Review of Lectures on Wave

Posted 2021-02-28 chs2020

Review of Lectures on Wave

2020-4-13 chs_2020 General Physics I (H)

Wave Motion

Linear Wave

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.

Superpostion Principle of Linear Equations

A linear combination of arbitary solns of a homogeneous linear equation is also a soln of the same linear equation.

Interference

The combination of separate waves in the same region of space to produce a resultant wave is called interference.

Reflection

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.

Transmission

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.

Model of Monoatomic Crystal

? 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.

1. We consider nearest-neighbor interations only. By Taylor Expansion,

[egin{align}phi(x_{n+1}-x_n)&=phi_0+frac{1}{2}K(x_{n+1}-x_n-a)^2+cdots&=phi_0+frac{1}{2}K(u_{n+1}-u_n)^2+cdotsend{align} ]

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.

[U=frac{1}{2}Ksum_{n}(u_{n+1}-u_{n})^2 ]

2. By Newton‘s 2nd law,

[egin{align}Mddot{u}_n=frac{mathrm{d}U }{mathrm{d}u_n}&=K(u_{n+1}-u_{n})-K(u_n-u_{n-1})&approx (Ka)frac{partial u}{partial x}Big|_{X_n+frac{a}{2}}-(Ka)frac{partial u}{partial x}Big|_{X_n-frac{a}{2}}&approx (Ka^2)Big(frac{partial^2u}{partial x^2}Big)_{X_n}end{align} ]

3. Rewrite the equation as

[frac{partial^2 u}{partial t^2}=frac{Ka^2}{M}cdotfrac{partial^2u}{partial x^2}qquad ext{where} v=asqrt{frac{K}{M}} ext{is the propagation speed of the wave}downarrow ]

Coefficient in the Wave Equation

[ ext{Linear Wave Equation:}qquadfrac{1}{c^2}frac{partial^2f}{partial t^2}-frac{partial^2f}{partial x^2}=0 ]

If a travelling wave (f=f(xpm 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.

Sinusoidal Wave

The Speed of Waves on Strings

Assume that the wavelength is much larger than the amplitude, then by Newton‘s second law and small-angle approx we have

[muDelta xfrac{partial^2y}{partial,t^2}=T(x+Delta x)frac{partial y}{partial x}Big|_{x+Delta x}-,T(x)frac{partial y}{partial x}Big|_{x} ]

It‘s reasonable to approx (T(x+Delta x)) and (T(x)) by (T), and then

[frac{partial^2y}{partial,t^2}=frac{T}{mu}frac{partial^2y}{partial,x^2} ,qquad v=sqrt{frac{T}{mu}} ]

Sinusodial Waves

The most important family of the solns for the Linear Wave Equation are (y=Asin(kx-omega t+phi)), where (k=frac{2pi}{lambda}) is the angular wave number, (omega=frac{2pi}{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=AsinBig[frac{2pi}{lambda}(x-lambda t)Big]), which is of the form (y=f(x-vt)).

From (y=Asin(kx-omega t+phi)) we derive (v_y=-omega Acos(kx-omega t)) and (a_y=-omega^2Asin(kx-omega t)), and hence (v_{y,max}=omega A) and (a_{y,max}=omega^2A).

Rate of Energy Transfer (Sinusoidal Wave on Strings)

[Delta U=frac{1}{2}(Delta m)omega^2y^2=frac{1}{2}(muDelta x)omega^2y^2 ]

[egin{align} dU&=frac{1}{2}muomega^2[Asin(kx-omega t)]^2mathrm{d}x&=frac{1}{2}muomega^2A^2sin^2(kx-omega t)mathrm{d}x end{align} ]

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

[P=frac{mathrm{d}E}{mathrm{d},t}=frac{frac{1}{2}muomega^2A^2mathrm{d}x}{dt}=frac{1}{2}muomega^2A^2v ]

Interference

Same frequency,wavelength,amplitude,direction. Different phase.

[egin{align} y_1&=Asin(kx-omega t)y_2&=Asin(kx-omega t+phi) y&=y_1+y_2=2AcosBig(frac{phi}{2}Big)sinBig(kx-omega t+frac{phi}{2}Big) end{align} ]

1. When (cos(phi/2)=pm 1), the waves are said to be everywhere in phase and thus interfere constructively. 2) When (cos(phi/2)=0), the resultant wave has zero amplitude everywhere, as a consequence of destructive interfere.

Beat

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).

[egin{align} y_1&=Acos(2pi f_1 t)y_2&=Acos(2pi f_2 t) y&=y_1+y_2=2AcosBig(2pifrac{f_1-f_2}{2}tBig)cosBig(2pifrac{f_1+f_2}{2}tBig) end{align} ]

The amplitude of the resultant wave varies in time.

[displaystyle A_{mathrm{resultant}}=2AcosBig(2pifrac{f_1-f_2}{2}tBig) ]

The two neighboring maxima in the envelop function are seperated by (displaystyleDelta t=frac{1}{f_1-f_2}). And here comes the Beat Frequency: (displaystyle f_b=|f_1-f_2|).

Standing Waves

Standing Waves - Same frequency,wavelength,amplitude. Different direction.

[egin{align} y_1&=Asin(kx-omega t)y_2&=Asin(kx+omega t) y&=y_1+y_2=2Asin(kx)cdotcos(omega t) end{align} ]

1. No sense of motions in the direction of propagation of either of the original waves. 2) Every particle of the medium oscillates in simple harmonic motion with the same frequency. 3) The amplitude of the individual waves depends on the location.

[egin{align} & ext{Nodes:}quad kx=nlambdaqquad &x=frac{lambda}{2},frac{2lambda}{2},frac{3lambda}{2},cdots & ext{Antinodes:}quad kx=(n+frac{1}{2})lambdaqquad &x=frac{lambda}{4},frac{3lambda}{4},frac{5lambda}{4},cdots end{align} ]

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).

Harmonic Series

In general, the wavelength of the various normal modes for a string of length (L) fixed at both ends are (lambda_{n}=2L/nquad n=1,2,3,mathrm{etc.}), with corresponding frequencies (f_n=v/lambda_{n}=ncdot v/2Lquad n=1,2,3,mathrm{etc.}) , which forms a harmonic series, and the normal modes are called harmonics.

Sound Wave

Sound waves in air

[egin{align} ext{Displacement}&&s(x,t) ext{Density}&& ho(x,t) ext{Pressure}&&P(x,t)end{align} ]

(initial density) ( ho_0longrightarrow ho= ho_0+ ho_e) (displaced density)

Displacement change vs density excess

[ ho_e=- ho_0frac{partial s}{partial x}qquad ]

The particle number do not change:

[egin{align} ho_0(x+Delta x-x)&= ho[x+Delta x+s(x+Delta x)-x-s(x,t)] ho_0Delta x&= ho[Delta x+frac{partial s}{partial x}Delta x]&= ho_e+ ho_0frac{partial s}{partial x}qquad ext{the density excess } ho_e ext{ is tiny}end{align} ]

Pressure change vs density excess

[P_e=kappa ho_eqquadkappa=frac{dP}{d ho}Big|_{ ho_0} ]

Wave equation for sound

[frac{1}{kappa}frac{partial^2s}{partial t^2}=frac{partial^2s}{partial x^2} ]

$$ ext{Force exerted on the volume (per unit area) = Pressure difference} $$

[egin{align} ho_0Delta xfrac{partial^2s}{partial t^2}=-frac{partial P}{partial x}Delta xend{align} ]

Note that (displaystylefrac{partial P}{partial x}=frac{partial P_e}{partial x}=kappafrac{partial ho_e}{partial x}=-kappa ho_0frac{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

[egin{align}v&=sqrt{kappa}P_e&=v^2 ho_eend{align} ]

Loudness

Take the periodic sound wave soln

[s(x,t)=s_{max}cos(kx-omega t) ]

Excess pressure wave is (pi/2) out-of-phase with the displacement

[P_e(x,t)= ho_0vomega s_{max}sin(kx-omega t) ]

Rmk Equilibrium atmospheric pressure (approx 1 imes 10^5mathrm{Nm}^{-2}). Audible excess pressure amplitude (approx 2 imes 10^{-5}-30,mathrm{Nm}^{-2}). We only hear the order-of-magnitude...

Decibel

The sound level is measured in the logarithmic scale

[eta=10logleft(frac{P_{e}}{P_{e,ref}} ight)qquad mathrm{where} P_{e,ref}=2 imes 10^{-5}mathrm{Nm}^{-2} ]

The scale is called decibels (dB). A decibel is (1/10) of a bel.

Sound Intensity

The intensity of a wave is roughly the power per unit area, more precisely,

[I=frac{mathscr{P}}{A}=frac{1}{2} ho v(omega s_{max})^2 ]

Sound wave in solids/liquids

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)

• Young‘s Modulus (displaystyle Y=frac{F/A}{Delta L/L_i})
• Shear Modulus (S=frac{shear stress}{shear strain})
• Bulk Modulus (displaystyle B=-frac{Delta P}{Delta V/V_i})

$$ Delta P=-Bfrac{Delta V}{V_i} $$

[frac{Delta V}{V}=frac{A[s(x+Delta x,t)-s(x,t)]}{ADelta x}=frac{partial s(x,t)}{partial x}qquad mathrm{as} Delta x o0 ]

Hence we have

[P_e(x,t)=-Bfrac{partial s(x,t)}{partial x} ]

[ ext{Force exerted on the volume (per unit area) = Pressure difference} ]

[egin{align} ho_0Delta xfrac{partial^2s}{partial t^2}=-frac{partial P}{partial x}Delta xend{align} ]

Note that (displaystylefrac{partial P}{partial x}=frac{partial P_e}{partial x}=-Bfrac{partial^2s}{partial x^2}), we get the wave equation

[frac{1}{B/ ho_o}frac{partial^2s}{partial t^2}=frac{partial^2s}{partial x^2} ]

and the speed of sound is

[v=sqrt{B/ ho_0} ]

Generally, the speed of all machanical waves follows an expression of the form

[v=sqrt{frac{elastic prop}{inertial prop}} ]

Sonar waves

Bulk modulus of water (B=2 imes 10^9mathrm{Pa})

Density of water ( ho=1 imes 10^3mathrm{kg/m}^3)

(implies) (v=sqrt{B/ ho}=1414mathrm{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.

Doppler Effect (non-relativistic)

1)

In the normal case,

[egin{equation}f=frac{1}{T}=frac{c}{lambda}end{equation} ]

2)

In this case, (lambda) is observed unchanged yet the speed is now (c+v_0). Hence

[egin{equation}f‘=frac{1}{T‘}=frac{c+v_0}{lambda}=frac{c+v_0}{c/f}=left(1+frac{v_0}{c} ight)fend{equation} ]

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

[f‘=frac{1}{T‘}=frac{c}{lambda-v_sT}=frac{lambda f}{lambda-v_s/f}=frac{1}{1-frac{v_s}{c}}f ]

4) In a Nutshell (Assuming (v_s<c))

$$ egin{align}f‘&=frac{1}{T‘}=frac{c+v_0}{lambda-v_sT}=frac{1+frac{v_0}{c}}{1-frac{v_s}{c}}fqquad ext{getting closer to each other}f‘&=frac{1}{T‘}=frac{c-v_0}{lambda-v_sT}=frac{1-frac{v_0}{c}}{1-frac{v_s}{c}}fqquad ext{S chasing O}end{align} $$ The other two cases are $$ egin{align}&f‘=frac{1-frac{v_0}{c}}{1+frac{v_s}{c}}fqquad ext{going away from each other}&f‘=frac{1+frac{v_0}{c}}{1+frac{v_s}{c}}fqquad ext{O chasing S}end{align} $$

Shock Waves ( (v_s>v) )

Macn Number: (v_s/v)

(displaystylesin heta=frac{v_s}{v})