Chapter2 机械波
2.1 Simple Harmonic Motion
Equilibrium（平衡态）
- Translational equilibrium The linear momentum of its center of mass is constant
- Rotational equilibrium The angular momentum about its center of mass, or about any other point, is also constant.
- Equilibrium An object is in equilibrium if it is in both translational and rotational equilibrium.
- static equilibrium If the object is at rest and so has no linear speed or angular speed, the object is in static equilibrium.
- stable static equilibrium If a body tends to return to a state of static equilibrium after having been displaced from that state by a force, the body is said to be in stable static equilibrium.
Simple Harmonic Motion（简谐运动）
Definition: An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely directed.
$$F_x=-kx=ma$$
$$\omega=\sqrt{\frac{k}{m}}$$
$$x=A\cos(\omega t+\phi)$$
$$E=\frac{1}{2}kA^2$$
Note
For a simple harmonic oscillator, the period and angular frequency is an intrinsic property of the system.
However, the amplitude and phase are two arbitrary constants that depend on the initial condition of the system.
Single Pendulum（单摆）
$$\omega=\sqrt{\frac{g}{L}}$$
Physical Pendulum（物理摆）
$$\tau=mgd\sin\theta=mgd\theta$$
$$\alpha=\frac{d^2\theta}{dt^2}$$
$$\tau=I\alpha$$
$$\frac{d^2\theta}{dt^2}=\frac{mgd}{I}\theta=\omega^2\theta$$
$$\omega=\sqrt{\frac{mgd}{I}}$$
Damped Oscillation（阻尼振动）
$$F_D=bv$$
（$b$: damping coefficient）
$$x=Ae^{-\frac{b}{2m}t}\cos(\omega t+\phi)$$
$$\omega=\sqrt{\frac{k}{m}-(\frac{b}{2m})^2}$$
Forced Oscillation（受迫振动）
When the object is doing damped oscillation, if we exert a force $F_{ext}=cos(\omega't)$, whose $\omega$ is determined by the environment, the the equation will be like this:
$$x=Ae^{-\frac{b}{2m}t}\cos(\omega t+\phi)+A'\cos(\omega't+\phi')$$
Note
$Ae^{-\frac{b}{2m}t}\cos(\omega t+\phi)$ is called transient solution.
$A'\cos(\omega't+\phi')$ is called steady solution.
$\omega=\sqrt{\frac{k}{m}-(\frac{b}{2m})^2}$
$\omega'$ is driving frequency,not the intrinsic frequency of the harmonic oscillator.
When $t$ is big enough, the result tends to be the steady solution.
- slow drive: $\omega'<\sqrt{\frac{k}{m}}$ The driving force is slow enough that the oscillator can follow the force after the transient motion decays.
- fast drive: $\omega'>\sqrt{\frac{k}{m}}$ The driving force is fast such that the oscillator cannot follow the force and lags behind ( $\pi$ out of phase). Note that the amplitude is smaller than that for slow drive.
- resonance（共振）: $\omega'=\sqrt{\frac{k}{m}}$ The amplitude quickly grows to a maximum. After the transient motion decays and the oscillator settles into steady state motion, the displacement $\frac{\pi}{2}$ out of phase with force.
Normal Mode（简正模）
A normal mode of an oscillating system is a pattern of motion in which all parts of the system oscillate harmonically with the same frequency and phase. The most general motion of the system is a superposition（叠加）of its normal modes.
Example
How to find normal mode?
- 1.Assume normal mode:
$x_i=A_i\cos(\omega t+\phi), i=1,2$ - 2.Find equations of motion:
$m\frac{d^2x_1}{dt^2}=-k'x_1-k(x_1-x_2)$
$m\frac{d^2x_2}{dt^2}=-k'x_2-k(x_2-x_1)$ - 3.Substitute into the formula:
$m\omega^2A_1=(k'+k)A_1-kA_2$
$m\omega^2A_2=-kA_1+(k'+k)A_2$ - 4.Transform the equations into matrix format:
$\begin{pmatrix}
k'+k-m\omega^2&-k\\
-k&k'+k-m\omega^2
\end{pmatrix}
\begin{pmatrix}
A_1\\
A_2
\end{pmatrix}=0$ - 5.To ensure there are solutions, the determinant（行列式）of the matrix should be 0:
$\omega_1=\sqrt{\frac{k'+2k}{m}}, A_1=-A_2$
$\omega_2=\sqrt{\frac{k'}{m}}, A_1=A_2$
Elastic Modulus（弹性模量）
$$Elastic~modulus\equiv\frac{stress}{strain}$$
Definition
Stress（应力）： A quantity that is proportional to the force causing a deformation; more specifically, stress is the external force acting on an object per unit cross-sectional area.
Strain（应变）： A measure of the degree of deformation.
2.2 Wave Motion
Reflection of Waves
Linear Wave Equation
$$\frac{\partial^2y}{\partial t^2}=v^2\frac{\partial^2y}{\partial x^2}$$
The linear wave equation applies in general to various types of linear waves.
Wave Function
$$y=A\sin(kx-\omega t+\phi)$$
Note
$k=\frac{2\pi}{\lambda}$（angular wave number）
$\omega=\frac{2\pi}{T}$（angular frequency）
The Speed of Waves on Strings
$$v=\sqrt{\frac{F}{\mu}}$$
Proof
$\Delta m=\mu\Delta x$
（$\mu$ is linear mass density）
$F_{1x}=F_{2x}\approx F$
（$F$ is tension in the string）
$\Delta ma_y=F_y$
$\mu\Delta x\frac{\partial^2y}{\partial t^2}=F\frac{\partial y}{\partial x}|{x+\Delta x}-F\frac{\partial y}{\partial x}|$
$\frac{\partial^2y}{\partial t^2}=\frac{F}{\mu}\frac{\partial^2y}{\partial x^2}$
according to the linear wave equation:
$v=\sqrt{\frac{F}{\mu}}$
Rate of Energy Transfer
$$P(x,t)=F_y(x,t)·v_y(x,t)$$
$$=-F\frac{\partial y}{\partial x}·\frac{\partial y}{\partial t}$$
$$=Fk\omega A^2\cos^2(kx-\omega t)$$
$$P_{avg}=\frac{1}{2}Fk\omega A^2=\frac{1}{2}\mu\omega^2A^2v$$
Interference
Same frequency, wavelength, amplitude, direction. Different phase.
$$\Delta r=|r_1-r_2|=\begin{cases} n\lambda:in~phase\\ (n+\frac{1}{2})\lambda:out~of~phase \end{cases}$$
Beating
Beating is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies.
Beat frequency:
$$f_b=|f_1-f_2|$$
Standing Waves
Same frequency, wavelength, amplitude. Different direction.
Variations of Pressure of Sound Waves
$$p(x,t)=-B\frac{\partial u(x,t)}{\partial x}$$
Note
$B$ is bulk modulus.
Speed of Sound in a Fluid
$$v=\sqrt{\frac{B}{\rho}}$$
Note
$\rho$ is the density of the medium in equilibrium.
Sound Intensity
$$I(x,t)=p(x,t)v(x,t)$$
Note
Here $v$ is the speed of phonon.
$$I=\frac{1}{2}\rho v(\omega A)^2$$
Note
Here $v$ is the speed of sound.
Decibel（分贝）
$$\beta=10\log_{10}(\frac{I}{I_0})$$
$$I_0=1.00\times10^{-12}W/m^2$$
Doppler Effect（多普勒效应）
$$f'=\frac{v+v_O}{v-v_S}f$$
Note
$v_O$ is the speed of observer, + if from $O$ to $S$.
$v_S$ is the speed of source, + if from $S$ to $O$.
Shock Wave（激波）
A shock wave happens when the source is moving faster than the speed of wave.
