CHAPTER 8: MECHANICAL WAVES TRANSMIT ENERGY IN A VARIETY OF WAYS DISCLAIMER FOR MOST QUESTIONS IN THIS CHAPTER Waves are always in motion, as they transmit energy and information from one point to another. When you sketch a wave or look at a photograph of one, remember that what you are showing is a single instant in time immediately after that instant, the wave will have changed and moved on. Remember WAVES MOVE! 8.1 The Properties of Waves EXERCISE 8.1 1. Compare how energy is transferred using matter to the way that energy is transferred using waves. (See page. 392.) Matter IS involved in two of the three waves identified on this page; matter that vibrates to carry wave energy is referred to as the medium. What is the medium for a sound wave? For an ocean wave? For a radio wave? (This could be a TRICK QUESTION!) 2. A common misconception about surfers is that they are carried into shore by the horizontal motion of water as waves rush up on a beach. Why is this not a good explanation? 3. Neatly sketch and clearly label the parts of a water wave, for three complete wavelengths. (The symbol λ is the Greek letter lambda; it stands for wavelength.) On your sketch, indicate two points that are in phase: indicate two points that have opposite phase. The height of the wave (above equilibrium) is called the wave s amplitude. Show the amplitude (a) on your sketch. Clearly label one complete wavelength on your sketch. 4. The sketch shows a water wave reflecting from a barrier in its path. How do the incident angle and reflected angles compare? (This is called the law of reflection for waves.) What name is given to the arrow that shows the direction of motion of each wave? Reflected angle Incident angle Reflected wave Incident wave PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 1
8.2 Transverse and Longitudinal Waves EXERCISE 8.2 1. Water waves at the surface are transverse; the wave travels across the surface of the water, while the water itself moves up and down, at right angles to the motion of the wave. Describe how you might move the end of a Slinky spring to generate a single transverse wave or wave pulse. 2. A second major type of wave is called a longitudinal wave. Describe how you might move the end of a Slinky spring to generate a longitudinal wave pulse. 3. Write a clear definition of transverse and longitudinal waves. 4. Sketch a longitudinal wave; label a compression and a rarefaction. 5. Both transverse and longitudinal wave pulses move through a variety of mediums. Generally, elastic solids (like the metal Slinky springs) can carry both types of waves. Liquids can carry transverse waves at their surfaces, but only longitudinal waves within the liquid. Gases can carry only longitudinal waves (which are sometimes called compression waves.) What type of wave do you suppose sound is? 6. If you ve ever been inside a building during an earthquake, you will know that the building can move up and down, and can also shake from side to side. What type of earthquake wave is likely providing each motion? (Transverse earthquake waves are called S-waves, and longitudinal earthquake waves are called P- waves.) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 2
7. What does the speed of a wave depend on? (p. 406) What is determined by the amplitude (usually shown by the letter a ) of a wave? 8. Explain the argument given on page 406 that ends with the equation l = v t. 9. How would you move the end of a spring to generate a periodic transverse wave? What would you change in order to change the frequency of this wave? To change the speed of the wave? To change the amount of energy carried by the wave? 10. The sketch below shows a periodic wave moving to the right along a coil spring. The wave is generated as the left end of the spring is moved up and down; the time for one complete up-and-down motion is 0.500 s. x v y 25.0 cm a) What is the wavelength of this wave? Label one complete wavelength on the sketch. (50.0 cm) b) How many wavelengths are generated by one complete up and down motion of the left end of the spring? c) How long does it take point x on the wave to reach point y? (1) (0.500 s) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 3
d) How fast is the wave moving? e) Use the motion equation v = d t (100 cm/s) to connect the speed of the wave, its period and its wavelength. f) Rewrite your equation from (e) using frequency. This is called the universal wave equation. 11. Periodic waves travel along a rope with a speed of 85.0 cm/s. The person creating the waves (by shaking the end of the rope) moves her hand back and forth once each 0.300 s. What is the wavelength of the resulting waves? (25.5 cm) 12. Waves are generated in a coil spring at a rate of 5 in 2.00 s. If the waves are 28.0 cm long, what is their speed along the spring? (70.0 cm/s) 13. The speed of sound in air is about 330 m/s. If the note middle C has a frequency of 256 Hz, find its wavelength. (1.29 m) 14. A sound wave traveling at 350 m/s has a period of vibration of 0.0250 s. Calculate the frequency and wavelength of the wave. (8.75 m) 15. Six ocean wave crests occur in a 29.0 m length. A cork in the water bobs 8.00 times in 6.00 seconds. Find the wavelength, frequency and speed of the waves. (5.80 m; 1.33 Hz; 7.73 m/s) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 4
8.3 Superposition and Interference Any medium that can carry a wave like air carrying sound waves and ripples on the surface of water, or waves traveling along a spring or rope can support the motion of more than one wave at the same time. When two waves meet, the medium must momentarily have the shape of the combination of both waves. The coils of a spring, for example, cannot be in two places at the same time, so that when two waves pass each other on the same part of the spring, the motion of the coils shows the sum of the two waves. Two or more waves passing each other in the same part of a medium are said to be superimposed; as they superimpose, they interfere (temporarily) with each other. The actual motion of the medium as the waves superimpose and interfere can be predicted using the principle of superposition. EXERCISE 8.1 1. Two superimposed sound waves have no lasting effect on each other. How does the example on page 413 (figure 8.26) show this? 2. Study figure 8.23 on page 411. Then sketch (on the dashed line, as accurately as possible) the superimposed wave for the two pulses shown below. What type of interference is shown? 3. Read and study page 412, then sketch the interference of the two pulses shown below. What type of interference is shown? 4. Refer to page 404. How does a wave pulse traveling along a rope or spring behave when it reflects from a fixed end? (Fixed means the end of the spring is restrained and can t move.) Draw a sketch. 5. Use the principle of superposition to explain your answer to the previous question (see page 413.) PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 5
6. Interference occurs with non-symmetrical waves as well as symmetrical ones. Accurate sketching of the medium as waves interfere can be done as illustrated in the example below. Use this technique to accurately draw the superimposed wave in each of a-e. (Note that for simplicity, many of the waves in this question have sharp corners. Real waves generally always have rounded corners, although they may have the overall shapes illustrated.) a) EXAMPLE Triangular wave Positive displacement caused by triangular wave Negative displacement caused by square wave Point on actual wave, found by graphically adding displacements; +2.5 4 = 1.5. Other points are found the same way. Square wave Note that a square wave when superimposed with another wave only moves the second wave vertically up or down. The shape of the second wave isn t changed by the square wave. b) c) Actual superimposed wave resulting from interference d) e) 7. Which of a-e in the previous question best show constructive interference? Destructive interference? 8. Superimposed waves that produce constructive or destructive interference can also be described using the term phase. Explain. PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 6
The discussion and sketches on pages 416-417 show what happens when identical periodic waves, traveling in opposite directions in the same medium, meet and interfere. The result is a standing wave; such a wave doesn t seem to move along a medium, but rather to vibrate about fixed points. Although the medium shows significant wave motion, the wave appears to be standing still. Standing waves occur in all media - in air, in water, for coil springs and ropes, and for large structures like the Tacoma Narrows Bridge. In musical instruments, standing waves represent the resonant frequencies of a stretched string in a guitar, the air column of a wind instrument, or a drum membrane (pages 422-424.) The lowest frequency for which a standing wave can occur in a given medium is called the fundamental frequency; for example, a guitar string vibrating at its fundamental frequency will demonstrate a half wavelength, with stationary points at either end and a point of maximum vibration amplitude at the centre. More than one standing wave may occur in a medium at one time. For musical instruments, these additional frequencies are called overtones. Drawing a standing wave in a rope, guitar string or coil spring (as shown at right) is done by showing both the wave trains, moving left and right and superimposing; of course, the spring can only be in one place at one time. Remember that these waves are moving, so in the sketch, only the points marked x are stationary, and along lines y, the rope or spring is flipping from side-to-side. y y y y y x x x x x 9. In the sketch above, what name is given to points labeled x? To points labeled y? 10. How does the distance between two x-points relate to the wavelength of the actual wave? 11. In the sketch above, the left end of the coil spring is moved up and down completely 10 times in 5.0 s. The distance between two adjacent x-points is 80.0 cm. Find the speed of the waves as they travel along the spring. (3.20 m/s) 12. Describe how the amplitude of a vibrating system changes when energy is added to the system at its resonant frequency; at some other frequency. PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 7
A sound can be amplified (made louder) through the resonance of a contained volume of air. All wind instruments such as the, clarinet, flute, trombone, and French horn make use of the resonance of a column or tube of air, which is set into vibration by the motion of the player's lips and (in some instruments) by a vibrating reed. Pipe organs also involve resonant air columns. Resonance in a tube of air is an example of a standing pressure wave. Superposition of the original wave and the same wave reflecting from the closed or open end of the tube results in regions of alternating constructive and destructive interference. A tube closed at one end will resonate (waves will undergo constructive interference) if the length of the tube is one-fourth of the wavelength (see p. 419.) A tube open at both ends will resonate if the tube length is one-half the wavelength (see p. 424.) As for other standing waves, additional resonances in either case occur if the tube length is increased by one-half a wavelength, by one wavelength, by one and one-half wavelengths, etc. TUBE OPEN AT ONE END Shortest resonant length : l = λ or λ = 4l 4 Additional resonant lengths : l = 3λ 4, 5λ 4, etc. (additional resonance for each extra length of λ 2 ) OPEN TUBE usually means open at both ends. CLOSED TUBE usually means closed at one end. TUBE OPEN AT BOTH ENDS Shortest resonant length : l = λ or λ = 2l 2 Additional resonant lengths : l = λ, 3λ 2, etc. (additional resonance each extra length of λ 2 ) NOTE THAT THE EXTRA LENGTH NEEDED FOR THE PIPE TO REACH ANOTHER RESONANCE POINT IS THE SAME FOR EITHER OPEN OR CLOSED PIPES: λ 2. EXAMPLES RESONANT AIR COLUMNS 1. A pipe open at both ends produces resonance at a length of 68.1 cm when used with a tuning fork of frequency 256 Hz. Assuming this is the shortest resonant length, find the speed of sound. Shortest resonant length is λ : 2 From v = fλ : λ = 0.681 2 λ = 1.362 m v = (256)(1.362) v = 348.67 m/s v = 349 m/s 2. At a certain temperature, the speed of sound in air is 335 m/s. What are the shortest two lengths of a plastic pipe, open at one end, which produce resonance when used with a 1024 Hz tuning fork? λ = v f λ = 335 1024 λ = 0.3272 m Pipe is open at one end, so l = λ 4 resonant length. is shortest l = 0.3272 4 l = 0.0818 m Next resonant length is an additional λ 2 : l 2 = 0.08179 + 0.3272 2 l 2 = 0.245 m PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 8
13. A student uses a tuning fork with a frequency of 1024 Hz to induce resonance in a length of plastic tubing with a movable plunger in one end. The plunger allows the effective length of the tube to be changed. What are the shortest two lengths of this tube for which resonance should occur, if the experiment is performed outside where the speed of sound is 334 m/s? 14. A 3.00 m long organ pipe open at one end resonates when air is blown against its opening. What is the frequency of the note produced if the speed of sound is 336 m/s? 15. A length of glass tubing with a movable plunger in one end produces resonance at points 29.1 cm apart. If this effect is noted in a summer-temperature room where the speed of sound is 349 m/s, what frequency of sound was used? 16. The spacing between points of resonance for a column of a mixture of gasses (open at one end) is 55.0 cm, using a 512 Hz tuning fork. What is the speed of sound in the gas mixture? 17. A tube open at both ends resonates at a frequency of 1200 Hz when the speed of sound is 341 m/s. What is shortest possible length for the tube? 18. A 1.52 m long length of pipe open at both ends resonates when used with a 450 Hz sound source. What is the speed of sound? PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 9
Two-dimensional waves, such as those shown on pages 427 and 428 and sketched at right, also show interference patterns. The analysis is for water waves in the ripple tank, but works equally well for other two-dimensional (or threedimensional) waves. In the sketch, the curved lines represent wave crests; the waves are created by two point sources S 1 and S 2, creating crests and troughs in unison. 19. Explain what interference is taking place at points I and II in the sketch above; use the term phase in your explanation. What name is given to points like point I? Like point II? I II S 1 S 2 20. Draw the central maximum and second order maximum lines on the pattern in the sketch above. What name is given to these maxima? 21. Draw the second order minimum line on this pattern. What name is given to this minimum? 22. Select any point on the second order maximum. How far (measured in wavelengths) is this point from wave source S 1? From wave source S 2? 23. Repeat the preceding question for a point on the second order minimum. PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 10
24. Generalize your results for the preceding two questions, using the term phase shift (as is done on page 427.) 8.4 The Doppler Effect EXERCISE 8.1 1. Relative motion between a source of waves and the point of detection of the waves results in an apparent change of frequency and wavelength called the Doppler effect. Explain how this happens; include a sketch. What happens to the apparent frequency of a sound source as it approaches you? As it moves away from you? What happens to the corresponding wavelengths? 2. State the general form of the Doppler equation; identify each variable. (You don t need to be able to derive this equation.) 3. A car with its horn blaring moves towards you at a speed of 15.0 m/s. The horn s frequency is 600 Hz. What frequency do you hear as the car approaches you? As the car passes and moves away, what frequency would you hear? Assume the speed of sound is 335 m/s. 4. The apparent frequency of a sound source is 450 Hz as the source approaches an observer at 25.0 m/s. If the speed of sound is 332 m/s, what is the actual frequency of the source? What is the actual wavelength of the source (the wavelength if the source were stationary)? 5. How is a shock wave created? 6. What is a sonic boom? PHYSICS 20N NOTES AND OUTLINE QUESTIONS CHAPTER 8 REVISED JANUARY 08 PAGE 11