin of a reference frame undergoing constant acceleration.

The slanted line OA is a line of simultaneity associated with the unaccelerated world line tangent to the accelerated world line at point A. This line of simultaneity goes through the origin, as is shown in figure 6.6. To demonstrate this, multiply equations (6.21) and (6.22) together and solve for (v∕c) :

From figure 6.6 we see that (ct∕x ) is the slope of the line OA, where ((x,) ( ct) ) are the coordinates of event A. Equation (6.23) shows that this line is indeed the desired line of simultaneity, since its slope is the inverse of the slope of the world line, (c∕v). Since there is nothing special about the event A, we infer that **all** lines of simultaneity associated with the accelerated world line pass through the origin.

We now inquire about the length of the invariant interval OA in figure 6.6. Recalling that (I) 2 = (x) 2 - (c) 2 (t) 2 and using equation 6.22>, we find that the length of OA is

[ 2 2 2 1 4 2 1 2 I = (x - c t ) = (c ∕a ) = c∕a, label<6.24>]

which is the same as the length of the interval OB. By extension, **all** events on the accelerated world line are the same invariant interval from the origin. Recalling that the interval along a line of simultaneity is the distance in the associated reference frame, we reach the astonishing conclusion that even though the object associated with the curved world line in figure 6.6 is accelerating away from the origin, it always remains the same distance (in its own frame) from the origin.

The analogy between this problem and the apparent paradox in which one remains a fixed distance from the center of the earth while accelerating away from it is not perfect. In particular, the earth case depends on the existence of the earth&rsquos mass.

### Gravitational Red Shift

Figure 6.7: Spacetime diagram for explaining the gravitational red shift. Why is the interval AC equal to the interval BC? (L ) is the length of the invariant interval OB.Light emitted at a lower level in a gravitational field has its frequency reduced as it travels to a higher level. This phenomenon is called the **gravitational red** **shift** . Figure 6.26 shows why this happens. Since experiencing a gravitational force is equivalent to being in an accelerated reference frame, we can use the tools of special relativity to view the process of light emission and absorption from the point of view of the unaccelerated or inertial frame. In this reference frame the observer of the light is accelerating to the right, as indicated by the curved world line in figure 6.26, which is equivalent to a gravitational force to the left. The light is emitted at point A with frequency (&omega ) by a source which is stationary at this instant. At this instant the observer is also stationary in this frame. However, by the time the light gets to the observer, he or she has a velocity to the right which means that the observer measures a Doppler shifted frequency (&omega) &prime for the light. Since the observer is moving away from the source, (&omega) &prime (< &omega), as indicated above.

The relativistic Doppler shift is given by

so we need to compute (U∕c). The line of simultaneity for the observer at point B goes through the origin, and is thus given by line segment OB in figure 6.26. The slope of this line is (U∕c), where (U ) is the velocity of the observer at point B. From the figure we see that this slope is also given by the ratio (X) &prime (∕X). Equating these, eliminating (X ) in favor of (L ) = ((X) 2 - (X) &prime 2 ) 1 ∕ 2 = (c) 2 (∕g), which is the actual invariant distance of the observer from the origin, and substituting into equation 6.25> results in our gravitational red shift formula:

If (X) &prime = 0, then there is no redshift, because the source is collocated with the observer. On the other hand, if the source is located at the origin, so (X) &prime = (X), the Doppler shifted frequency is zero. In addition, the light never gets to the observer, since the world line is asymptotic to the light world line passing through the origin. If the source is at a higher level in the gravitational field than the observer, so that (X) &prime (< ) 0, then the frequency is shifted to a higher value, i. e., it becomes a &ldquoblue shift&rdquo.

Equation (6.26) works for more complex geometries than that associated with an accelerated reference frame, e.g., for the gravitational field (g ) associated with a star, as long as | (X) &prime|≪ (c) 2 (∕g). In this case (L ) is no longer the distance to the center of the star but remains equal to (c) 2 (∕g).

### Event Horizons

The 45 ∘ diagonal line passing through the origin in figure 6.6 is called the **event horizon** for the accelerated observer in this figure. Notice that light from the &ldquotwilight zone&rdquo above and to the left of the event horizon cannot reach the accelerated observer. However, the reverse is not true &mdash a light signal emitted to the left by the observer can cross the event horizon into the &ldquotwilight zone&rdquo. The event horizon thus has a peculiar one-way character &mdash it passes signals from right to left, but not from left to right.

### Problems

- An object moves as described in figure 6.8, which shows its position (x ) as a function of time (t).
- Is the velocity positive, negative, or zero at each of the points A, B, C, D, E, and F?
- Is the acceleration positive, negative, or zero at each of the points A, B, C, D, E, and F?

Figure 6.8: Position of an object as a function of time.

- Sketch the object&rsquos velocity vectors at points A, B, and C.
- Sketch the object&rsquos acceleration vectors at points A, B, and C.
- If the string breaks at point A, sketch the subsequent trajectory followed by the object.

Figure 6.9: Object in circular motion.

Express your answer as the speed of light minus your actual speed. Hint: You may have a numerical problem on the second part, which you should try to resolve using the approximation (1 + (ϵ) ) x &asymp 1 + (xϵ), which is valid for | (ϵ) |≪ 1.

- Find the object&rsquos velocity as a function of time.
- Using the above result, find the slope of the tangent to the world line as a function of time.
- Find where the line of simultaneity corresponding to each tangent world line crosses the (x ) axis.

- What is the net force on a 100 kg man in the car as viewed from an inertial reference frame?
- What is the inertial force experienced by this man in the reference frame of the car?
- What is the net force experienced by the man in the car&rsquos (accelerated) reference frame?

- What would the rotational period of the earth have to be to make this person weightless?
- What is her acceleration according to the equivalence principle in the earth frame in this situation?

- Describe qualitatively how the hands of the watch appear to move to the Zork as it observes the watch through a powerful telescope.
- After a very long time what does the watch read?

Hint: Draw a spacetime diagram with the world lines of the spaceship and the watch. Then send light rays from the watch to the spaceship.

- From the perspective of Chimborazo, does the clock in Guayaquil appear to be running faster or slower than the Chimborazo clock? Explain.
- Compute the fractional frequency difference ((&omega ) - (&omega) &prime )(∕&omega ) in this case, where (&omega ) is the freqency of the Guayaquil clock as observed in Guayaquil (and the frequency of the Chimborazo clock on Chimborazo) and (&omega) &prime is the frequency of the Guayaquil clock as observed from Chimborazo. You may wish to use the results of the previous problem.

## Amplitude of wave stretched as well as red shift?

Electromagnetic waves are not transverse vibrations of a medium. There is no actual sideways motion. It's not like photographically enlarging an image of a water wave, where the amplitude grows by the same scale factor as the wavelength.

Although this question can be answered purely classically, I think it's easier to analyze if you think in terms of photons. The photon's frequency is reduced by a factor z, so E=hf is also reduced by that factor. The volume occupied by the wave is increased by a factor of z^3, so the energy density is reduced by a factor of z^4. Since the energy density is reduced by z^4, the electric and magnetic fields are also reduced, by factors of z^2.

[EDIT] Marcus pointed out that z should be replaced with 1+z everywhere above in order to be consistent with standard notation.

Are you suggesting a distant star is "brighter" than a nearer star?? Is that what you think you observe. no.

Also, cosmic redshift does not occur over relatively small galactic distances but rather much larger interstellar distances. in the smaller regions gravity keeps everything pretty much in the same relative position. and nearby galaxies can be moving towards one another as well as away from each other. this could result in some redshift or blue shift.

No medium, I get that..what does the second part mean. "actual sideways motion" ??

The field oscillates that way, right.Wiki has a (correct) concise explanation and associated diagram here:

Although this question can be answered purely classically, I think it's easier to analyze if you think in terms of photons. The photon's frequency is reduced by a factor z, so E=hf is also reduced by that factor. The volume occupied by the wave is increased by a factor of z^3, so the energy density is reduced by a factor of z^4. Since the energy density is reduced by z^4, the electric and magnetic fields are also reduced, by factors of z^2.

This is a clear complete response to Steve, so there's really nothing to add, but it illustrates an ambiguity in the English language---how we talk about fractional increase and increase "by a factor".

Or, in this case the reverse: "decrease by a factor".

The energy density of the photons is actually decreased by a factor of (1+z) 4

in the sense that that is what you divide the old density by to get the new density.Beginners can get confused by this ambiguity in English.

For example if the redshift of some light is z = 0.5, then the energy of each photon has been reduced by a factor of 1.5 (you divide by that to get the present energy).

And the volume has been increased by a factor of 1.5 3

so the number of photons per unit volume is reduced by 1.5 3 (they are spread out in a larger volume).So the whole effect on the energy density is to reduce it by a factor of 1.5 4 = a bit over 5.

So you divide the old energy density by a factor of 5 or so.This does not have any simple intuitive relation to 0.5 4 = 1/16 so a beginner might get confused by saying "reduced by a factor of z 4 = 1/16

## Physics of Sound – Everything you must know about Sound Waves

The mechanical wave that propagates vibration in a medium is Sound. The propagation medium can be either liquid, solid, or gaseous. It is the fastest in solids, then liquidates and slowest in gas.

This vibration is often audible because of wave pressure in gaseous, liquid, and solid medium. The vacuum does not allow sound to travel. Psychology defines sound as the perception by the brain of the sound pressure reception.

Propagation is a term to describe traveling sound. Three is often a disturbance in the sound pattern which causes energy to travel away from the source and this pattern is the sound wave.

The sound waves are longitudinal and thus the propagation of particles is parallel to the direction of propagated energy waves. Another type of wave is Transverse waves, they travel perpendicular to the direction of propagated waves.

The vibration causes atoms to move back and forth creating high pressure and low pressure in the medium. These pressure zones are compressions and rarefactions respectively.

The transportation of these regions to the surrounding medium leads travel of sound waves from one medium to another. The nature of sound is different because of the source of origin. A guitar will have a different sound than a drum.

### Nature Of Sound

The nature of sound mainly depends on five factors – Frequency, Wavelength, Amplitude, Time, and Velocity.

#### Frequency of Sound

The frequency of the sound wave is the total number of rarefactions and compressions that take place per unit time. The sound of a single frequency creates tone while a mixture of them creates a note. The formula of the frequency of a sound wave is –

f = 1T (f – frequency of a sound wave and T – time period)#### Wavelength of Sound

The wavelength is the distance between compression and a followed rarefaction. The formula for the wavelength of the sound is –

λ=vF (f – frequency of the sound wave and v – velocity of the sound wave)#### Amplitude of Sound

The maximum disturbance in the sound wave is the magnitude of the sound. And the magnitude of the sound is the amplitude of the sound which is also responsible for measuring energy. Higher amplitude means the energy is higher in the sound wave.

#### Human and Sound

- Humans have a limited hearing range of sound frequencies. 20 Hz and 20,000 Hz is the frequency range of the human ear. Under ideal laboratory conditions, it can go as low as 12 Hz and high as 20,000 Hz.
- The ear is the organ through which sound reaches the body. The outer shape of the ear resembles a funnel through which the sound enters. There is a canal for sound to pass that ends at a thin membrane called the eardrum.
- The eardrum vibrates as the sound vibration reaches it. It is a rubber sheet-like structure. The eardrum sends the message of sound to our brand through the inner ear making us hear the sound.
- Humans produce sound from the larynx. There are two vocal cords inside through which air passes and produces sound. Men have the longest vocal cord of 20mm approx and for women, it is 15mm.

#### Time Period

The time period is the total time of completing one vibration. The T denotes time takes in different formulas. The seconds measure the time takes and the Denton is ‘s’. It is usually a reciprocal of wave frequency.

#### Velocity

The total distance covered by a wave in one second becomes its velocity. Meter, per second measures this quantity.

**Velocity = Wavelength × Frequency or v = λv**### Speed of Sound

The sound waves propagate a medium at a certain speed and that is the speed of a sound. It is different for different mediums and remains highest in solid as the atoms are in a compact setting.

The distance decides the interaction of atoms in a particle. The energy transfer is faster when there is higher interaction. This is the reason behind the faster sound in solid mediums.

The speed of sound follows this formula –

**C = dt (d – distance and t – time taken)****Medium****Speed of sound**Water 1481 m/s Air 343.2 m/s Copper 4600 m/s Hydrogen 1270 m/s Glass 4540 m/s #### Intensity of Sound

The sound intensity in simple words refers to the passing of energy in one second through per unit area. Watt per meter square measures the sound intensity. It has no relation to human ear sensitivity. And it is an objective quantity.

This is not the same as loudness as loudness looks at the response to the sound by ears. And a pitch is the sound flatness.

The difference between music or noise is also due to intensity. The pleasant sound is music while the unpleasant sound is noise. The formula for Intensity of sound is –

**I = PA (I – intensity, P – power and A – area)**#### Reflection Of Sound

Laws of reflection are applicable to sound as well because it is similar to light reflection. The laws of reflection are –

- The incident and the normal sound are present in the same place.
- The angle of incidence = Angle of reflection

When the sound comes in contact with a hard surface, it reflects back to the source. This reflection of sound is the echo. The hard surfactant reflects the sound while the soft one absorbs it.

But in case of low sound frequency, the sound can not reflect back. And when there are multiple echoes from one source, it is reverberation.

### Echoes

The sound repetition due to reflection produces an echo. This is common when we find ourselves alone in an empty room and hear our own voice due to sound reflection. Hard surfaces like cliff, mountains, and wall reflect the sound and thus we hear an echo.

The sound surfaces are capable of absorbing the sound. The echo time interval must be 1/10th of a second so that it is audible as two different sounds to human ears.

These intervals allow us to identify the difference between the original and the reflected sound. And the distance between a human and the reflecting surface has to be a minimum of 17.2 meters for an audible echo.

And the air temperature is 20 degrees as the air temperature changes the distance. This is the reason why echos are more on a hot or humid day.

In case the reflecting medium is water, the distance must be 75 meters.

#### Uses of Echo

- Measuring the sea depth
- Locating the underwater objects
- Investigating the insides of a human body

### Reverberation

The multiple reflections of sound from the reflecting surface give rise to Reverberation. There is a building of each reflection which eventually falls as the objects absorb it in closed spaces.

Some may infuse it with echo because of similarities but in the case of reverberation, the distance between sound sources is less an obstacle in reflection is also comparatively low.

Reverberation time is the parameter to measure reverberation quantitatively. It usually looks at the length of time taken in sound decaying by 60 decibels from the starting level.

The time delay in this is not less than 0.1 seconds and the reflection time from surface to the observer is not more than 0.1 seconds. Though the original sound stays in the memory even after all the repetition.

#### Advantages and Disadvantages of Reverberation

They are responsible for soundproofing musical halls so that there is good music quality in the hall. The sound engineers are responsible for the construction of these specialized halls.

The disadvantage is when the room has no sound-absorbing surface which leads to dying of sound. The listener will have trouble listening to the sound and there will be a loss of articulation.

#### Application of Reverberation

- Recording Studios
- Chamber reverberator
- Loudspeakers
- Microphones
- Plate reverberator

#### Reverberations Reduction

The best idea to reduce it is to use sound-absorbing material covering on hard objects. This will allow the reflected sound to decay faster and the listener will be able to hear the original sound. The mineral wool or fiber glasses are porous material that converts sound energy to heat energy by friction.

### Vibration of Air Columns

The wind instrument produces sound by vibration inside the air pipes. There is motion inside these tubes making air particles move parallel to the pipe wall. This results in longitudinal wave creation inside the pipe.

This does not depend on closed or open nodes. The nodes are the end of strings to prevent vibrating. This means air doesn’t leave the pipe.

The air particles thus don’t show any motion thus creating a displacement node. There is high pressure inside the tube because of this, thus often referred to as pressure antinodes. In the case of an open pipe end, the air easily leaves and enters.

But the pressure at the end must be equal to room pressure and result in the back and forth movement of particles with high magnitude. This leads to a displacement antinode.

#### Supersonic and Shock Waves

A sonic boom is a situation when the source speed is faster than the sound speed. It is a type of shock wave where the source generates multiple waves and they get together over time creating a strong sum wave. This phenomenon is very unusual or rare.

An example is a motorboat, the speed of the boat is faster than the wave speed creating a v-shaped bow. A march cone creation takes place in the air when the aircraft has a higher speed than the sound speed. A higher march cone is a sign of faster aircraft speed and vice versa.

#### Refraction of Sound Waves

As we read above, sound travels faster in warm air. This is because the sound waves bend when they cross air layers of different temperatures. The bending of these sound waves is refraction. The air is warmer near the ground on a summer day and thus the speed of sound near the ground is faster.

The sound bends away from the ground because of refraction. And the case is opposite on a colder winter day. The refraction of sound depends upon the atmosphere density. The density often decreases when there is a temperature rise. This phenomenon has a lot of similarities to the refraction of light.

#### Diffraction of Sound Waves

The reason why we are able to hear a sound behind the closed door is because of the diffraction of sound. This phenomenon allows sound waves to bend around obstacles. This means that the small keyhole on the door allows sound waves to pass.

We see the diffraction of waves in our daily lives. This is why the thunder lightning sound at a distance sounds higher than the closer one.

The deeper sound wave tones bend near obstacles making you hear a deep rumbling of thunder.

### Resonance

Every object has a natural frequency according to its features or characteristics. Resonance is the situation when the object moves at its natural frequency because of vibration received from another system with the same frequency.

Acoustic, mechanic, electric, and optic types of systems are capable of resonance. There is a rise in vibration amplitude due to resonance. It is also very risky in some cases.

Soldiers crossing a suspension bridge always break to avoid resonance and bridge falling. This is also the reason why radios have a particular frequency to tune in.

#### Interference of Sound Waves

The principle of superposition allows two or more sound waves to get together or combine. This combining of waves is Interference. The superposition principle states “ when two waves are in the same place at the same time, their amplitudes are combined”.

- The constructive interference – total wave magnitude > individual wave
- The Destructive interference – total wave magnitude < individual wave

#### Electromagnetic Waves

The formation of electromagnetic waves is due to the change in magnetic and electric fields mutually. 299,792,458 m/s is their speed in vacuum. Microwaves and X-rays have electromagnetic waves.

### Doppler Effect

Planetary science uses this effect very regularly. It says that “frequency of the type of wave increases with decreasing distance between the source of the wave and observers”. Christian Johann Doppler was behind this concept.

- “Waves emitted by a source travelling towards an observer gets compressed”
- “Waves emitted by a source travelling away from an observer get stretched out”

#### Doppler Effect Formulas

In simpler words, the change in frequency because of relative motion between the observer and the source. The formula for this effect is –

**f’ = (v +- vo / v +- vs) f**(Here, v – sound wave velocity, vo – velocity of the observer, vs, the velocity of the source, f’ – observed frequency, and f- actual frequency.) This is the only equation of this effect but it is capable of changing in case of velocity change.

**Source Moving Towards the Observer at Rest**

The observer velocity is zero thus vo becomes zero. The equation for the observer at rest is –**f’ = (v / v – vs) f****Source Moving Away from the Observer at Rest**

The observer velocity is again zero and the source velocity is negative. The equation for this situation is –**f’ = (v / v – (-vs)) f****Observer Moving Towards a Stationary Source**

In this situation, the source velocity is zero. This equation for this situation is –**f’ = (v + vo / v) f****Observer Moving Away from a Stationary Source**

In this case, the source velocity is zero, and the observer’s velocity is negative. The equation is –**f’ = (v – vo / v) f**#### Uses of Doppler Effect

- Sirens
- Radar
- Astronomy
- Medical Imaging
- Blood Flow Measurement
- Satellite Communication
- Vibration Measurement
- Developmental Biology
- Audio
- Velocity Profile Measurement

#### Doppler Effect Limitations

- It is applicable when the velocity of sound is more than the source and observers’ velocities.
- The motion of the observer and source has to be in the same line.

#### Doppler Effect In Light

The Doppler Effect on light is an apparent change in light frequency because of light’s relative motion between the source and the observer. The sound wave goes under fluctuations spending on the source, medium, and observer.

But light needs no medium and even in vacuum relative speed impacts the effect.

#### Red Shift and Blue Shift

When the observer is looking at a light source moving away, he receives a lower frequency compared to the original frequency. This results in a shift towards the red end of the spectrum and is the redshift.

And when the light source is moving towards him, the frequency received is higher. He will see the high end of the light spectrum and is the blueshift.

### Conclusion

Sound is one of the most important physics topics for multiple competitive exams. These exams are UPSC, RRB, SSC, etc. Some important topics were the nature of sound, the doppler effect, waves, and more.

The General Science paper will definitely have some questions about this topic. And this is important for Prelims as well. This is because it comes under the 9th physics and is present in NCERT books as well.

Candidates will definitely receive reliable and easy-to-understand information from the article above. Aspirants will be better prepared after reading this.

Stay with DataFlair for more such interesting articles.

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## K.i.s.s.

Over looked is the density of light in the inter galactic space. When the theories of light we are now believing our understanding of light was more limited than it is now.

Our theorests did not know that light bent with the density of gravity, demonstrated by the occlusion of stars.

This is "The Molly principal of light"

Since light is effected with gravity then we may surmise that light will also effect light, Not much but some. Over the millions of years and more of travel the only thing we know is between the stars is light. Light is energy, and or particles traveling fast and thru proximity the light absorbs some of the energy from light coming the other direction. This loss of energy is noticed as a red shift in spectral signature of the light (in both directions). thou we can only measure one.

From some article on stellar occlusion I remember some formula that if gravity effects mass at 1, gravity effects light at the power of negative26. I contend that light may effect light at the power of negative676.. Thats a small number, but remember that the light has been traveling a long time and the light is in very close contact with the light from the other direction.

Molly was my Great grandmother, she was a small woman but if she wanted to move the mountain she started pushing, The mountain moved!

Also I wonder if redshifted light is more prone to these effects than blue shifted light, where the effects become somewhat exponential? the more the other light effects the light we are observing the more redshifted it becomes and the more prone it is to the effect .

Over looked is the density of light in the inter galactic space. When the theories of light we are now believing our understanding of light was more limited than it is now.

Our theorests did not know that light bent with the density of gravity, demonstrated by the occlusion of stars.

This is "The Molly principal of light"

Since light is effected with gravity then we may surmise that light will also effect light, Not much but some. Over the millions of years and more of travel the only thing we know is between the stars is light. Light is energy, and or particles traveling fast and thru proximity the light absorbs some of the energy from light coming the other direction. This loss of energy is noticed as a red shift in spectral signature of the light (in both directions). thou we can only measure one.

From some article on stellar occlusion I remember some formula that if gravity effects mass at 1, gravity effects light at the power of negative26. I contend that light may effect light at the power of negative676.. Thats a small number, but remember that the light has been traveling a long time and the light is in very close contact with the light from the other direction.

Molly was my Great grandmother, she was a small woman but if she wanted to move the mountain she started pushing, The mountain moved!