Physics
Harmonics
Harmonics refer to the multiples of a fundamental frequency that are produced when a wave undergoes periodic oscillations. In physics, harmonics are commonly observed in sound waves, where the fundamental frequency corresponds to the pitch of the sound, and the higher harmonics contribute to the timbre or quality of the sound. Understanding harmonics is essential in the study of wave behavior and resonance phenomena.
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6 Key excerpts on "Harmonics"
eBook - PDF- Stanley A. Gelfand(Author)
- 2016(Publication Date)
- Thieme(Publisher)
The period (or the time needed for one complete replication) of a complex periodic wave is the same as the period of its fundamental frequency. Harmonics are whole number or integral multiples of the fundamental fre-quency. In other words, the fundamental is the larg-est whole number common denominator of a wave’s Harmonics, and the Harmonics are integral multiples of the fundamental frequency. In fact, the fundamen-tal is also a harmonic because it is equal to 1 times itself. In the case of wave f1 + f2 + f3, 1000 Hz is the fundamental (first harmonic), 2000 Hz is the second harmonic, and 3000 Hz is the third harmonic. Another example of combining sinusoids into a complex periodic wave is given in Fig. 1.13 . Here, the sine waves being added are odd Harmonics of the fun-damental (1000 Hz, 3000 Hz, 5000 Hz, etc.), and their amplitudes get smaller with increasing frequency. The resulting complex periodic wave becomes squared off as the number of odd Harmonics is increased, and is called a square wave for this reason. Waveforms show how amplitude changes with time. However, the frequency of a pure tone (sine wave) is not directly provided by its waveform, and the frequencies in a complex sound cannot be deter-mined by examining its waveform. In fact, the same Waveform Spectrum 1000 Hz Tone 3000 Hz Tone 5000 Hz Tone 7000 Hz Tone Square wave Time (ms) 10002000300040005000 Frequency (Hz) 60007000 8000 Amplitude Fig. 1.13 Waveforms ( left ) and cor-responding spectra ( right ) of odd har-monics combined to form a square wave (bottom). Notice that the spec-trum of a pure tone has one vertical line, whereas the spectrum of a com-plex periodic sound has a separate ver-tical line for each of its components. 1 Acoustics and Sound Measurement 17 in duration. It is aperiodic by definition because its waveform is not repeated (Fig. 1.14a) .
eBook - PDF- Stanley A. Gelfand, Lauren Calandruccio(Authors)
- 2022(Publication Date)
- Thieme(Publisher)
In fact, the fundamental is also a harmonic because it is equal to 1 times itself. In the case of wave f1 + f2 + f3, 1000 Hz is the funda- mental (first harmonic), 2000 Hz is the second harmonic, and 3000 Hz is the third harmonic. Another example of combining sinusoids into a complex periodic wave is given in Fig. 1.13. Here, the sine waves being added are odd Harmonics of the fundamental (1000 Hz, 3000 Hz, 5000 Hz, etc.), and their amplitudes get smaller with increasing frequency. The resulting complex periodic wave becomes squared off as the number of odd har- monics is increased, and is called a square wave for this reason. Waveforms show how amplitude changes with time. However, the frequency of a pure tone (sine wave) is not directly provided by its waveform, and the frequencies in a complex sound cannot be determined by examining its waveform. In fact, the same frequencies can result in dramatically different-looking complex waveforms if their phase relationships are changed. Hence, another kind of graph is needed when we want to know what fre- quencies are present. This kind of graph is a spec- trum, which shows amplitude on the y-axis as a function of frequency along the x-axis. Several ex- amples are given in Fig. 1.12 and Fig. 1.13. The fre- quency of the pure tone is given by the location of a vertical line along the horizontal (frequency) axis, and the amplitude of the tone is represented by the height of the line. According to Fourier’s theorem, complex sounds can be mathematically dissected into their constituent pure tone components. The process of doing so is called Fourier analysis, which results in the information needed to plot the spec- trum of a complex sound. The spectrum of a com- plex periodic sound has as many vertical lines as there are component frequencies. The locations of the lines show their frequencies, and their heights show their amplitudes, as illustrated in Fig.
eBook - PDF- Stanley A. Gelfand(Author)
- 2011(Publication Date)
- Thieme(Publisher)
The period (or the time needed for one complete replication) Fig. .0 Combining sinusoids that have the same fre-quency and amplitude when they are ( a ) in-phase (showing complete reinforcement); ( b ) 180 ° out-of-phase (showing cancellation); ( c ) 90 ° out-of-phase. 1 1 Acoustics and Sound Measurement of a complex periodic wave is the same as the period of its fundamental frequency. Harmon-ics are whole number or integral multiples of the fundamental frequency. In other words, the fundamental is the largest whole number common denominator of its Harmonics, and the Harmonics are integral multiples of the funda-mental frequency. In fact, the fundamental is also a harmonic because it is equal to one times itself. In the case of wave f l + f2 + f3, 1000 Hz is the fundamental (first harmonic), 2000 Hz is the second harmonic, and 3000 Hz is the third harmonic. Another example of combining sinusoids into a complex periodic wave is given in Fig. 1.12. Here, the sine waves being added are odd har-monics of the fundamental (1000 Hz, 3000 Hz, 5000 Hz, etc.), and their amplitudes get smaller with increasing frequency. The resulting com-plex periodic wave becomes squared-off as the number of odd Harmonics is increased, and is called a square wave for this reason. Waveforms show how amplitude changes with time. However, the frequency of a pure-tone (sine wave) is not directly provided by its waveform, and the frequencies in a complex sound cannot be determined by examining its waveform. In fact, the same frequencies can result in dramatically different looking com-plex waveforms if their phase relationships are changed. Hence, another kind of graph is needed when we want to know what frequen-cies are present. This kind of graph is a spec-trum , which shows amplitude on the y -axis as a function of frequency along the x -axis. Several examples are given in Figs. 1.11 and 1.12 . The Fig.
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Now for standing waves, there are additional requirements determined by the spatial limitation of where the waves can occur, specifying the wavelengths or frequencies, which can only occur in discrete values (which are quantized). The value n = 1 corresponds to the fundamental (or basic) frequency, as follows: ω 1 = 2π ν 1 = π L v, λ 1 = 2L The higher Harmonics correspond to the numbers n = 2, 3 . . ., that is, multiples of the fundamental frequency, as we have already noted. The first harmonic differs from the fun- damental vibration in frequency by a factor of two. In music, this step from the fundamental frequency to the first harmonic corresponds to increasing the tone by one octave. In the case of nonresonant excitation, as occurs, for example, when drawing, striking, or plucking a string instrument, not only standing waves of a defined frequency are excited, but a whole spectrum thereof. The general oscillation state is then a superposition of harmonic oscillations, u(x, t) = n A n cos(ω n t + δ n ) sin k n x This is something we have just seen recently when treating Fourier transforms. 129 The Principle of Superposition Figure 4.16 Basic and higher vibrational modes of a tuning fork. Membranes, Hollow Bodies As already mentioned, standing waves also occur in two- and higher-dimensional systems. Figure 4.16 shows the basic and higher vibrational modes of a tuning fork. The tuning fork is often connected to a wooden hollow body, the dimensions of which are adjusted in such a way that the fundamental frequencies of this sound body and the tuning fork are the matched (see also Section 4.7.3). The hollow body thus acts as a resonator, which transforms the sound wave produced in the tuning fork into a sound wave in a larger space and therefore is perceptible in all directions. A tuning fork without a resonating body is very difficult to hear if it is not right next to the ear.
eBook - PDF- Dave Benson(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
There is a resonant frequency corresponding to the maximal response of the damped system to the incoming sine wave. The third inaccuracy is that for loud enough sounds the restoring force may be nonlinear. This will be seen to be the possible origin of some interesting acoustical phenomena. Finally, most musical notes do not consist of a single sine wave. For example, if a string is plucked, a periodic wave will result, but it will usually consist of a sum of sine waves with various amplitudes. So there will be various different peaks of amplitude of vibration of the basilar membrane, and a more complex signal is sent to the brain. The decomposition of a periodic wave as a sum of sine waves is called Fourier analysis, which is the subject of Chapter 2. 1.5 Harmonic motion Consider a particle of mass m subject to a force F towards the equilibrium position, y = 0, and whose magnitude is proportional to the distance y from the equilibrium position, F = −ky . Here, k is just the constant of proportionality. Newton’s laws of motion give us the equation F = ma , where a = d 2 y dt 2 1.6 Vibrating strings 19 Figure 1.11 A vibrating string. is the acceleration of the particle and t represents time. Combining these equations, we obtain the second order differential equation d 2 y dt 2 + ky m = 0. (1.5.1) We write ˙ y for d y dt and ¨ y for d 2 y dt 2 as usual, so that this equation takes the form ¨ y + ky / m = 0. The solutions to this equation are the functions y = A cos( k / m t ) + B sin( k / m t ). (1.5.2) The fact that these are the solutions of this differential equation is the explanation of why the sine wave, and not some other periodically oscillating wave, is the basis for harmonic analysis of periodic waves. For this is the differential equation governing the movement of any particular point on the basilar membrane in the cochlea, and hence governing the human perception of sound.
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
D Vibrations, Waves, and Sound Introduction A vibration is an oscillatory repetitive motion that occurs at a specific location over time. Common examples of vibrations are a pen- dulum swinging back and forth, a weight bobbing on a spring, or a plucked string on a musical instrument. Vibrations lead to the propagation of waves. A vibrating string creates a sound wave, electromag- netic waves are produced by the vibrations of electric charge, and seismic waves come from vibrations from within the Earth. All waves can be described by common char- acteristics such as wavelength, frequency, and period. This chapter begins the study of vibrations by examining simple harmonic motion and follows this with an examina- tion of the general principles that apply to all waves. Different types of waves will be examined, concluding with a more compre- hensive examination of sound. Simple Harmonic Motion A simple model used to examine vibra- tions is that of a simple harmonic oscillator. A simple harmonic oscillator is a system consisting of a "to and fro" motion around an equilibrium position. A condition for simple harmonic motion is that the restoring force is proportional to the displacement from the equilibrium position. For example, con- sider a weight attached to a vertical spring (Figure 8.1). If the spring is stretched by pulling on the weight and then released, the weight will bob up and down around the initial equilibrium position. As the weight moves down past the equilibrium position, stretching the spring, it will decelerate in proportion to a restoring force acting upward that attempts to return the weight to its equi- librium position. Once the weight "bottoms out," it will then move upward through the equilibrium position, compressing the spring. In the absence of friction, this up and down motion would continue forever, but in actuality, it eventually damps out due to friction.
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