Chapter 16: Wave Motion

 

We start Ch. 16 in the middle of the Monday, April 23 lecture.

 

We see ocean waves, hear sound waves, see light waves, and receive radio waves.

 

In this chapter we learn about mechanical waves that are the vibrations of materials. The best examples are musical instruments. Stringed instruments (guitars, violins, pianos, etc) have strings that vibrate and these cause the air around them to move which creates sound waves.

 

Kinds of waves:

1. Mechanical: these cause the substance they are in to vibrate

Sound waves

Earthquake waves

2. Electromagnetic: these move through vacuum (no substance!) and are oscillations of electric and magnetic fields

Microwaves (inside of ovens)

Radio waves (radio, TV, cell phones, etc)

3. Quantum mechanical waves (electrons in atoms, for example)

 

Descriptive terms

 

Standing Transverse Waves (guitar strings):

Imagine a wave on a guitar string. The drawing below shows one position of the string as a solid line and another position of the string as a dotted line. The points on the string oscillate up and down.

 

The peak-to-peak amplitude is from the top of the oscillations to the bottom. The equilibrium position of the string is the straight line in the middle. The amplitude is from the equilibrium position to the maximum.

 

The string is tied down at the ends so the end positions do not move. On the string shown here, there is one wavelength so there is an additional place (a node) in the middle where the string does not move.

 

The peaks and nodes stay at the same place (do not move along the string) so this is called a standing wave.

 

The up and down oscillations can be described by a frequency f in cycles per second (Hertz) or w=2pf, radians per second. The distance that encompasses one "up and down" is one wavelength, l.

 

This wave is called transverse because the mechanical motion is perpendicular to the direction along the string.

Every point on the string is oscillating up and down the way a mass on the end of a string would oscillate (except at the nodes where the local amplitude is zero).

 

Traveling Transverse Waves:

 

The figure at left shows a wave moving along the surface of the ocean. The three waves show the wave at t = 0 and two later times. The wave is moving to the right. The vertical axis represents the height of the wave. From top to bottom (also called "crest" to "trough") is the peal-to-peak amplitude.

 

If I sit in one place and watch the number of wavelengths that go by in a second, that is the frequency, f.

 

Above is a plot of the y-displacement as a function of x and is a sine wave. One snapshot looks like y(x) = A sin Cx, where C is a constant.

If I am standing on a fishing pier, and can run along keeping up with a wave crest, that is the speed of the wave. The ocean wave is called a traveling wave because the crests are moving unlike the standing wave where the crests stay in the same place.

 

Examples of traveling transverse waves are ocean surface waves, electromagnetic waves, and some earthquake waves.

Longitudinal waves (traveling or standing):

 

These waves are easily illustrated with a slinky spring. Stretch the spring and then oscillate one end parallel to the direction of the spring. The wave moves down the spring as is usually traveling. The wave is called a longitudinal wave because the mechanical motion is in the direction that the wave is moving (along the spring). (It is also possible to make a transverse wave on a Slinky.)

The sound wave is a longitudinal wave and some earthquake waves are longitudinal.

What the Slinky longitudinal wave looks like:

 

Parts of the spring are compressed (closer together) and parts are decompressed (further apart). If I paint a spot on the Slinky and plot its position as a function of time, it looks like a sine wave.

The Slinky itself looks like a sine wave but that is not the wave motion we are talking about.

 

This plot (at left) is x-displacement as a function of x and is a sine wave. One snapshot looks like Dx(x) = A sin Cx, where C is a constant. In this formula x is the original equilibrium position, and Dx is how far the point has moved from this position.

 

 

 

 

Pulses

 

If you hold the end of a rope and jerk the end back and forth once, it launches a single "ripple" down the rope. This is called a pulse. A single back and forth pulse is sometimes called a solitary wave or a soliton.

 

The solid line shows the wave at t = 0 and the dotted line shows the wave a little later. The arrows show the direction of the material carrying the wave which is transverse. Solitary pulses are traveling waves.

 

Solitary pulses are easily observed in shallow water like in a bathtub. Place both hands in the water, palms forward and push once. A pulse of water will move away from your hands.

 

Wavetrains

 

A continuous wave is one that continues forever (no beginning and no end). It is more common for oscillations to start and stop. The figure shows a wavetrain with a gradual beginning and end. This might describe a child in a swing that starts to swing then gets bored and stops. It could also describe the waves from a distant earthquake.

 

 

The math

 

The time dependence:

 

At a fixed location: Displacement = A sin (wt + f).

 

A is amplitude, w is frequency in radians/second, and f is a starting phase. This also applies to the simple harmonic oscillator.

Displacement could be longitudinal or transverse, depending on the kind of wave.

The amplitude of the wave is usually measured by an observer at rest. This observer sees a displacement that varies sinusoidally with time.

 

The spatial dependence:

 

For the vibrating string or rope we have to take a snapshot that freezes the motion. The snapshot looks like:

 

Displacement = A sin kx = A sin (2px/l).

 

where k = 2p/l is a new constant called the wavenumber (NOT the spring constant). l is the wavelength.

The combination kx should take the sine function through two pi radians every time we go a wavelength. So when x = one wavlength l, the combination kx is equal to 2p. Notice that x/l is the number of wavelengths in a distance x.

The units of k are 1/meters, or m-1, or /m (pronounced "per meter").

 

Time and space dependence of standing waves: (section 17-6 in the textbook)

 

If we put together the time and space dependence, we get for standing waves:

 

Displacement = A sin (kx) sin (wt).

 

The two sine functions for the x and t dependence multiply one another. This is the combination that "works". Adding the sine functions doesn't work. The A sin (kx) part tells me that locally the amplitude of oscillations is A sin (kx) and the sin (wt) part says that everywhere the string is moving up and down with frequency w.

 

The nodes in the standing waves are places where the amplitude is zero. For musical instruments, the string cannot move at the ends where it is attached. So the two ends are automatically two of the nodes. If one wavelength fits on the string there is a node also in the middle. So we could have a wave with only half a wavelength on the string. The figure shows waves with 1/2, 1, and 1 1/2 wavelengths.

 

We can express this mathematically. For string of length L

L = m l /2, where l is the wavelength and m is an integer. The length of the string is a whole number of half wavelengths. The wave with m =1 is called the fundamental and the others are called the harmonics. When you pluck a guitar string, you get the fundamental and some of the harmonics all occurring together.

 

Time and space dependence of traveling waves: (section 17-6 in the textbook)

 

The traveling wave is a sine wave that is moving horizontally. If I move along with a traveling wave, it is NOT oscillating up and down. Imagine that you are surfing, traveling along with the ocean wave. Is it oscillating up and down under your feet? (no!).

 

What we need to do to convert the simple sine wave sin (kx) to a traveling wave is make a change of variables:

Xnew = x - vt. The variable Xnew is "traveling" at velocity v and the wave looks stationary:

 

Displacement for traveling wave (tentative): A sin k(Xnew) = A sin [k(x-vt)].

 

The time dependence is sin (kvt). But we already know the time dependence is sin wt.

For this solution to "work" it must be true that kvt = wt, then w = kv, and then

 

Velocity of traveling wave: v = w/k.

 

Now I can eliminate the v from A sin [k(x-vt)], using the relation above

 

Displacement for traveling wave = A sin (kx-wt).

 

Other things that work:

This solution for the traveling wave is not the only one. The cosine can be used instead of the sine and I can add any starting phase. For example A cos[kx-wt+f] is a solution and has a starting phase that is not zero.

 

Exercise: prove to yourself that the wave A cos[kx+wt+f] is traveling in the -x direction.

 

Relation of speed, frequency, and wavelength:

 

Above we found that v = w/k.

Recall the definitions k = 2p/l and w = 2pf. Use these in the above formula and get:

 

Speed-velocity-wavelength relation: v = f l.

 

Remember that a string can have half an integer number of half wavelengths, or:
L = m l /2, or: l = 2 L/m.

Using that f = v/l, we find that f = mv/2L. The different values of m tell us the frequencies of the different harmonics we can get from a string.

Speed of waves on a string:

 

The figure shows a pulse moving along a rope. Imagine that you are running along side of the pulse. There is a tension T in the rope. A point on the rope follows the arc of the rope that locally has a radius R. The centripetal acceleration of the point at the top of the pulse (the end of the arrow) is v2/R.

 

A little length of the rope dx has mass dm = (M/L) dx where dx is the length of the little bit of rope and M/L is the mass per unit length. This little bit of rope is pulled downward by the force at the end of the little length. These forces are the dotted arrows at angle q. For this little bit of rope we use F = ma where F is the force in the y direction and a is the y-acceleration which is v2/R.

Fy = 2 T sin q 2 T q.

Here I have used the small angle approximation for which sin q = q.

 

Let's consider a short length dx on both sides of center, then the angle
q = dx / R or dx = Rq.

The mass on both sides is 2 (M/L) dx So F = ma becomes

 

Fy = 2 T q = 2 (M/L) (dx) a = 2 (M/L) (Rq) (v2 /R)

 

Divide by 2 q to get: T = (M/L) v2, which can be solved for v

 

Speed of waves on a string: v2 = T/(M/L) = T/m

 

where m = M/L is the mass per unit length. Or, v = (T/m)

This derivation is "cool" because to get started we have to suppose some radius R and some angle q, and then at the end the R and q drop out.

 

When waves add: Beats and interference

 

Beats: the sum of waves in time:

What happens when I launch two sound waves?

The result is that they add. Two see the answer, consider two simple sinusoids y(x) = sin w1t and y(x) = sin w2t

The sum is y(x) = sin w1t + sin w2t.

The first sine wave has 28 peaks. The second has 32 peaks.

 

Where two peaks of amplitude 1.0 overlap, there is a big peak of amplitude 2.

However, where a peak of amplitude 1.0 overlaps a valley of amplitude -1, the SUM IS ZERO.

If this were a sound I was listening to, it would get louder and softer.

For example, if the frequencies were 400 Hz and 402 Hz, I would hear a sound with the average frequency 401 Hz but the

amplitude would increase and decrease once per second.

Why once per second? Because there is a trigonometric identity: sin A + sin B = 2 sin [(A+B)/2] sin [(A-B)/2].

The first sine gives the average frequency and the second MULTIPLIES the first causing the amplitude to go up and down. The ups and downs in the amplitude of the sum is called the beat.

This figure illustrates the sum of two waves in time:

 

Interference: the adding of waves in space

Consider two point sources (black circles in the figure below), like stereo loudspeakers, emitting the same sinusoidal sound waves into the air. If the two distances from the two speakers are the same, then the waves will add in-phase. If the distances are different by half a wavelength, the waves will be out of phase and will subtract. The pattern in the drawing below shows white where the sum of the waves is large and black where the sum is small. The strip at the right shows the variation down the right side of the figure. The pattern of "loud and soft" is called an interference pattern.

 

 

The condition for "loud" is that the distance D2 and D1 should be different by a whole number N of wavelengths.

 

Loud: D2 - D1 = Nl

 

The condition for "soft" is that the difference is different by (N + 1/2) wavelengths.

 

Soft: D2 - D1 = (N + 1/2) l.

 

 

We have skipped Wave Power and Intensity, section 16-5, and the Wave Equation, section 16-7.

We have jumped ahead and covered 17-6, Standing Waves.