Objectives. Applications Of Waves and Vibrations. Main Ideas

Transcription

1 Applications Of Waves and Vibrations Unit 9 Subunit 2 Page 41 Objectives 1. Describe what's meant by interference of waves. 2. Describe what's meant by "superposition of waves." 3. Distinguish between constructive and destructive interference. 4. Describe what's meant by a "standing" wave. 5. Identify examples of wave interference. 6. Define natural frequency of an object. 7. Define resonance. 8. Identify examples of wave resonance. 9. Identify workplace applications where technicians measure and control waves and vibrations. Main Ideas When two or more waves in a medium pass through the same position at the same time, the waves overlap and "interfere" with each other. When two or more waves in a medium interfere at a certain position, the overall wave displacement is equal to the sum of the displacements caused by the individual waves. This is called "superposition of waves." When two waves of equal frequency and amplitude are in phase, and interfere at a given position, they produce the same effect as that of a single wave of the same frequency but DOUBLE the amplitude. This is called "constructive interference." When two waves of equal frequency and amplitude are 18O~ out of phase, and interfere at a given position 1 they produce the same effect as NO WAVE at all. This is called "destructive interference." The natural frequency of an object is that frequency at which the object will vibrate on Its own, when set into motion. Resonance between a source of vibration and another object exists when the frequency of the vibrating source matches the natural frequency of the object. 1

2 Wave Interference When two or more waves in a medium pass through the same point at the same time, the medium responds to each wave. When two or more waves interfere, their overall effect is either greater or less than the effect of any one of the waves alone. If the overall effect is greater, the interference is called constructive interference. The overlapping of waves at a point is called interference. If the overall effect is less, it's called destructive interference. The Effects of Interfering waves Whenever two or more waves meet and overlap, the resulting effect equals the sum of the effects caused by each wave alone. What does this mean? The amplitude of the resultant pulse equals the sum of the two pulses (Figure 9-1 8b). Constructive interference has occurred. After passing each other, continue along unaffected by one another. The process of ADDING the two waves together algebraically when they coincide is called superposition. Figure 9-19 (a, b and c) shows two pulses of opposite amplitude approaching one another. Note that when they overlap, they cancel each other. The superposition leads to destructive interference at the point of overlap. After passing each other, the two pulses are restored and travel onward with the same shape as before. 2

3 The principles of interference and superposition also apply to continuous waves. Figure 9-20 shows two sine waves A and B that overlap. Wave B is drawn with long dashes and circles (0); Wave B with short dashes and crosses (+). Example 9-H: Constructive Interference of Two Sine Waves Given: The two overlapping sine waves, A and B, shown here at an instant of time. Find: Resultant wave. Solution: Add each pair of waves (A + B) to obtain the resultant. Do this by drawing several vertical guidelines, as shown in the second graph to the right. Mark a circle (0) where sine wave A intersects the vertical line. Mark a cross (+) where sine wave B intersects the vertical line. Add the vertical positions represented by the circle and cross. Indicate the sum as a solid dot (e). Draw a bold line through the solid dots. Sum A + B is a curve of greater displacement than either A or B alone, so we call it constructive interference. The two waves reinforce each other. Example 9-I: Destructive Interference of Two Sine Waves Given: The two overlapping sine waves, A and B, shown here at an instant of time. Find: Resultant wave. Solution: Add each pair of waves (A + B) to obtain the resultant. Do this, as in Example 9-H, by drawing several vertical guidelines as shown in the second graph to the right. Again. for each place where wave A intersects the vertical line draw a circle (0). Similarly mark a cross (+) where wave B intersects the vertical line. Add the vertical positions represented by the circle and cross and plot the sum with a solid dot (e). 3

4 Draw a bold line through the solid dots. Sum A + B is a horizontal line along the neutral position. The resultant wave's displacement is less than either wave A or wave B, and so this is destructive interference. In fact. at the instant of time shown here, the resultant has zero displacement. At this instant, the two waves cancel each other. STANDING WAVES Sound waves that reflect back and forth and produce a very useful pattern of interference called standing waves. The tones you hear from air blowing across the openings found in whistles, bottles, pipe organs, flutes, and other musical instruments like these are the result of standing waves. Sound waves from one end of the tube interfere with the sound waves reflected from the opposite end of the tube. When the wavelength of the sound waves and the length of the tube are properly matched, standing waves within the tube will cause rapid air movement at the open end(s) of the tube. Consequently, you will hear a loud tone from the tube. The frequency of the tone will be one of the natural frequencies of that tube. (The lowest natural frequency is generally the loudest. Ifs often called the fundamental frequency.) To understand standing waves, we need to examine these sums at several consecutive instants of time. You've probably noticed by now the tiny set of sine waves printed at the bottom edge of the right-hand pages in this text. You also may have discovered that you can "see" the motion of these waves by quickly but smoothly fanning through the pages from back to front. Each tiny graph shows the result of adding two sine waves to produce a resultant. Let's look more closely at one of these graphs, as shown below. Notice that three waves are shown. One is drawn with dashes the arrow beside it tells us that this wave is traveling to the right. (Fan through the pages again, starting at the back of the book, to see the dashed wave "moving" to the right.) Another is drawn with dots-the arrow beside it shows that it is moving to the left. (Can you see it moving when you fan through the pages?) Suppose these waves represent "ripples" traveling down a rope tied to a table. You could think of the dashed curve as the incident wave and the dotted curve as the reflected wavetraveling in the opposite direction. 4

5 The resultant curve doesn't appear to move left or right. There are several locations along the path of travel where the resultant is always zero. The amplitude of the resultant curve varies from a maximum (twice the amplitude of either traveling wave) to a minimum (near zero). The frequency and wavelength of the resultant are the same as those of either the incident wave o the reflected wave. These locations are known as nodes. Conversely, there are several points where the resultant swings rapidly from a maximum positive displacement to a maximum negative displacement. These are known as antinodes, or points of maximum movement. FORCED VIBRATIONS The vibrations of a ship's propellers are transmitted to the turbine in a ship. The vibrations can cause misalignment in the close fit of the turbine parts. The turbine can be damaged. The "pounding" of a heavy air compressor in a building can be transmitted by waves to delicate electronics in a nearby laboratory. Because of the vibrations, electronic devices can malfunction. In both of these examples, the propeller vibrations and air compressor vibrations are called forced vibrations. Forced vibrations can be either good or bad. For example, many musical instruments use vibrating strings or reeds to produce a sound. Forced vibrations in mechanical systems usually are unwelcome. That's because forced vibrations coming from other sources cause the machine to vibrate at unwanted frequencies. Rubber mounts or shock absorbers are used to isolate mechanical systems from forced vibrations. Shock absorbers and rubber mounts absorb vibrations and protect the machinery. Rubber mounts are checked frequently. NATURAL FREQUENCY OF AN OBJECT The natural frequency of an object is that particular frequency at which the object vibrates when set into motion and left to vibrate or oscillate on its own. Think about a child's swing. If the swing is set in motion, it moves back and forth at a certain frequency. If you stop the swing and then start it again, it moves back and forth at the same frequency. That's its natural frequency. 5

6 FORCED VIBRATION and NATURAL FREQUENCY The Tacoma Narrows Bridge disaster is a classic example of what happens when an outside force sets up a motion that matches the natural frequency of an object. In this famous (but tragic) example, the blowing wind caused the bridge to twist along its entire length. The back-and-forth twisting motion matched the natural frequency of the suspended bridge. As the blowing wind gave more and more energy to the torsional motion of the bridge, the bridge twisted back and forth with larger and larger amplitude. When a forced vibration doesn't match the natural frequency of an object or mechanical system, very little energy is transferred from the vibration to the object. You can prove this by trying to pump a swing by pushing it at a frequency (forced vibration) that's different from the natural frequency of the swing. You'll find that you can't build up much amplitude in the swing. That's because your successive pushes are out of phase with the natural frequency of the swing. Finally, the bridge tore itself apart. When forced vibrations match the natural frequency of an object that vibrates, the object vibrates with increasing amplitude. Energy is transferred efficiently from the source of forced vibrations to the object. This condition is called resonance. Resonance is very important in mechanical systems. Usually technicians work to either avoid or control resonance in mechanical systems. Vibrations are almost always produced in machines when they operate. Structures also vibrate when forced vibrations are present. Vibrating machines, earthquakes, passing trucks, and sonic booms are a few examples of forced vibrations that can affect structures and machines. These vibrations don't usually represent a major problem-unless they coincide with a natural frequency of the machine or structure. 6

7 Any Questions? Complete: -Student Exercise, page 52 -Math Skills Lab, page 53 -Lab 9-3, Resonance of Sound Waves -Lab 9-4, Measuring Phase Diference 7