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Physics of Oscillations and Waves: With use of Matlab and Python: (Undergraduate Texts in Physics)

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Nguyễn Gia Hào

Academic year: 2023

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We have made efforts to provide a logical and reader-friendly structure of the book. Many parts of the book have been modified after the translation, so don't blame Prof.

Introduction

The Multifaceted Physics

We are therefore not satisfied with a purely mathematical description of the movement of the guitar string. Second, there are only a handful of simplified cases we are able to handle, and most of the other equations are intractable by analytical means.

Fig. 1.1 Oscillations and waves are woven into a host of phenomena we experience every single day
Fig. 1.1 Oscillations and waves are woven into a host of phenomena we experience every single day

Numerical Methods

  • Supporting Material
  • Supporting Literature

With numerical calculations, it is easier to focus on algorithms, basic equations, than with analytical methods. For example, we can use multiple transducers located in an array for ultrasonic diagnostics, oil leaks, sonar and radar technology.

Free and Damped Oscillations

Introductory Remarks

Kinematics

In our case, this reference point is the position of the mass at rest. Then the x component of the vector indicates the instantaneous amplitude of the harmonic at a given instant.

Fig. 2.1 A harmonic oscillation is characterized by amplitude, frequency and phase; see the text
Fig. 2.1 A harmonic oscillation is characterized by amplitude, frequency and phase; see the text

Going from One Expression to Another

  • First Conversion
  • Second Conversion
  • Third Conversion
  • Fourth Conversion

Phasors are very useful when multiple contributions to a motion or signal of the same frequency need to be summed. This is a fraction whose numerator is the y-component and denominator the x-component of the phasor att =0.

Fig. 2.3 Sketch of a phasor that rotates around the origin with an angular velocity ω .
Fig. 2.3 Sketch of a phasor that rotates around the origin with an angular velocity ω .

Dynamical Description of a Mechanical System

Important: The extension of the spring from length L0 to L1 is due to the force of gravity. We were able to "explain" oscillatory motion using a combination of Hooke's law and Newton's second law.

Fig. 2.4 Definition of different lengths of the spring with and without an attached mass; see the text
Fig. 2.4 Definition of different lengths of the spring with and without an attached mass; see the text

Damped Oscillations

The kinematic description given in Sect.2.1 is identical to the solution of the dynamic equation we set up in this section based on Newton's law. Note that, for certain initial conditions, A1 and A2 may have different signs, and the time course of the displacement may contain surprises.

Superposition and Nonlinear Equations

In other words, when we include a second-order term to complete the description of friction, we see that the principle of superposition no longer holds. Even if we find a possible solution for such an oscillating system and then another solution, the sum of these individual solutions will not necessarily be the solution of the differential equation.

Electrical Oscillations

Thus, the charge on the capacitor decreases exponentially and goes to zero (the reader would be familiar with this). We see again that there are two constants that must be determined by means of the initial conditions.

Fig. 2.6 The voltage across a charged capacitor will decrease exponentially to zero after the capac- capac-itor is connected to a resistor
Fig. 2.6 The voltage across a charged capacitor will decrease exponentially to zero after the capac- capac-itor is connected to a resistor

Energy Considerations

Although the energy of the capacitor and inductor varies from zero to a maximum value and back in an oscillating manner, these variations are time-shifted. The energy calculations we just completed only apply if there is no loss in the system.

Learning Objectives

Note that each oscillation must contain the two terms given in the equation in the previous paragraph, but other terms may be included. Be able to explain why the principle of superposition does not apply when nonlinear terms are included in the equation of motion.

Exercises

How large is the period of time for a mass at the end of the semispring compared to the period of the mass in the original spring. At time=2.0 s, its position is +2.4 cm above the equilibrium position and the velocity of the mass is -16 cm/s.

Introductory Remarks

The amplitude of the oscillations, which has been shown to depend on the frequency of the external force, reaches its peak value when the frequency of the applied force is close to the natural frequency of the system, a phenomenon called resonance. Finally, some relevant details regarding the physiology of the human ear are briefly mentioned.

Forced Vibrations

The amplitude of the oscillations is then given by Eq. 3.4), and the phase difference between the output and the applied force (or input) is given by Eq. Figure 3.1 shows schematically how the amplitude and phase change with the frequency of the applied force.

Resonance

  • Phasor Description

This happens when the frequency of the applied power (input frequency) coincides with the natural frequency of the (undamped) system. However, if ωFL is much larger than 1/(ωFC)(C“shortened”), the current will be offset 90◦ to the voltage (In a calculation exercise at the end of the chapter you are asked to show this.).

Fig. 3.2 A close-up view of the relative amplitude in a forced oscillation as a function of the frequency of the applied force
Fig. 3.2 A close-up view of the relative amplitude in a forced oscillation as a function of the frequency of the applied force

The Quality Factor Q

The curves in Fig.3.7 show that after an applied force is turned on, the amplitude of the oscillations increases, without becoming infinite. The frequency of the applied voltage is equal to the resonant frequency on the left and slightly lower on the right.

Fig. 3.6 When the frequency of the applied force changes relative to the system’s own natural frequency, the amplitude will be greatest when the two frequencies are nearly equal.
Fig. 3.6 When the frequency of the applied force changes relative to the system’s own natural frequency, the amplitude will be greatest when the two frequencies are nearly equal.

Oscillations Driven by a Limited-Duration Force

The loss of energy is independent of the strength of the force after it has disappeared. We see that for too small σ (the effect lasts only a few oscillation periods), the maximum amplitude increases roughly proportional to the duration of the force.

Fig. 3.9 The force F ( t ) for centre frequency 100 Hz and pulse width σ equal to 0.025 and 0.10 s.
Fig. 3.9 The force F ( t ) for centre frequency 100 Hz and pulse width σ equal to 0.025 and 0.10 s.

Frequency Response of Systems Driven by Temporary Forces *

If, on the other hand, we apply longer and longer "power pulses", the frequency response of the system will reach a limiting value. When the force lasts for a short time (few oscillations) the frequency of the force is ill-defined.

Fig. 3.13 The correlation between the frequency response of a system and the frequency of the driving force when the duration of the force changes
Fig. 3.13 The correlation between the frequency response of a system and the frequency of the driving force when the duration of the force changes

Example: Hearing

Consequently, if we hear a dark sound (low frequency), only the inner part of the basilar membrane will vibrate. We can hear both a bass sound and a disc rhythm at the same time, because both sound stimuli stimulate different parts of the basilar membrane.

Fig. 3.15 The inner ear has a three-channel structure that stretches almost three rounds from bottom to top
Fig. 3.15 The inner ear has a three-channel structure that stretches almost three rounds from bottom to top

Learning Objectives

The prevailing opinion is that the sound impression becomes indifferent to the phase of the various frequency components of the sound signal. Know the gross anatomy of the ear well enough to explain how we can hear multiple tones simultaneously.

Exercises

An alternating voltage V(t)=V1cos(ωFt) is applied to an electrically oscillating circuit, ωF is equal to the resonant (angular) frequency of the circuit. After a long time, the oscillations in the circuit stabilize and the amplitude of the current fluctuations is I1.

Fig. 3.17 Sensitivity curve of a “single-photon detector”
Fig. 3.17 Sensitivity curve of a “single-photon detector”

How wide is the circuit's frequency response for a "long-term" applied voltage. f) How "long" must the applied voltage actually last for the circuit to reach a nearly steady state (this amplitude no longer changes appreciably with time). g) Assume that the circuit is subjected to a force pulse with a center frequency equal to the resonant frequency and that the force pulse has a Gaussian amplitude envelope function [Eq. 3.19)], where σ has a value equal to twice the time period corresponding to the mean frequency of the circuit.

Numerical Methods

Introductory Remarks

Introduction

Parts of the chapter were written by David Skålid Amundsen as a summer assignment for CSE 2008. First, a quick overview is given of the simplest numerical methods used to solve differential equations.

Basic Idea Behind Numerical Methods

This is followed by a practical example, and finally we will include code examples that can be used to solve the problems given in the following chapters.

Euler’s Method and Its Variants

The figure also shows a plot of the solution found using Euler's method (red curve) with very large time steps (0.2 s). Then we calculate the gradient in the middle of the interval and use this for the entire interval.

Fig. 4.2 Euler’s simple method of calculating a function numerically. The top (blue) curve is the exact analytic solution
Fig. 4.2 Euler’s simple method of calculating a function numerically. The top (blue) curve is the exact analytic solution

Runge–Kutta Method

  • Description of the Method

Furthermore, we can find an estimate of the second derivative at mid-step using v2, n, x2, and Eq. There is another solution to the differential equation that passes through this point (thin green line).

Partial Differential Equations

The right-hand side of the above equation gives(x,t0+t), the value of the function at a later time t+t. Boundary conditions determine the state of the system at the endpoints of the calculations at all times.

Example of Numerical Solution: Simple Pendulum

If we wanted to include friction in the description of the movement of the pendulum, it would represent a more complex expression of the effective force than we had in our case. This is an added bonus of numerical solutions: the force at work—and thus the actual physics of the problem—becomes more central to our search for a solution.

Test of Implementation

In the case of the simple pendulum, there happens to be a trick up our sleeve. If we reduce the amplitude to 1/100 of the original, the maximum difference is reduced to 10−6 of the original difference.

Fig. 4.6 Comparison between analytical and numerical solution of a shuttle movement. For expla- expla-nations: See the text
Fig. 4.6 Comparison between analytical and numerical solution of a shuttle movement. For expla- expla-nations: See the text

Reproducibility Requirements

We often need to do a set of calculations to make sure the "resolution" in the calculation is appropriate and manageable (neither too high nor too low). If we adhere to these rules, we will always be able to go back and reproduce the results obtained in the past.

Some Hints on the Use of Numerical Methods

However, do not overdo it, because it hinders a survey and the readability of the program. It is not always easy to find out where such an error is located in the program code.

Summary and Program Codes

  • Suggestions for Further Reading

Use the new position and speed to find an estimate for the acceleration, a2, at the middle of the interval. Use this new acceleration and speed (a2 and v2) to find a new estimate for position and speed (v3) in the middle of the interval.

Learning Objectives

Exercises

  • An Exciting Motion (Chaotic)

Try to get a plot corresponding to the initial part of each of the time courses we find in Fig.3.7. In this description, we work with "normalized velocity" γ (n), which is proportional to the initial velocity of each bounce, and with φ(n), which is the phase of the floor movement just as the bounce starts.

Fig. 4.7 Motion of a simple pendulum. Position vs time is shown to the left and phase space presentation of the motion to the right
Fig. 4.7 Motion of a simple pendulum. Position vs time is shown to the left and phase space presentation of the motion to the right

Fourier Analysis

Introductory Examples .1 A Historical Remark.1A Historical Remark

  • A Harmonic Function
  • Two Harmonic Functions
  • Periodic, Nonharmonic Functions
  • Nonharmonic, Nonperiodic Functions

In the left part of Fig.5.1, we have shown a section of an arbitrary harmonic function of time. Since we have time along the thex axis in the left part of Fig.5.1, we call this a "time-domain" representation of the function.

Fig. 5.1 Section of a harmonic function plotted, in the left part, as a function of time (“time domain”) and, in the right part, as a function of frequency (“frequency domain”)
Fig. 5.1 Section of a harmonic function plotted, in the left part, as a function of time (“time domain”) and, in the right part, as a function of frequency (“frequency domain”)

Real Values, Negative Frequencies

An analysis similar to the one we have made in the previous examples gives the coefficients (and amplitudes) indicated in the right part of the figure. If we actually do a Fourier analysis of the first harmonic function we examined, the frequency domain image will have the appearance shown in the right part of Fig.5.6.

Fig. 5.5 Common phasor description (in red) of a harmonic function C cos (ω t + φ) at time t
Fig. 5.5 Common phasor description (in red) of a harmonic function C cos (ω t + φ) at time t

Fourier Transformation in Mathematics

  • Fourier Series

The particular case of the Fourier transform is of particular interest, especially when studying Chapter 7 for the analysis of sound from musical instruments. If f(t) is a periodic function with period T, the Fourier transform can be more efficient than the general transform in Eq.

Frequency Analysis

A periodic signal that is not sinusoidal (harmonic) will automatically lead to overtones in the frequency range. The reason it does not become a pure sinusoid is that the physical process involved in producing the sound is complicated and turbulence is involved.

Fig. 5.8 An example of sound from a flute displayed both in the time domain and frequency domain.
Fig. 5.8 An example of sound from a flute displayed both in the time domain and frequency domain.

Discrete Fourier Transformation

  • Fast Fourier Transform (FFT)
  • Aliasing/Folding

Using the expression in Eq. 5.17), we have shown that the expression for the discrete Fourier transform in Eq. 5.18) is quadratic based on the same expression as we had in the original Fourier transform. When we perform inverse Fourier transform with IFFT, the negative frequencies are expected to be positioned in the same way as they are after a simple FFT.

Important Concrete Details .1 Each Single Point.1Each Single Point

  • Sampling Theorem

All remaining frequency components are complex conjugates of lower frequency components (assuming f(t) is real). We simply need to ensure that there are no contributions with frequencies above half the sampling frequency of the sampled signal.

Fig. 5.10 A function sampled N = 8 times (left) along with the Fourier transform of the function (right) consisting of N = 8 complex numbers
Fig. 5.10 A function sampled N = 8 times (left) along with the Fourier transform of the function (right) consisting of N = 8 complex numbers

Fourier Transformation of Time-Limited Signals

We see that the integral (sum) of the product between the red and blue curves in the adheb will be approximately the same. The Fourier transform of the inc signal (the longer term signal) is shown in the lower right corner outside.

Fig. 5.12 Fourier transformation of a cosine signal multiplied with a Gaussian function
Fig. 5.12 Fourier transformation of a cosine signal multiplied with a Gaussian function

Food for Thought

If we say that "several frequencies are present simultaneously" in the motion that lies at the back of the Fourier spectrum in Fig.5.13, the statement does not accord well with the underlying physics. The reason we get a series of "harmonics" in this case is that the planetary motion is periodic, but not a pure sinusoid.

Figure 5.13 shows the Fourier transform of a periodic motion. In essence, this figure resembles Fig
Figure 5.13 shows the Fourier transform of a periodic motion. In essence, this figure resembles Fig

Programming Hints

  • Indices; Differences Between Matlab and Python
  • Fourier Transformation; Example of a Computer ProgramProgram

But if we think in terms of "several frequencies existing at the same time", this is equivalent to saying that the motion of the planet must be described with multiple circular motions occurring at the same time. Fourier analysis can be performed for virtually all physical time variables, since the sine and cosine functions included in the analysis form a complete set of basis functions.

Appendix: A Useful Point of View

  • Program for Visualizing the Average of Sin–Cos ProductsProducts
  • Program Snippets for Use in the Problems

The integral of the product function now receives contributions only in the time interval where f differs from zero. The real part of a section of the Fourier spectrum of the function appearing in Eq.

Fig. 5.14 Two functions included in the integral in Fourier integral Eq. (5.24).
Fig. 5.14 Two functions included in the integral in Fourier integral Eq. (5.24).

Learning Objectives

The Fourier transform is a great help in studying stationary time-varying phenomena in much of physics. Usually when performing Fourier transform numerically, we use ready-made functions within the programming package we are using.

Exercises

Is there a match between the peak heights in the time domain image and the amplitude of the frequency spectrum. Calculate what the signal sent from an FM transmitter looks like and find the frequency spectrum of the signal.

Fig. 5.16 A drawing of epicycles in an old Arabic document written by Ibn_al-Shatir [1], Public Domain
Fig. 5.16 A drawing of epicycles in an old Arabic document written by Ibn_al-Shatir [1], Public Domain

Perform the Fourier transform of this small portion of the signal (doesn't have to have 2n data points). Describe the difference between the inverse of the complex Fourier transform and the one you just found.

Waves

Introduction

We can use a "movie" (animation) that shows how the wave propagates in space over time. Therefore, solving the wave equation requires that we know the initial conditions as well as the boundary conditions.

Fig. 6.1 Waves that form on water
Fig. 6.1 Waves that form on water

Plane Waves

  • Speed of Waves
  • Solution of the Wave Equation?
  • Which Way?
  • Other Waveforms
  • Sum of Waves
  • Complex Form of a Wave

In other words, the peak of the wave moves to larger x values ​​as time increases. For a left wave, exactly the same applies, but "in front" of the wave must now mean to the left of the wave.

Fig. 6.3 Snapshot of
Fig. 6.3 Snapshot of

Transverse and Longitudinal

That an electromagnetic wave is transverse means that the electric and magnetic fields are in a direction perpendicular to the direction in which the wave propagates. They only indicate the magnitude and direction of the abstract amounts of electric and magnetic fields at the various positions in the room.

Derivation of Wave Equation

  • Waves on a String
  • Waves in Air/Liquids
  • Concrete Examples
  • Pressure Waves

The string has a linear mass density (mass per length) equal to μ, and the length of the segment isx. Deriving the wave equation for motion in air/liquids is more complicated than the case discussed in the previous section.

Fig. 6.4 Forces that act on a small segment of a string suffering transverse motion.
Fig. 6.4 Forces that act on a small segment of a string suffering transverse motion.

Learning Objectives

6.20) The result shows that wave motions in a compressible medium can be described both as displacements of small volumes of the medium and as pressure variations. If the amplitude of the displacement of the small volumes (with a thickness significantly smaller than the wavelength) is equal to η0, the amplitude of the pressure wave is equal to Kη0.

Exercises

Try to make a corresponding list for the derivation of the wave equation in air/water. You may use program snippet 2 at the end of Chapter 5 for part of the program.

Fig. 6.7 Experimental setup
Fig. 6.7 Experimental setup

Sound

Reflection of Waves

  • Acoustic Impedance *
  • Ultrasonic Images

The wave is completely reflected and the result takes the opposite sign of the input. In this case, however, some of the wave (and energy) will also propagate along the thin string.

Fig. 7.1 A transverse wave reaches an interface between two media. The wave is drawn for four successive instants
Fig. 7.1 A transverse wave reaches an interface between two media. The wave is drawn for four successive instants

Standing Waves, Musical Instruments, Tones .1 Standing Waves.1Standing Waves

  • Quantized Waves
  • Musical Instruments and Frequency Spectra
  • Wind Instruments
  • Breach with Tradition
  • How to Vary the Pitch
  • Musical Intervals

We can see that the pitch of the root tone here can be continuously varied. In Fig.7.12 we saw that the frequency of the fundamental tone for a B trumpet was approximately 231.5 Hz.

Fig. 7.4 A travelling (left) and a standing wave (right) as a function of position at various times.
Fig. 7.4 A travelling (left) and a standing wave (right) as a function of position at various times.

Sound Intensity

Hình ảnh

Fig. 1.1 Oscillations and waves are woven into a host of phenomena we experience every single day
Fig. 3.9 The force F ( t ) for centre frequency 100 Hz and pulse width σ equal to 0.025 and 0.10 s.
Fig. 3.12 The frequency response (actually only maximum amplitude) of the oscillating system for different durations ( σ ) of the force pulse (left part)
Fig. 3.15 The inner ear has a three-channel structure that stretches almost three rounds from bottom to top
+7

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