Waves A mechanical wave is a traveling disturbance in a material.

Waves A mechanical wave is a traveling disturbance in a material. It is the disturbance that travels, not the material. A given particle of the material moves only slightly—“in place;” the disturbance temporarily displaces it from its original position—it is an oscillator around a fixed point. And it, in turn, disturbs its neighbor; and so on.… So the motion of the wave is very different than the motion of an individual particle in the wave. And it is the energy of the particle’s motions—not any particle itself—that is carried along at the speed of the wave’s travel. OSU PH 212, Before Class #22

Some waves pass quickly; their peaks and valleys happen only once (or a few times). Most of the significant waves in physics (including those we will talk about here) have many repeating peaks and valleys. These are periodic waves. In fact, most waves of interest in physics are certain kind of periodic wave: a sinusoidal wave—a wave that can be described by a sine or cosine function, because each point in the wave is an oscillator undergoing Simple Harmonic Motion. So unless otherwise stated here, when we say “waves,” we mean periodic, sinusoidal waves. OSU PH 212, Before Class #22

What’s Waving? Mechanical waves must travel through some material medium. These don’t exist in the absence of matter. Electromagnetic waves are an oscillation of electric and magnetic fields and, since these fields can exist in a vacuum, these waves can travel through a vacuum. Matter waves are those of really tiny particles of matter – electrons for example. In quantum mechanical world, particles have wave properties! (What’s “waving?” The probability of its presence at a given location at a given time.) OSU PH 212, Before Class #22

OSU PH 212, Before Class #22

Transverse Waves In transverse waves, each particle in the material oscillates in a direction perpendicular to the wave’s travel. Examples: Guitar strings EM waves* “Stadium” waves “Ripples” through solids—e.g. earthquake (shear) waves On liquid surfaces—but not within fluids. OSU PH 212, Before Class #22

Longitudinal Waves • In longitudinal waves, each particle in the material oscillates in a direction parallel to the wave’s travel, producing a series of condensations and rarefactions, resulting in a pressure wave in fluids and solids. • Examples: • Sound in air, water, and earth (earthquake pressure waves) • Springs (e.g. “Slinky) • In other, more complicated waves, each particle of the material has components of motion in directions both perpendicular and parallel to that of the wave’s travel. Water waves are an example of this. OSU PH 212, Before Class #22

Sound Waves OSU PH 212, Before Class #22

Wave Math We can’t really describe a physically moving wave with a stationary graph. After all, its appearance changes with time (t). Moreover, the appearance (the oscillation position) of a particle depends also on which particle you look at—its position(x) in the wave. Suppose you have such a wave and you just stare at one particle—at one x-position—while the wave goes by. The time graph of the particle’s oscillation could look like y = cos(t) or y = sin(t) Or, suppose you “freeze” the wave—take a photo of all oscillating particles— at one time. How would the pattern of their phases look as you stroll along beside the frozen wave? It could be described by y = cos(x) or y = sin(x) OSU PH 212, Before Class #22

So to describe the wave in general, we need to allow for both variables that affect it: y = cos(x + t) With this formula, you can “see” the wave’s “peaks and valleys” either because you’re looking at different positions along the wave, or because you’re looking at one position at different times—or both. Of course, we need to include more information about a wave than just y = cos(x + t) in order to describe it accurately. For example, a cosine function varies only between 1 and –1. But what if the wave’s particles have a maximum displacement of 3.0 m? Or 1 x 10-6 m? We need a coefficient to adjust this maximum displacement properly: y = Acos(x + t), where A is the maximum displacement—the amplitude. OSU PH 212, Before Class #22

So we have the wave’s basic “height” calibrated. Now, what about the its shape? If you freeze it in time and look at its shape, is it stretched out and elongated—say, one full wavelength (one peak and one valley) in a kilometer? Or does it have many wavelengths in a single millimeter? And how about its period? If you let it travel by you and stare at one particle in one position along the wave, how long does it take that particle to go through one full displacement cycle (back-and-forth once or up-and-down once)? A minute? A year? A nanosecond? How do we fix these things? OSU PH 212, Before Class #22

If we simply use y = Acos(x + t) to describe the wave, we are saying that its wavelength is 2 meters and that its period (the time of one particle’s displacement cycle) is 2 seconds. Why? Because the cosine function completes a cycle every time its argument (the number inside the parentheses of the function) is a multiple of 2. But our wave has some other wavelength, , and some other period, T. We must adjust the math description so that: At x =  meters, the position portion of the argument must complete a multiple of 2. At t = T seconds, the time portion of the argument must complete a multiple of 2. In other words, we need this: y = Acos[(2/)x + (2/T)t] OSU PH 212, Before Class #22

Note: The period, T, is the time it takes for a particle’s motion cycle to complete once. The inverse of T is the frequency, f (and has units of sec-1): f = 1/T So we can describe a wave mathematically in either of two ways: y = Acos[(2/)x + (2/T)t] or y = Acos[(2/)x + (2f)t] Note also: The above formulas describe a wave moving in the negativex-direction. For a similar wave moving in the positivex-direction, use these: y = Acos[(2/)x – (2/T)t] or y = Acos[(2/)x – (2f)t] OSU PH 212, Before Class #22

Finally, notice that there is a very convenient relationship between the two values, f and : A wave is a traveling disturbance, and that travel has a speed. What do we mean by that? We mean that if a wave is traveling past you at a speed of v, it means that the wave peaks (or any other corresponding points on successive cycles) are passing you at that speed. In other words, one wavelength ( meters) is going past you in one period (T seconds). So the speed, v, of the passage of each such wave peak is just /T m/s: v = /T But recalling that f = 1/T, (and it’s often more convenient to measure f than T), we can also calculate v this way: v = f· OSU PH 212, Before Class #22

What is the frequency of this traveling wave? • 0.1 Hz • 0.2 Hz • 2 Hz • 5 Hz • 10 Hz OSU PH 212, Before Class #22

A summary of traveling wave math For a traveling wave, moving in the positive x direction (to the right), the displacement of the medium as the wave goes by can be written: D(x, t) = Acos(kx – ωt + φ0) Or: D(x, t) = Asin(kx – ωt + φ0 + p/2) where ω = 2πf = 2π/T and k = 2π/λ φ0 = initialphase angle (at x = 0 when t = 0) k is the wave number, the reciprocal of the spatial period (the wavelength) times 2π. The wave speed v = λf = ω/k OSU PH 212, Before Class #22

Put it all together. Build a traveling sinusoidal transverse wave of vertical displacement that fits this description: It travels in the –x direction at a speed of 12 m/s. It has an amplitude of 1.4 m and a period of 0.5 s. When t = 0, the displacement at the origin is zero. A.y = (1.4)cos[(/3)x + 4t – /2] B.y = (1.4)cos[(/6)x – 2t – /2] C.y = (1.4)cos[(/3)x + t + /2] D.y = 2cos[(/3)x + (1.4)t + /2] E. None of the above. (Are there other possible solutions—something not listed above?) OSU PH 212, Before Class #22

Wave Speed for Traveling Wave • Important: The wave speed depends on the medium. If the medium is unchanging, the speed is constant, and therefore: Frequency and wavelength are inversely proportional to each other. • Also important: Even if the medium does change, the frequency of the disturbance (which, after all, originated somewhere else and is just being passed along) does not change, but wavelength does. OSU PH 212, Before Class #22

Speeds and spectra of sound and light • Speed of sound in air = 343 m/s (in 20O C air) • vair = (331.3 + 0.61OC–lT) m/s where T is temp C • Sonic spectrum >0 Hz to 109 Hz • Sound spectrum 20 Hz to 20 kHz • Infrasonic vs. ultrasonic • Frequency determines pitch • Speed of light in vacuum = 3.00 x108 m/s • Electromagnetic spectrum > 0 Hz? To 1023 Hz. • Visible light spectrum 400 to 790 THz (1012) • - Frequency determines color OSU PH 212, Before Class #22

Wave speed on a string In a stretched string (a guitar string perhaps), the speed of a transverse wave depends on the tension of the string, Ts, and the linear mass density, μ. μ = m/L A fatter string has a higher μ than a thinner string. OSU PH 212, Before Class #22

Extra Material needed for Lab 8 The rest of the slides in this set are a summary of what you’ll need to know for Lab 8. Please read also sections 33.1-33.4 in your textbook. OSU PH 212, Before Class #22

Interference: Demonstrating the Wave Nature of Light Long before James Maxwell worked out the math to connect electric and magnetic fields to light, the diffraction behavior of light had been observed (and it’s the same reasoning as for sound or water or any other disturbance, though the opening must be much smaller than for sound). So it was well understood that light was some kind of wave, because it could “bend” around obstacles. Indeed, this “bending,” called diffraction, is demonstrated clearly with the help of interference patterns. How does this work? OSU PH 212, Before Class #22

Diffraction As we have seen, a wave is a traveling disturbance in a material. Each particle disturbed then behaves like the original disturbance—thus disturbing its neighbors, and so on. But notice that any such disturbance disturbs its neighbors in all directions—without preference. For example, sound waves are pressure waves; given a chance, pressure pushes in all directions equally—spherically symmetrical patterns. Why, then, do wave fronts form? Because the speed of the disturbance is a property of the medium, all neighbors that are equidistant from the original disturbance “get disturbed” at the same time—so the subsequent disturbances that they send to one another essentially cancel out (sum to zero)—so that the “net” disturbance turns out to be only in directions away from the very first point of disturbance: radially outward. OSU PH 212, Before Class #22

Now what happens when those wavefronts encounter an obstacle—some material that does not get disturbed (and subsequently “become a disturber”) like the particles in the medium? Suddenly, the wavefront is interrupted; it has an “end”—where the last particle on this end no longer has neighbors that can be disturbed on that side. This is true, of course, only while the wavefront is passing that obstacle. After that, the last particle again has “disturbable” neighbors in all directions, but this time the ones on the outside (where there used to be an obstacle) are now undisturbed—so there is no “canceling” disturbance coming from them—so the wavefronts are free to move in that direction. Thus the wave directions “bend” around the obstacle. OSU PH 212, Before Class #22

Look at the outermost points of disturbance in the diagram below. The row of disturbance sources represent points on a wavefront that are just now emerging from an opening (the obstacles on the left and right) as the wavefront advances (up the page here). So the sum of all their disturbances are now free to propagate as wavefronts toward the sides as well as in the direction of the original wave front. This effect is diffraction. OSU PH 212, Before Class #22

All of this is due to the fact that every point in a wave become a new source of disturbance. Look more closely here at the summing of all the overlapping wave fronts. This is Huygens’ Principle: “Each point of a wave is the center of a fresh disturbance—the source of a new wave. And its disturbance is the sum of all the waves arising from the points in the medium already traversed.” OSU PH 212, Before Class #22

The point is, the wavefronts from these various “doorway” particles —individual sound or light sources that are distributed across the width of the aperture (“doorway”)—overlap in a regular pattern of constructive and destructive interference. If you stand somewhere in the vicinity of the aperture, you will observe a certain level of intensity (loudness for sound, brightness for light); move elsewhere and you can detect a change in intensity as you cross through zones of alternating constructive and destructive interference. These bright/dim or loud/muffled places of alternating interference can be observed and measured easily. In fact, the pattern has a characteristic angle, called the dispersion angle, , that marks the first minimum (the first “dim” or “muffled” location—destructive interference) on either side of the wave’s original path. OSU PH 212, Before Class #22

Light from a single source moves toward an opaque barrier which has only one thin rectangular opening. OSU PH 212, Before Class #22

OSU PH 212, Before Class #22

For a rectangular aperture (a slit) of width a, this dispersion angle for light is given by sin  = /a. Notice that in each of these relations,  becomes significant only when  becomes comparable to the aperture (a ). And what happens when  is greater than a? And when a is much greater than ,  becomes very close to zero; the first minimum is pretty much “straight ahead.” (Why, then, don’t we notice a diminished intensity directly in front of a large aperture? Because the separations between adjacent minima and maxima are so tiny.) OSU PH 212, Before Class #22

What is the angular width of the central maximum created by an opening 1.00 μm wide if the wavelength of the light is 500 nm? • 0.115 degrees • 0.200 degrees • 2.00 degrees • 30.0 degrees • 60.0 degrees • none of the above OSU PH 212, Before Class #22

An early demonstration of the wave nature of light (notable because it also allowed good measurement of wavelength) was Young’s double-slit experiment. Light from a single source (a narrow beam) was sent through two adjacent slits onto a wall. The result was a pattern of light and dark fringes, indicating that the waves from the two slits were interfering with one another (either constructively or destructively, depending on the difference in the distances they had traveled). And there were brighter and more defined secondary minima (and maxima). OSU PH 212, Before Class #22

Light from a single source moves toward an opaque barrier which has two thin rectangular parallel openings. The barrier creates two distinct light waves which are close together and in-phase. Place a viewing screen on the other side of the barrier from the source. Orient the barrier and the screen so that they are parallel to each other. On the screen there will be a series of bright and dark bands called fringes which result from the interference of the two waves. The fringes are collectively known as an interference pattern. OSU PH 212, Before Class #22

From the geometry of the path-length differences, it was observed that for the pair of slits separated by a distance d: The angles  formed by the bright fringes are: sin  = m/d, where m = 0, 1, 2, … The angles  formed by the dark fringes are: sin  = (m + 1/2)/d, where m = 0, 1, 2, … OSU PH 212, Before Class #22

Example: In lab, Sally Sue sets up the laser, the double slit, and the screen such that the screen is 2.00 m from the slits. On the screen, she measures the distance from the central maximum to the 5th bright fringe to be 3.60 cm when the slit separation is 0.175 mm. Based on Sally’s data, what is the wavelength of the laser? (see After class 22.) OSU PH 212, Before Class #22

More examples: Suppose you send red light ( = 664 nm) through a single rectangular slit that is 2.00 x 10-6 m (2 µm) wide, then displayed on a screen. At what angle would you expect to see the bright central section go dark (i.e. the angle of that first big minimum)? Suppose you send red light ( = 664 nm) through a pair of slits that are separated by a distance of 1.20 x 10-4 m (120 µm). At what angles would you expect to see light and dark fringes displayed on a screen? (see After class 22.) OSU PH 212, Before Class #22

Multiple-Slit Diffraction: A Diffraction Grating We know how single-slit diffraction works. And Young’s experiment use two slits. How about more than two—how about many hundreds or thousands of slits? The overlapping is more complicated, but the idea is just the same: When waves from all the individual sources arrive at certain points in phase, we see a bright maximum; if they’re out of phase, we see a dark minimum. The geometry gives results much like Young’s double-slit case: Bright maxima occur at angles  where sin = m/d, with m = 0, 1, 2, etc., and d is the separation between adjacent slits. (There is a notable difference in the multi-slit pattern though: The bright areas are narrower and sharper than in the double-slit case.) OSU PH 212, Before Class #22

Diffraction Gratings OSU PH 212, Before Class #22

Why do we care about diffraction gratings? X-ray diffraction allows us literally to “map” the arrangements of atoms in many solids. Also, we use gratings to sort out the various wavelengths of light we’re receiving from unknown sources. The science of measuring the various emitted wavelengths of light from atoms or molecules is called spectroscopy. Identifying chemical composition from emitted light is spectral analysis. Light emission (from an atom for example) is dependent on the electron configuration of that atom, and every element has a unique electron configuration, so every element has its own unique, characteristic spectrum. This works across even great distances—our universe! Transmission versus reflection gratings…. OSU PH 212, Before Class #22

Example: A diffraction grating is illuminated with yellow light at normal incidence. The pattern seen on a screen behind the grating consists of three yellow spots, one at zero degrees (straight through) and one each at ±45°. You now add red light of equal intensity, coming in the same direction as the yellow light. The new pattern consists of … 1. red spots at 0° and ±45°. 2. yellow spots at 0° and ±45°. 3. orange spots at 0° and ±45°. 4. an orange spot at 0°, yellow spots at ±45°, and red spots slightly farther out than ±45°. 5. an orange spot at 0°, yellow spots at ±45°, and red spots slightly closer in than ±45°. OSU PH 212, Before Class #22

A Diffraction Summary Single slit (rectangular) of width a: Large central bright fringe, then dark/light alternating fringes, with angles for dark fringes given by sin  = p/a, p = 1, 2, 3, …. Between each pair of dark fringes is a bright fringe—but it’s fainter than the central fringe. Double-slit; slits separated by a distance, d: Bright fringes form at each  given by sin  = m/d, where m = 0, 1, 2, …. Dark fringes form at angles given by sin  = (m + 1/2)/d, where m = 0, 1, 2, … Many slits (a grating); adjacent slits separated by d: Bright fringes form at ’s given by sin  = m/d (m = 0, 1, 2…) Fringes are sharper than in the double-slit case. (For X-ray ’s, many solids are natural gratings.) OSU PH 212, Before Class #22

Waves A mechanical wave is a traveling disturbance in a material.