SECTION 3 SECTION 3 Objectives Plan and Prepare Differentiate between the Harmonics harmonic series of open and closed pipes. Preview Vocabulary Key Terms Calculate the harmonics of a fundamental frequency timbre Latin Word Origins Often terms used vibrating string and of open and harmonic series beat in physics take their meaning from their closed pipes. Latin roots. Fundamental, for instance, Relate harmonics and timbre. Standing Waves on a Vibrating String comes from the Latin fundamentum, Relate the frequency difference As discussed in the chapter “Vibrations and Waves,” a variety of standing waves can occur when a string is fixed at both ends and set into vibration. meaning “foundation.” Apply this root between two waves to the The vibrations on the string of a musical instrument, such as the violin in to the definition of fundamental number of beats heard per Figure 3.1, usually consist of many standing waves together at the same second. frequency (i.e., the foundation is the time, each of which has a different wavelength and frequency. So, the lowest point of a structure, just as sounds you hear from a stringed instrument, even those that sound like fundamental frequency is the lowest a single pitch, actually consist of multiple frequencies. frequency of a standing wave). Figure 3.2, on the next page, shows several possible vibrations on an idealized string. The ends of the string, which cannot vibrate, must always FIGURE 3.1 be nodes (N ). The simplest vibration that can occur is shown in the first row of Figure 3.2. In this case, the center of the string experiences the most Stringed Instruments The displacement, and so it is an antinode (A). Because the distance from one Teach vibrating strings of a violin produce node to the next is always half a wavelength, the string length (L) must standing waves whose frequencies equal λ1/2. Thus, the wavelength is twice the string length (λ1 = 2L). depend on the string lengths. As described in the chapter on waves, the speed of a wave equals the Demonstration frequency times the wavelength, which can be rearranged as shown. v = f λ, so f = _ v SEEING SOUNds λ Purpose Observe sound waves from By substituting the value for wavelength found above into this a variety of sources. equation for frequency, we see that the frequency of this vibration is equal to the speed of the wave divided by twice the string length. Materials oscilloscope, microphone, _v _v fundamental frequency = f1 = = small amplifier (if needed), assorted λ1 2L sound sources This frequency of vibration is called the fundamental frequency of the vibrating string. Because frequency is inversely proportional to wave- Caution Consult the oscilloscope’s length and because we are considering the greatest possible wavelength, user’s manual for instructions on the the fundamental frequency is the lowest possible frequency of a standing proper use of the oscilloscope. wave on this string. Procedure Connect the microphone or fundamental frequency the lowest Harmonics are integral multiples of the fundamental frequency. the output from the microphone and frequency of vibration of a standing wave The next possible standing wave for a string is shown in the second row amplifier to the input of the oscillo- of Figure 3.2. In this case, there are three nodes instead of two, so the scope. Display the sound pattern of the string length is equal to one wavelength. Because this wavelength is human voice by having several students half the previous wavelength, the frequency of this wave is twice that of the fundamental frequency. speak or sing into the microphone. Have f = 2f the students note the patterns on the 2 1 ©RubberBall/Getty Images screen of the oscilloscope. Show the Differentiated418 Chapter 12 Instruction characteristic of a single frequency sound by inputting the sound from an BELOW LEVEL oscillator, sine wave generator, tuning Untitled-99Students 418 may be confused as to why the 5/18/2011 6:42:48 AM fork, or a student’s voice at a single addition of one node equals half the wave- pitch. length and double the frequency. It may help Then have several students play the them to visualize folding the string in half with same note on various instruments one the second node as the center. at a time. Have the students note the characteristic harmonics of each instrument as they are displayed on the oscilloscope screen. 418 Chapter 12 FIGURE 3.2 THE HARMONIC SERIES Teaching Tip A A Have students visualize the harmonics A fundamental frequency, or A λ = 2L f N N 1 1 first harmonic to help them recall the appropriate N N N N equations. Both ends of a string fixed at A A A A each end must be nodes. For the first A A N N λ = L f = 2f second harmonic harmonic (one loop), the length L of the N N 2 2 1 N _1 _1 N N N string must equal ​ 2 λ. Thus, ​ 2 λ = L and A NA A A A A λ = 2L. For other harmonics, students A A A N N __2 can use this same technique with the N N λ = L f = 3f third harmonic N N N N 3 3 3 1 _1 N N appropriate multiples of ​ λ. A NA AN A 2 A A A A A A A A N N N N N N 1 N N N __ 4 fourth harmonic N N λ4 = 2 L f4 = f1 N N N TEACH FROM VISUALS N N N FIGURE 3.2 Be sure students understand that each loop corresponds to half a ThisHRW • pattern Holt Physics continues, and the frequency of the standing wave shown PH99PE-C13-003-002-AHRW • Holt Physics wavelength. inPH99PE-C13-003-002-A theHRW third • Holt rowPhysics of Figure 3.2 is three times the fundamental frequency. PH99PE-C13-003-002-A More generally, the frequencies of the standing wave patterns are all Ask Find the wavelength (λ5) and integral multiples of the fundamental frequency. These frequencies frequency (f5) for the next possible case harmonic series. harmonic series a series of frequen- form what is called a The fundamental frequency (f1) in the harmonic series. corresponds to the first harmonic, the next frequency (f ) corresponds cies that includes the fundamental 2 frequency and integral multiples of the _2 to the second harmonic, and so on. Answer: λ = ​ ​ L, f = 5f fundamental frequency 5 5 5 1 Because each harmonic is an integral multiple of the fundamental frequency, the equation for the fundamental frequency can be general- ized to include the entire harmonic series. Thus, fn = nf1, where f1 is the v __ fundamental frequency (f1 = 2L ) and fn is the frequency of the nth harmonic. The general form of the equation is written as follows: Did YOU Know? Harmonic Series of Standing Waves When a guitar player presses down on on a Vibrating String a guitar string at any point, that point f = n _v n = 1, 2, 3, . becomes a node and only a portion of n 2L the string vibrates. As a result, a single string can be used to create a variety (speed of waves on the string) frequency = harmonic number × ___ of fundamental frequencies. In the (2)(length of vibrating string) equation on this page, L refers to the portion of the string that is vibrating. Note that v in this equation is the speed of waves on the vibrating string and not the speed of the resultant sound waves in air. If the string vibrates at one of these frequencies, the sound waves produced in the surrounding air will have the same frequency. However, the speed of these waves will be the speed of sound waves in air, and the wavelength of these waves will be that speed divided by the frequency. ©RubberBall/Getty Images Sound 419 PRE-AP dependence on string length and speed, supply students with multiple scenarios that encourage Untitled-99 419 Point out to advanced students that frequency 5/18/2011 6:42:49 AM depends on both string length and wave speed, them to substitute values into the harmonic as shown by the equation for the harmonic series equation as well as solve for unknowns. series. Thus, two strings of the same length will not necessarily have the same fundamental frequency. The string’s tension and mass per unit length affect the speed of waves on the string, so the fundamental frequency can be changed by varying either of these factors. To test their understanding of frequency’s Sound 419 FIGURE 3.3 Standing Waves in an Air Column Waves in a Pipe The harmonic Standing waves can also be set up in a tube of air, such as the inside of a Teach continued series present in each of these organ trumpet, the column of a saxophone, or the pipes of an organ like those pipes depends on whether the end of shown in Figure 3.3. While some waves travel down the tube, others are the pipe is open or closed. reflected back upward. These waves traveling in opposite directions Teaching Tip combine to produce standing waves. Many brass instruments and The memory device used for a vibrating woodwinds produce sound by means of these vibrating air columns. string can be used for open and closed pipes as well. Ask students to visualize If both ends of a pipe are open, all harmonics are present. The harmonic series present in an organ pipe depends on whether the the standing wave, remembering that reflecting end of the pipe is open or closed.