Physics: Wave Mechanics and Vibration

[ComradeRed]

So my class is on vibrations and waves and sound right now. Any tricky concepts or difficult applications that you can think of? I don't really understand the Doppler effect all that well, particularly with all the equations such as f'=f((v+or-v0)/v). You know the observed frequency equations including observer in motion, source in motion, and observer and source both in motion.

So if you could just go over all of these concepts in an online classes thread that would be extremely helpful 'cause I'm not sure if I have enought time to read the entire chapters. One possible tricky application is coming up with a loud enough sound to knock over dominos precisely, though this would require such energy it would be impractical.

Maybe something involving convalution or interference of waves? Then predicting the frequency of the outcome? I dunno, it's been a while and I may be a little rusty ;)

The Doppler Effect
Have you noticed that a biker is louder when he is coming than going? This is because of the doppler effect where the sound waves of his motor is "stacked" when he is coming yet when he leaves they are no longer stacked. This is because he is moving while constantly "emitting" noise from his engine.

This is the crux of the doppler effect. :) The perceived frequency is higher when the biker is coming than when he is leaving.

This can be represented mathematically by the equation:

f=f_{0}(1+\frac{v_{0}}{v})

for the original frequency f_{0}, the velocity of the observer v_{0}, and the velocity of the observed v.

Remember: I am using LaTeX syntax!!!

Example Problem[\b]

I was asked to solve this problem for an example:

3. A train at rest blows a whistle to alert passengers that it is about to depart from a subway station. The pitch of this whistle is 1.14E4 Hz. The speed of sound in the air in that subway tunnel is 342 m/s. The speed of sound in iron is 5130 m/s. So we know the medium, and we know the sound.

b. What is the distance between consecutive areas of compression and of rarefaction in the spherical sound waves spreading from the whistle in that air?

Well, first of all, in a longitudinal wave -- that is a waves that are parallel to their direction of travel -- there are rarefaction and compression.

That is sections of space where the wave is "more dense" than other parts (the other parts are called "rarefaction" pronounced rar-ree-faction). You can think of this as the wave is compressed in the "compression" and the rarefaction is the distance between two compressions.

The wavelength is the distance between two centers of compression.

The wavelength has a fantastic relationship to frequency and velocity, demonstrated by the beautiful formula:

[b](wavelength)(frequency)=(velocity)

So recall from the question "The pitch of this whistle is 1.14E4 Hz. The speed of sound in the air in that subway tunnel is 342 m/s. The speed of sound in iron is 5130 m/s." There are 1.14*10^4 cycles per second and the velocity 342 m/s.

With simple algebra we get

wavelength = (342 m/s)/(1.14*10^4 s^-1)

this reduces to

wavelength = .3 meters

provided we ignore "sig-figs"(it is actually "0.300meters" because of the measurements, but don't worry about this, it's all tangential).

The distance is the wavelength, just to let ya know ;)

c. Assuming that the sound was loud enough to be heard from the end of the 1200 m long tunnel, when was it heard though air? Though the rails?

This is rather simple from mechanics, distance divided by velocity equals the time it took.

That is to say the velocity in the air was 342 meters/second, thus

(1200m)/(342m/s) = 3.50877192982456140350877192982456 s after it rang out through air.

On the other hand we also have the metallic medium with a different velocity. It is "The speed of sound in iron is 5130 m/s."

Now we just use the same formula:

(1200m)/(5130 m/s) = 0.233918128654970760233918128654971 s after it rang out through iron.

d. What was the apparent frequency of the sound waves that reached the end of the tunnel?

Just to warn you, my coffee supply is running low, so might too the coherency of my reply hereon end.

In reality, this would be impossible to figure out because we would need to know the velocity and orientation therein of the air, etc.

However, ignoring these things it is considerably easier (anything doable is easier than something that is impossible!).

IF we were to asssume that this was an ideal experiment, then the frequency would remain the same. But I am lazy and as I stated I'm out of coffee (:o).

e. As the train left the station, did the frequency appear to change for a listener on the platform? inside the train? at the other end of the tunnel? We are dealing with three observers: one on the train, one off of the train watching it leave, and one watching the train approach.

The doppler effect explains that the biker is louder coming than going. So, for the observers not on the train the frequency changed. The observer on the train believes that the frequency did not change.

The observer watching the train approach, like observing a biker approach, believes the frequency is getting faster and faster.

On the other hand we have an observer watching the train leave, possibly missing it because he was late and had to get coffee (<_<); this poor fellow believes that the frequency is slower than the frequency observed by the other two observers.

Hope that helps (in my caffeine deprived state) :)

[Connolly]

Have you noticed that a biker is louder when he is coming than going?

Why does it not seem this way with aircraft - particularly jet aircraft, where it seems to be the opposite than this. :blink: :huh:

[ComradeRed]

Why does it not seem this way with aircraft - particularly jet aircraft, where it seems to be the opposite than this. It&#39;s because of the position of the jets, and where the noise from the jets come from (the exit at an incredible velocity leaving) the aircraft, the aircraft moves around the speed of the waves.

On a bike, the waves are faster than the bike. Thus they stack when he is coming than leaving.

With an airplane it is the opposite (provided it isn&#39;t a biplane or triplane :lol:).


Simple harmonic motion: frequency and period(T) For a wave with a frequency f, a simple harmonic motion to it would be any wave with a frequency that is an integer multiple of f.

Mass-spring, pendulum, transverse and longitudinal waves Basically, longitudal waves work through the medium. That is it travels "parallel" to the medium, so we know when it passes because the medium "compresses" around the peaks. Think of a sound wave, the air (as the medium) "compresses" around the peaks as the wave travels.



Interference (constructive and destructive) Well, you can think of waves as circles. These circles are called phasors. Remember when we had the units of frequency: cycles per second? Well, the circle contains a radius.

This radius is actually a vector, the length describes the amplitude of the wave.

The "amplitude vector" moves around to complete a cycle. This is the cycle described by the frequency.

This also makes more intuitive sense when one says that a wave is "90 degrees out of phase", the vector for one wave forms a right angle with the vector for the other wave.

Constructive Interference is when the two vectors are in the same phase and add together.

Destructive Interference is when the two vectors are "180 degrees out of phase", and if the amplitudes are equal, then the two waves cancel out.



Wave equation The wave equation is

A sin (2 pi f(t - r/c))

for the amplitude A, the frequency f, the time observed t, the distance r, and the speed of sound c.

Anatomy of wavesWaves are characterised by crests (highs) and troughs (lows), either perpendicular (in the case of transverse waves) or parallel (in the case of longitudinal waves) to wave motion.


Compression waves This is the peak of the waves, where the medium is "more compressed" than at the troughs. At the troughs, it is considered to be "not compressed".

Tuning forksA tuning fork is a simple metal two-pronged fork with the tines formed from a U-shaped bar of elastic material (usually steel). A tuning fork resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and after waiting a moment to allow some high overtones to die out. The pitch that a particular tuning fork generates depends on the length of the two prongs, with two nodes near the bend of the U.

ResonanceResonance is the tendency of a system to absorb more energy when the frequency of the oscillations matches the system&#39;s natural frequency of vibration (its resonant frequency) than it does at other frequencies. It is "constructive interference" essentially, except rather than the two waves merging together, one absorbs the other.

BeatsA beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies.

Hearing (range and intensity) For the net power radiated P, the intensity is:

absolute(I) = P/(4*pi*r^2)

for some radius r, and the absolute power of intensity.

Decibels A measure of the ratio between sound power and intensity ratios. It is dimensionless like a percent.

EchoesAn echo (plural echoes) is a reflection of sound, arriving at the listener some time after the direct sound. The time delay is the extra distance divided by the speed of sound.

Speed of sound It is described by the equation

c = sqrt(k*R*T)

for the gas constant R, the adabiatic index k, and the absolute temperature T in kelvins.

The adabiatic index is usually given or in a huge book of all (or most) of the adabiatic indices of many particles, mediums, etc.

Harmonics Given the frequency of a wave, a harmonic of that wave is an integer multiple of the frequency. So for some frequency f, a harmonic to that wave would be nf for some integer n.

4 characteristics of a string that will change the frequency. Direct or inverse relationship? How long it is, how taut it is, how thick it is, and the temperature of the string (at least, those are that which effect the violin string&#39;s frequency ;)).

[Janus]

The wave equation is

A sin (2 pi f(t - r/c))

for the amplitude A, the frequency f, the time observed t, the distance r, and the speed of sound c.

I think we&#39;re talking about different wave equations. The equation I was originally talking about was v=f(wavelength). I don&#39;t know why my teacher called it the wave equation.

Also, could you explain the spherical and plane waves? As well as standing waves?

[ComradeRed]

I think we&#39;re talking about different wave equations. The equation I was originally talking about was v=f(wavelength). I don&#39;t know why my teacher called it the wave equation. That one&#39;s easy. In a given time period, there are a number of cycles completed. That is frequency: cycles per second.

Wavelength deals with distance per cycle.

Thus (frequency)(wavelength) = (cycles / second)(distance / cycle) = (distance / second) = velocity :)



Also, could you explain the spherical and plane waves? A plane wave is assumed to be constant with respect to one coordinate. That is, if we have a three dimensional coordinate system: x,y,z. We have a plane wave.

That means on one of the planes (xy, yz, zx) the coordinates do not change. Think of it like the top of a water tank. The level of water remains constant. No amount of waves will change that provided that the water remains in the tank. An interesting note is that plane waves usually have constant frequency.

On the other hand spherical waves are rather interesting. Imagine if I had lit a match. The light of the match radiates spherically, no?

The frequency is not constant, it is usually not langitudinal either.


As well as standing waves? A standing wave is like a jump rope: it doesn&#39;t move forward or backwards but it rotates.

It occurs because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

[mikelepore]

To the person who asked about the Doppler effect.

I taught physics for a while. I found it helpful to visualize that you&#39;re standing still, I&#39;m running toward you, and throwing baseballs at you at a rate of ten per minute. Maybe you catch them at a rate of eleven per minute. To go the other way: you&#39;re standing still, I&#39;m facing you, but running backwards away from you. I throw you baseballs at a rate of ten per minute. Perhaps now you catch them at a rate of nine per minute. Maybe someone can visualize that first. This is basically what waves are doing when there is relative motion between source and observer.

Mike Lepore -- lepore at bestweb dot net

[Severian]

Originally posted by The [email protected] 6 2006, 08:50 AM
Why does it not seem this way with aircraft - particularly jet aircraft, where it seems to be the opposite than this. :blink: :huh:
Because when you&#39;re behind the plane, the engine is pointing at you?

[mikelepore]

It&#39;s not based on which way an aircraft engine is pointed, but only on it velocity vector. No matter which way a noisemaker is oriented, it&#39;s still a point source of wave fronts radiating spherically outward.

[mikelepore]

Originally posted by [email protected] 7 2006, 04:38 AM

On the other hand spherical waves are rather interesting. Imagine if I had lit a match. The light of the match radiates spherically, no?

It&#39;s also helpful for learners to realize why everything that radiates spherically has an inverse-square relationship with distance, such as the intensity of light or sound some distance from the source, the gravitational field at some distance from a mass, and the electric field at some distance from a point charge. In all these cases, something that is conserved is being distributed over continuously-enlarging spheres, so that the amount of flux passing through any given surface size on that sphere has to decrease. It will decrease by whatever factor represents distance from the center of the sphere. The formula for the area of a sphere contains the square of the radius. This is where all the inverse square laws come from.

Mike Lepore - lepore at bestweb dot net