Period, Amplitude & Energy

1. Packet #3 Period, Amplitude & Energy 2008, new book

2. PERIOD The period of a wave is the shortest time interval during which the motion of the wave repeats itself . Notice that the period is a time, thus it has a symbol T and a unit of [sec]. Recall this picture with the bobbing cork. The time it takes for the cork to go up & then all the way back down again is the period. Cork bobs up & down as waves pass by Direction of the wave

3. PERIOD continued How would you measure the period of a water wave if you were in a boat? Again, wait until you just reach the top of a peak. Start your stopwatch and time how long it takes for you to go down & all the way back up again. If it takes you 5 seconds, then the period is 5.0 seconds. As you should know, since “a bigger sample size is better”, it’s actually more accurate to time ten up-&-down-bobs & then divide by 10.

4. PERIOD versus FREQUENCY Notice the similarities between period & frequency. They both concern the bobbing of the cork - or of you if you are in a boat. Recall that ….. frequency is the number of full bobs/vibrations/waves in one second, & period is the number of seconds in one full bob/vibration/wave. What do you notice? Don’t frequency & period seem somehow opposites of each other? (Examine the sentence above until you can convince yourself of this.)

5. PERIOD vs FREQUENCY cont. Frequency: number of full bobs/vibrations/waves in one second. Period: number of seconds in one full bob/vibration/wave. Both period & frequency are ratios, where you divide to get the answer. You can make the math easy on yourself when you’re taking your measurements and simply divide by one (second or full bob, depending on the case). Or you can be more accurate as discussed previously and use a bigger sample size, where you’d have to divide by something other than one.

6. PERIOD vs FREQUENCY cont. Frequency: number of full bobs/vibrations/waves in one second. Period: number of seconds in one full bob/vibration/wave. Simply written, the ratios are: Frequency (n) = vibrations per second = vibrat/sec = [Hz] Period (T) = seconds per vibration = sec/vibrat = [sec]. It is obvious that, not only are these opposites of each other, they are in fact reciprocals or inverses of each other. Thus we obtain the formula: T = 1 / n (or n = 1 / T if you prefer).

7. AMPLITUDE The amplitude of a wave is its maximum displacement from its equilibrium position. Thus, you can measure either the height of a crest OR the depth of a trough. It is a common mistake to think that the amplitude is the entire distance from peak to crest. It is NOT!

8. AMPLITUDE continued The amplitude of a wave is directly proportional to the energy of the wave or the work that the wave can do. Actually, the relationship is: E µú Aú2 (The energy is directly proportional to the square of the absolute value of the amplitude.) Just remember that bigger amplitude means more energy. In the picture below, wave A has more energy & could do more work (or damage to a boat or house if it was an ocean wave).

9. AMPLITUDE vswavelength, frequency, velocity, period The amplitude of a wave is completely independent from all of these other properties. It has absolutely no relationship to them. That does not mean that it is constant, as velocity is. It just means that you can easily change the amplitude of a wave - for instance by simply shaking the slinky further back & forth than you were before - and not have to change any of those other properties. (Consider as a counter-example wavelength, frequency, and period. These three are definitely not independent from each other. You already know that if you change one of these three, the other two must necessarily change as per their indirect relationships.)

10. AMPLITUDE vswavelength, frequency, velocity, period If the amplitude of a wave is completely independent from all of these other properties & has absolutely no relationship to them, then (unlike the counter-example where changing one property must necessarily change the others) the amplitude of a wave could be easily changed without effecting any of the other properties. Notice that even though the amplitude decreased from Wave A to Wave B, that the wavelength is unchanged. Thus, the frequency and the period will be unchanged also.

11. AMPLITUDE vswavelength, frequency, velocity, period Let’s talk about slinkies again! Many people think that the slinky is slowing down as it travels from one partner to the other, but we’ve already stated that is impossible because the velocity of a wave does NOT change unless the medium changes - and it wasn’t! So what was really happening? Friction was deceptively causing a decrease of the amplitude. We say the wave “dies off” (which does not mean slows down). Remember from our earlier discussions that friction does negative work, or decreases the total amount of energy in a system. Thus, since only the amplitude of a wave is proportional to energy, then it is only the amplitude that is effected by friction, not the speed!

12. Self-Check Questions • Use the definition, examples of how you would measure it, & sketches to explain period. • What is the symbol AND the unit for period? • What is the math term for the relationship between the frequency & the period of wave? Describe in words what such a relationship means. (If the frequency gets _____ then the period gets _____.) • What formula could be used to determine the period or frequency of a wave if the other was known? • What could you do to change the period of a slinky wave? Be specific: what specific effect would a specific action have? (Hint: think about its inverse relationship with frequency.)

13. More Self-Check Questions (Show any math work!) #6, p399#93: • **Try book page 399 #93 a&b about period. (¼th, 0.72 sec) • Look at book p.398 #82 and determine the PERIOD of the wave given by using the formula or units for period. (T = 1.67 sec/wave or simply 1.67 seconds) • Use the fact that T and  are inverses to solve p398 #77b and 79b. (0.21 sec; 1 x 106 sec) • Take the new formula for the velocity of a wave and substitute period in for frequency according to their relationship. I do not recommend that you memorize this formula, as it is unnecessary, although some people prefer to. • Using the new formula above for the speed of a wave (you now have 3 total, do you remember the other 2?) do book page 876 #9. (2.9 m/s)

14. Even More Self-Check Questions • Draw a sketch of a wave train on which amplitude is indicated in two different ways. • What is the symbol AND the unit for amplitude? • What could you do to change the amplitude of a slinky wave? • In the question above, how would what you did to the slinky affect the wavelength, frequency, period, and speed of the wave? (Is this a trick question?) • Explain how the comparative energy content of a wave is ascertained. Give an example of why the amount of energy that a wave transmits is an important characteristic to know. (Hint: think of Japan’s tsunamis and/or the Hayward fault.) • What effect does friction (or any kind of resistance) have on the various wave properties?