标签：use normal serve newton init mes mic max original
2020-4-13 chs_2020 General Physics I (H)
Waves that obey the superposition principle are called linear waves and are generally characterized by small amplitudes. Waves that violate the superpostion principle are called nonlinear waves and are often characterized by large amplitudes. The rigorous definition of linear/non-linear waves follows.
Def A wave is called linear/non-linear if the DE governing this wave is linear/non-linear.
A linear combination of arbitary solns of a homogeneous linear equation is also a soln of the same linear equation.
The combination of separate waves in the same region of space to produce a resultant wave is called interference.
By Newton‘s third law, the support must exert an equal and opposite reaction force on the string, which causes the pulse to invert upon reflection.
Free Boundary Condition
When it reaches the post, the pulse exerts a force on the free end of the string, causing the ring to accelerate upward. Afterwards, the downward component of the tension force pulls the ring back down.
Situation in which the boundary is intermediate between these two extremes - part of the incident pulse is reflected and part undergoes transmission, i.e. some of the pulse passes through the boundary.
? Equilibrium positions: \(X_n=na\)
? Deviations from the equilibrium: \(u_n=x_n-X_n\)
? Hence \(u_{n+1}-u_n=x_{n+1}-x_{n}-a\)
Our aim is to give the Wave Equation. First derive the expression for Interatomic Potential, then by differentiation, get the Equation of Motion for every single particle.
Here \(K\) is given by the second derivative of \(\phi\). Note that the Fisrt Derivative Vanishes At The Equilibrium, since there is no force between two atoms that are at their equilibrium positions.
If a travelling wave \(f=f(x\pm vt)\) is a soln of this equation, then we easily have \(c^2=v^2\), and hence $c $ is the propagation speed of the wave.
Assume that the wavelength is much larger than the amplitude, then by Newton‘s second law and small-angle approx we have
It‘s reasonable to approx \(T(x+\Delta x)\) and \(T(x)\) by \(T\), and then
The most important family of the solns for the Linear Wave Equation are \(y=A\sin(kx-\omega t+\phi)\), where \(k=\frac{2\pi}{\lambda}\) is the angular wave number, \(\omega=\frac{2\pi}{T}\) is the angular frequency and \(\phi\) is the phase const. Note that \(v=\frac{\lambda}{T}\), we rewrite the Sinusodial Wave Equation as \(y=A\sin\Big[\frac{2\pi}{\lambda}(x-\lambda t)\Big]\), which is of the form \(y=f(x-vt)\).
From \(y=A\sin(kx-\omega t+\phi)\) we derive \(v_y=-\omega A\cos(kx-\omega t)\) and \(a_y=-\omega^2A\sin(kx-\omega t)\), and hence \(v_{y,max}=\omega A\) and \(a_{y,max}=\omega^2A\).
Recall that for simple harmonic oscillation, the total energy \(E=K+U\) is const, which means \(\mathrm{d}E=\mathrm{d}K+\mathrm{d}U\), and therefore
Same frequency,wavelength,amplitude,direction. Different phase.
Beating is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies (\(\,\)i.e. \(|f_1-f_2|\) is very small).
The amplitude of the resultant wave varies in time.
The two neighboring maxima in the envelop function are seperated by \(\displaystyle\Delta t=\frac{1}{f_1-f_2}\). And here comes the Beat Frequency: \(\displaystyle f_b=|f_1-f_2|\).
Standing Waves - Same frequency,wavelength,amplitude. Different direction.
The distance between adjacent nodes/antinodes is half the wavelength, and the distance between a node and an adjacent antinode is one fourth the wavelength.
No energy is transmitted along the string across a node, and energy does not propagate in a standing wave.
Consider a string fixed at both ends (Transverse Version). Consider standing waves in air columns (Longitudinal Version).
In general, the wavelength of the various normal modes for a string of length \(L\) fixed at both ends are \(\lambda_{n}=2L/n\quad n=1,2,3,\mathrm{etc.}\), with corresponding frequencies \(f_n=v/\lambda_{n}=n\cdot v/2L\quad n=1,2,3,\mathrm{etc.}\) , which forms a harmonic series, and the normal modes are called harmonics.
(initial density) \(\rho_0\longrightarrow\rho=\rho_0+\rho_e\) (displaced density)
Displacement change vs density excess
The particle number do not change:
Pressure change vs density excess
Wave equation for sound
$$
\text{Force exerted on the volume (per unit area) = Pressure difference}
$$
Note that \(\displaystyle\frac{\partial P}{\partial x}=\frac{\partial P_e}{\partial x}=\kappa\frac{\partial \rho_e}{\partial x}=-\kappa\rho_0\frac{\partial^2s}{\partial x^2}\), hence we obtain \(\displaystyle \frac{1}{\kappa}\frac{\partial^2s}{\partial t^2}=\frac{\partial^2s}{\partial x^2}\).
Velocity of sound wave
Take the periodic sound wave soln
Excess pressure wave is \(\pi/2\) out-of-phase with the displacement
Rmk Equilibrium atmospheric pressure \(\approx 1\times 10^5\mathrm{Nm}^{-2}\). Audible excess pressure amplitude \(\approx 2\times 10^{-5}-30\,\mathrm{Nm}^{-2}\). We only hear the order-of-magnitude...
Decibel
The sound level is measured in the logarithmic scale
The scale is called decibels (dB). A decibel is \(1/10\) of a bel.
The intensity of a wave is roughly the power per unit area, more precisely,
The elastic property of solid/fluid plays the role of the restoring force to support sound waves.
Stress the external force acting on an object per unit cross-sectional area
Strain A measure of the degree of deformation
Elasstic modulus The constant of proportionality.(depends on the material and the nature of the deformation)
$$
\Delta P=-B\frac{\Delta V}{V_i}
$$
Hence we have
Note that \(\displaystyle\frac{\partial P}{\partial x}=\frac{\partial P_e}{\partial x}=-B\frac{\partial^2s}{\partial x^2}\), we get the wave equation
and the speed of sound is
Generally, the speed of all machanical waves follows an expression of the form
Bulk modulus of water \(B=2\times 10^9\mathrm{Pa}\)
Density of water \(\rho=1\times 10^3\mathrm{kg/m}^3\)
\(\implies\) \(v=\sqrt{B/\rho}=1414\mathrm{m/s}\)
Speed (wave length) in water \(>\) Speed (wave length) in air
The sonar wave at this frequency can sense objects that are roughly as small as the wave length.
1)
In the normal case,
2)
In this case, \(\lambda\) is observed unchanged yet the speed is now \(c+v_0\). Hence
3)
In this case, \(v\) is observed unchanged yet the wave length observed is not \(\lambda\), as we can see in the picture below, but \(\lambda-v_sT\). Hence
4) In a Nutshell (Assuming \(v_s<c\))
$$
\begin{align}f‘&=\frac{1}{T‘}=\frac{c+v_0}{\lambda-v_sT}=\frac{1+\frac{v_0}{c}}{1-\frac{v_s}{c}}f\qquad\text{getting closer to each other}\\f‘&=\frac{1}{T‘}=\frac{c-v_0}{\lambda-v_sT}=\frac{1-\frac{v_0}{c}}{1-\frac{v_s}{c}}f\qquad\text{S chasing O}\end{align}
$$
The other two cases are
$$
\begin{align}&f‘=\frac{1-\frac{v_0}{c}}{1+\frac{v_s}{c}}f\qquad\text{going away from each other}\\&f‘=\frac{1+\frac{v_0}{c}}{1+\frac{v_s}{c}}f\qquad\text{O chasing S}\end{align}
$$
Macn Number: \(v_s/v\)
\(\displaystyle\sin\theta=\frac{v_s}{v}\)
标签：use normal serve newton init mes mic max original
原文地址：https://www.cnblogs.com/chs2020/p/12723167.html
