Vibrations and Waves

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There are 10 kinds of people: those who know binary and those who don’t.

For vibrations we restrict our discussion to simple harmonic motion (SHM): back and forth sinusoidal motion,
x = Asin(w t) or x = Acos(w t). (A is called the amplitude, the distance from the center to an extreme point of the oscillation, w is “angle”/time.) SHM is more general than it appears, because any vibratory motion (repetitively moving back and forth) can be expressed as the sum of sines, cosines, or both, called Fourier series. If you go into physics or engineering, you will study Fourier series. The w is the Greek lower case omega. It will look like a curly w on most computers; it is the angular frequency, the number of radians per second. (There are 2p radians in a cycle.) Other Greek letters: a, b = alpha, beta. If those show up as a, b, then I think your browser does not have symbol font, and this will cause some reading difficulties.

The waves will be of the type, y = Asin(kx-w t), plane waves. These are the special cases that everyone learns first. If you want to go beyond that, go elsewhere.

Vibrations and waves are very important in physics, engineering and many other fields. The building you are in is vibrating a little due to earthquakes somewhere, traffic nearby, the wind, people moving around in it, and various electrical devices being used. You are constantly being bombarded with various waves, and if you can hear and see, you are using waves.  This document will cover the basics, and there are links to related topics.  The topics you really must know have * in front of their links.

*Simple Harmonic Motion (SHM)

One approach is to base SHM on circular motion. If you know calculus, you will know how to avoid this at first, but later when we get into phasors, circular motion cannot be avoided, so you might as well look into it from the beginning. Some of the derivations here will use calculus, and I have concocted a little primer on calculus that might help.

The sketch below shows the circular path of an object, say a ball on the end of a string. The radius is A, and it has constant speed v = w A, where w is in radian/s. As you can see,
x = Acos(w t) is the x component of its position. So its x (or y) component undergoes simple harmonic motion. Note that the angle of v above the negative x direction is the complement of w t, hence
vx = -vsin(w t) = -Aw sin(w t). Recall that in circular motion the acceleration is toward the center and has magnitude
v2/radius or w 2 times radius, so in this case,
ax = -Aw 2cos(w t).

Simple harmonic motion, then, is what you see if you look at the shadow of something moving in a circular path in sunlight, in a plane parallel to the sun's rays. If we call x (and -x) the direction of motion, and let x be a maximum (x = A) when t = 0, then

x = Acosw t

v = -Aw sinw t

a = -Aw 2cosw t

If you know calculus, you can easily verify these: v is dx/dt and a is dv/dt. Also, we could make x = 0 at t = 0, then
x = Asinw t,
v = Aw cosw t and
a = -Aw 2sinw t.

Or for some problems it may be necessary to make x = something other than 0 or A at t = 0, so in general,
x = Acos(w t + f ), where f is an angle in radians called the phase angle.

Newton's second law will tell us the conditions needed for SHM. First, note that x and a are proportional:

a = -xw 2.

Then replace a with S F/m according to the 2nd law, and we see that S F needs to be proportional to x, and the negative sign means that if x is to the right, S F is to the left.

The simplest example is a mass attached to a linear spring, a spring for which we can write spring force f = -kx, where k is a constant. (This is known as Hooke's law.) Then f = ma becomes
-kx = -mxw 2, so we find that

w = (k/m)1/2. And if you prefer to deal with T, the period, use
T = 2p /w. Incidentally, if the spring mass is a significant fraction of m replace m with m+ms/3. The way to see that this is so makes use of calculus: Note that when m has speed v, the parts of the spring have speed that vary linearly from v to zero along the length of the spring, so write vs = (v/L)x, where L is spring length and x is distance from the stationary end. The kinetic energy of a portion of spring is dK= 1/2mvs2dx, where m= mass per length, ms/L. Combining these equations,
dK= 1/2 (ms v2 x2/L3) dx. Integrating from zero to L, we find that kinetic energy of the spring is K = 1/2msv2/3.  Maximum PE = maximum KE , or
1/2kA2 = 1/2(m+ms/3)vm2. w is vm/A, and if you do the algebra, you get w = (k/[m+ms/3])1/2  .

If a mass is hanging on a spring, the gravitational force must be considered, yet it can be shown that we get the same w.

Similarly we can find w for the *simple pendulum, the *physical pendulum and the *torsion pendulum. (* = you probably need to know it.)

Waves

In wave motion, energy is transmitted from hither to yon without matter being transmitted. (An exception to this, matter waves, is found in quantum mechanics.) As an example, give the end of a garden hose a rapid up and down shake. A wave pulse will travel along the hose like this:

If this pulse is traveling to the right, then on the pulse left of the peak the hose is moving down and on the right of the peak it is moving up. Someone holding the right end of the hose will receive a little “kick,” so energy has been transferred, although no material moved from left to right.

One cool thing about waves is that one equation that fits all, whether we are dealing with waves on a string or hose, sound, or light. Consider a continuous sine wave traveling on a horizontal string. If you fix your attention on one point on the string (x constant), you will see that point oscillate up and down with SHM, and, borrowing the language of calculus, its
velocity is y/ t
(The means small change of, with constant x. It should look like a curly d.) and its
acceleration is 2y/ t 2.

Similarly if you take a snapshot of the wave, you see how y changes with changes of x, while holding t constant. The analogous quantities are
y/ x and 2y/ x 2. ( still means small change of, but now with t constant.) It can be shown (using calculus) that if the change of slope per x distance is proportional to the acceleration
(i.e. 2y/ x 2 is proportional to 2y/ t 2), a wave will be propagated if given a start, and the constant of proportionality is the square of the speed of the wave (or the inverse of v2, depending on how you write the equation-- figure it out by the units).

The reason that
2y/ x2 is proportional to 2y/ t2 in a wave traveling in the x direction is that if you plot a graph of y vs x at a particular time and y vs t at a particular x, they will have the same general shape. It is also certainly true that the slopes,
y/ x and y/ t, are proportional, but we get our equation from f = ma, so we use the 2 types.

Thus if F 2y/ x 2 = m 2y/ t 2, the wave speed is
v = (F/m )1/2. We will show that this is indeed the case for a low amplitude wave on a string with tension F and mass/length = m . OK *whip that sucker over to me. Speaking of whips, the loud crack of a bull whip is the tip breaking the sound barrier.

You might want to look into *standing waves on a string, especially if you are into stringed musical instruments.

For compressional waves on a spring of length L, spring constant k, and mass m, we will show that the speed is
v = (k/m)1/2L, again with the one equation fits all. (Note that the time it takes for a pulse to go from one end of the spring to the other,
L/v, is (m/k)1/2, a constant. Stretch it to twice the length and the pulse obediently goes twice as fast.)

For a sound wave we will show that if s is the displacement of a molecule from its home position,
B 2s/ x2 = r 2s/ t2, where B is the bulk modulus and r is the density, so the velocity of sound is
v = (B/r )1/2. Also we will show that B/r can be written as
g RT/M, for sound in a gas with ratio of specific heats g=Cp/Cv , gas law constant R, absolute temperature T and molecular mass M. There is calculus involved with the derivation
of v = (B/r )1/2 = (g RT/M)1/2. Try to get in touch with your inner genius, and you will understand this stuff. (Last time I tried that, I ran into my inner dumbth.) OK hit me with these sound bites.

For longitudinal waves in a rod, use Young's modulus instead of the bulk modulus. Take an aluminum rod, hold it between two fingers at the center, and tap on the end with a hammer. Repeat, holding it halfway between the center and one end. That tone is an octave higher.

Then look into *sound standing waves in a tube (tooting a flute, for example). And you might think that blowing over the top of a beer bottle is the same kind of thing, but try it, and you will find that the frequency is much lower than what you would expect from the theory of the standing waves in a tube. So what's the deal? It is a Helmholtz resonator- a little tube connected to a large cavity, and it acts like a mass on a spring. There are caves (called breathing caves) that act this way. A shift in the wind can set them oscillating, and the period of oscillation might be an hour or more for a large cave.

Recall the wave equation, 2y/x2 = (1/v2)2/t2.  For a light wave replace y with E or B, electric or magnetic field (both are in the wave), and from E&M theory we show
2E/ x 2 = m o e o 2E/ t 2, and similarly for B, so the
speed of light c = 1/(m o e o)1/2.Want the derivation? OK beam it up to me, Scottie.

In SI units, E = cB.  It can be shown that if EM waves are absorbed, the pressure on the surface is I/c where I is intensity, power/area.  If reflected, pressure = 2I/c.

Now we shed some light on *Snell's law, n1sinq1 = n2sinq2. This is the mother of refractive optics (lenses, mostly). It turns out that one equation fits all mirrors and thin lenses: 1/do + 1/di = 1/f. (1/Object distance+1/image distance=1/focal length.)
*mirrors and thin lenses, and what to do about thick lenses. Armed with lenses, we can build telescopes and microscopes.

Then interference, using light, microwaves, sound, or whatever. The concept of superposition is a fancy way of saying that when two or more waves intersect, get in the right additude, dude, and add them. A useful tool is the *phasor diagram.

The way the eye sends signals to the brain results in what is known as color theory. White is a mixture of red, green and blue, for example. That is a property of the eye-brain team, not a property of light. Get a powerful magnifier and look at the white on your computer screen, and you will see the red, green and blue dots.

The *double slit is the grandmother of the *grating, which is an important device for measurements. For the double slit and grating, you need to know about
dsinq = ml. The single slit is actually more difficult conceptually! Take a grating and let d approach zero and the number of slits approach infinity, and voila! We have a *single slit. We use *phasors to work with these things.

Then we get into *photons, and we find that light is a particle as well as a wave. A pulse of light must have energy equal to a whole number multiple of the photon energy hf (a photon is a quantum of light; h is Planck’s constant and f is frequency), and in that sense it acts like, or is, a particle. A possible Nobel Prize is in store for you if you can find a fraction of a photon.  More quantum capers on my main quantum page (see below).

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composed by fredrick.gram @tri-c.edu where you can ask questions or tell me things or tell me off.