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Wavelength, Frequency, & Velocity (Pay close attention!) Packet #2 2008, new book

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WAVELENGTH The wavelength of a wave is defined to be the shortest distance between points where the wave pattern repeats itself. Notice that on a transverse wave, this is the displacement between two successive crests, between two successive troughs, or twice the displacement between two successive equilibrium points. Wavelength

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WAVELENGTH continued Notice that on a longitudinal wave, the wavelength is the distance between two successive compressions or between two successive rarefactions. Wavelength compression rarefaction

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FREQUENCY The frequency of a wave is defined to be the number of vibrations per second of a part of the wave. Since the unit Hertz means “per second”, frequency has the unit [Hz] or [sec -1 ]. In the picture below, the number of times the cork bobs up & back down again in one full second would be the frequency. Direction of the wave Cork bobs up & down as waves pass by

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FREQUENCY continued How would you measure the frequency of a water wave if you were in a boat? Wait until you just reach the top of a peak. Start your stopwatch, which should be set to beep at exactly one second. Count how many times you go down&up. (Each “down&up” together counts as 1 wave.) If you only made it all the way down - but not back up - in a second, the frequency would be 0.5 Hz or 0.5 sec -1. As you should know, since “a bigger sample size is better”, it’s actually more accurate to set your watch for ten seconds & then divide by 10; or a minute & then divide by 60.

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SPEED/VELOCITY The speed or velocity of a wave is defined to be the distance the wave (it is easiest to consider an exact portion of the wave such as the crest) travels divided by the time it takes to travel this distance. Let’s say for convenience that the wave/crest shown in the picture below traveled one meter in one second. Then the speed or velocity of the wave would be 1.0 m/s. Distance wave/crest has moved in a specific time Direction of the wave

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SPEED/VELOCITY continued How would you measure the velocity of a water wave if you were in a boat? Well, a boat is tough, but how about “Goin’ surfin’, Dude!” Ride along the top of a crest - but don’t forget to bring your tape measure and stopwatch! Simply measure how far you go in what amount of time and divide. It’s exactly how we always measure “how fast” something is traveling.

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FREQUENCY versus SPEED/VELOCITY It is extremely important - and the most commonly missed question in the entire wave chapter - that the difference between the frequency of a wave and the speed of a wave be completely understood. They are both “rates”, something per unit of time, but notice from the picture below that their motions on a transverse wave are perpendicular to each other: Direction of the wave SPEED of the wave (as the crest travels a certain distance to the right in one second) FREQUENCY of the wave (as the cork bobs up & down a certain number of times in one second)

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FREQUENCY vs SPEED/VELOCITY cont. Remember the slinky lab. The frequency was how fast the slinky was being vibrated, or how fast the hand was vibrating back & forth. The person doing the vibrating could alter this very easily at will. The speed was how fast a single pulse traveled the distance to the person holding the other end of the slinky. Again, notice that these rates were perpendicular to each other. FREQUENCY of the wave (as the hand shakes back & forth a certain number of times in a specific amount of time) SPEED of the wave (as a pulse travels a certain distance to the right in a specific amount of time

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WAVELENGTH ( , FREQUENCY ( or f), and VELOCITY (v) for an unchanging medium The velocity or speed of a wave is unalterable if the medium is unchanging. In other words, there is absolutely NOTHING anyone could do to change the speed of the slinky wave, except use a different slinky - a different width or radius of coil, a different metal, etc. Whether you believe it or not, if you had actually measured it with strobe- photography, every single one of the slinky wave pulses would have ALWAYS reached your lab partner in exactly the same amount of time, no matter what either of you did! A.

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WAVELENGTH ( , FREQUENCY ( or f), and VELOCITY (v) for an unchanging medium cont. What could you do to change the frequency of the slinky wave? Shake your hand back & forth faster of course! Look at the picture below. In wave 1 the hand was shaken somewhat slower, in wave 2 the hand was shaken somewhat faster. Thus, from wave 1 to wave 2 the frequency increased. Note again that the speed of the wave was unchanged by this! B. WAVE 1 WAVE 2

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WAVELENGTH ( , FREQUENCY ( or f), and VELOCITY (v) for an unchanging medium cont. What could you do to change the wavelength of the slinky wave? Again, you could shake your hand back & forth faster. Recall the picture below, where in wave 1 the hand was shaken somewhat slower and in wave 2 the hand was shaken somewhat faster. Notice that the wavelength changed, being longer in wave 1 and shorter in wave 2. Thus, from wave 1 to wave 2 the wavelength decreased. C. WAVE 1 WAVE 2

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WAVELENGTH ( , FREQUENCY ( or f), and VELOCITY (v) for an unchanging medium cont. Reexamine what happened from wave 1 to wave 2 again. In wave 1 the hand was shaken somewhat slower and in wave 2 the hand was shaken somewhat faster. From wave 1 to wave 2 the frequency increased. From wave 1 to wave 2 the wavelength decreased. This is an indirect relationship (for an unchanging medium). D. WAVE 1 WAVE 2

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WAVELENGTH ( , FREQUENCY ( or f), and VELOCITY (v) for an unchanging medium cont. Recall from previous courses that an indirect relationship has the general formula: = constant And, as discussed previously, there is a ready-made constant to use in this case (of an unchanging medium) - the velocity. Thus, the formula relating the wave properties of velocity, wavelength, and frequency is: v = This is called the “wave formula”. (Original, huh?) E.

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Self-Check Questions (Show any math work!) 1.What are the three most important properties of waves? What are their symbols AND what are their units? (Note that although your text uses the second symbol for frequency, the first one is actually preferred. Thus, two of the symbols are the Greek letters “lambda” and “nu”. Similarly, Hz is the preferred unit.) 2.Draw a sketch of a long wave train on which wavelength is indicated in two different ways. 3.Look at book page 398 #82 and determine the WAVELENGTH of the wave given. Note the distance they are talking about is shown at right, which is 3.0 m. How much of a wavelength does that represent? (Since its ½ a wavelength, the answer is the wavelength is 6.0 m) How would the answer differ if the distance given was between trough & equilibrium? ( =12.0 m) What about between successive equilibrium points? ( = 6.00 m) If you have difficulties with these, try drawing your own sketch. #3, p398#82:

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More Self-Check Questions 4.Distinguish between frequency & velocity. Use both the definitions/units AND the example of how you would measure these things if you were on a boat/in the water. Be extremely clear. Remember that this is an extremely important question, and the one most commonly missed by physics students on the unit test! 5.Look at book p.398 #82 and determine the FREQUENCY of the wave given. ( = 0.60 waves/sec or 0.60 Hz) 6.(a) Write the old formula for determining the velocity of something as was done back in chapter 3 (and that can still be used for a wave/pulse). THEN (b) write the new formula that can be used to determine the velocity of a wave specifically, if the wavelength & frequency of the wave are known.

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Even More Self-Check Questions 7.Try book p398 #78a using the old velocity formula from ch.3 (The period is unnecessary. vel = 1.9 m/s) 8.Try book p398 #77a using the new “wave equation” formula for velocity (0.29 m/s) 9.Also do #76, #79a, and then #82 where you should use the & that you found earlier. (4.0 m/s, 1.50 x 10 3 m/s, 3.6 m/s) 10.Do #81 where you need to divide the time by 2, since it’s a reflection. (1350 m.) 11.What could you do to change the frequency of a slinky wave? Be specific: what specific effect would a specific action have? 12.What could you do to change the wavelength of a slinky wave? Again, be very specific! 13.What could you do to change the speed or velocity of a slinky wave? (Is this a trick question?)

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And Even More Self-Check Questions 14.What is the math term for the relationship between the frequency & the wavelength of wave in an unchanging medium? Describe in words what such a relationship means….. (If the frequency gets _____ then the wavelength gets _____.) 15.What formula could be used to determine the wavelength or frequency of a wave if the other was known along with the velocity? 16.Try book p398 #89, to use a wave’s speed & frequency to get wavelength. (5.9 x 10 -9 m) 17.Try book p398 #90, to use a wave’s speed & wavelength to get frequency. (550 kHz = 545 m, 1600 kHz = 188 m, 88 MHz = 3.4 m, 108 MHz = 2.78 m) 18.Do all the factoids about wavelength, frequency, & velocity that were discussed hold true for longitudinal waves as well as the transverse waves that were used in most of the previous slides? (Why do you think we use transverse waves?)

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