Wave Motion

Wave Motion

Wave Function

• A wave that causes the particles to move parallel to the direction of wave motion is called a longitudinal wave.
• A wave that causes the particles to move perpendicular to the wave motion is called a transverse wave.

The displacement \(y\) depends on both \(x\) and \(t\). For this reason, it is often written as \(y(x,t)\), which is called wave function. - Right moving : \(y(x, t) = f(x - vt)\) - Left moving : \(y(x, t) = f(x + vt)\)

Wave Equation

Set $y = f(x + vt) = f(u) $, then

\[ \begin{gathered} \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial u^2} \\\\ \frac{\partial^2 y}{\partial x^2} = \frac{\partial^2 y}{\partial u^2} \\\\ \Rightarrow \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \end{gathered} \]

Linear Wave

Phenomenon

• Superposition Principle : If two or more traveling waves are moving through a medium, the resultant wave function at any point is the algebraic sum of the wave functions of the individual waves.
• Linear Waves :
  • Waves that obey the superposition principle are called linear waves.
  • Waves that violate the superposition principle are called nonlinear waves.
• The combination of separate waves in the same region of space to produce a resultant wave is called interference.

• Reflection :
  • When the string's end is fixed, the pulse moves back in the opposite direction.
  • When the string is tied to a light ring that is free to slide vertically on a smooth post, again, the pulse moves back in the same direction.
• Transmission :
  • When the boundary is intermediate between these two extremes, part of the incident pulse is reflected and part undergoes transmission.
  • When from soft to hard, the reflected wave is invertd.
  • When from hard to soft, the reflected wave is not invertd.

Speed of Wave in a Solid

Set \(x_n\) to be the positions of atoms, and $X_n = na $ to be the equilibrium positions. Let the wave function be $u_n = u(X_n, t) $, then we can get \(\(U^{\text{harm}} = \frac{1}{2}K\sum_n (u_{n+1}-u_n)^2\)\)

So for every atom, there are

\[ \begin{aligned} M\frac{\partial^2 u_n}{\partial t^2} &= -\frac{dU^{\text{harm}}}{d u_n} \\ &= K(u_{n+1} - u_n) - K(u_n - u_{n-1}) \\ &= Ka\left.\frac{\partial u}{\partial x}\right|_{X_n} - Ka\left.\frac{\partial u}{\partial x}\right|_{X_{n-1}} \\ &= Ka^2\left.\frac{\partial^2 u}{\partial x^2}\right|_{X_n} \end{aligned} \]

So we can get \(\(\frac{\partial^2 u}{\partial t^2} = \frac{Ka^2}{M}\frac{\partial^2 u}{\partial x^2}\)\)

Therefore \(\(v = a\sqrt\frac{K}{M}\)\)

Sinusoidal Wave

Sinusoidal Wave Function

\[ \begin{aligned} y &= A\sin(kx - \omega t + \phi) \\ &= A\sin\left[2\pi \left(\frac{x}{\lambda} - \frac{t}{T}\right) + \phi \right] \\ &= A\sin\left[ \frac{2\pi}{\lambda}(x - vt) + \phi \right] \end{aligned} \]

\[ k = \frac{2\pi}{\lambda} = \frac{2\pi f}{v}, \quad \omega = \frac{2\pi}{T} = 2\pi f, \quad v = \frac{\omega}{k} \]

Speed of Wave on a String

Set \(\mu\) to be the linear density, then we can get

\[ \begin{gathered} \Delta ma_y = \mu \Delta x \frac{\partial^2 y}{\partial t^2} = F \left.\frac{\partial y}{\partial x}\right|_{x+\Delta x} - F \left.\frac{\partial y}{\partial x}\right|_{x} = F \frac{\partial^2y}{\partial x^2} \cdot \Delta x \\\\ \Rightarrow \frac{\partial^2y}{\partial t^2} = \frac{F}{\mu} \frac{\partial^2y}{\partial x^2} \end{gathered} \]

Therefore \(\(v = \sqrt{\frac{F}{\mu}}\)\)

Energy Transfer

The power of energy transfer at a particular position and time is

\[ \begin{aligned} P(x, t) &= F_y(x, t)v_y(x, t) \\ &= -F \frac{\partial y(x, t)}{\partial x} \frac{\partial y(x, t)}{\partial t} \\ &= Fk\omega A^2\cos^2(kx - \omega t) \end{aligned} \]

So The average rate of energy transfer is \(\(P_{avg} = \frac{1}{2}Fk\omega A^2 = \frac{1}{2}\mu\omega^2 A^2v\)\)

Different Cases of Superposition

Interference Case

\[ \begin{aligned} y = y_1 + y_2 &= A\left[ \sin(kx - \omega t) + \sin(kx - \omega t + \phi) \right] \\ &= 2A\cos\left( \frac{\phi}{2} \right) \sin(kx - \omega t + \frac{\phi}{2}) \end{aligned} \]

\[ \begin{gathered} \Delta r = v\Delta t = \lambda f \Delta t \\ \phi = \omega \Delta t = 2\pi f \Delta t \end{gathered} \]

So we can get \(\(\Delta r = \frac{\phi}{\pi}\cdot \frac{\lambda}{2}, \quad \frac{\phi}{2} = \pi \frac{\Delta r}{\lambda}\)\)

When \(\displaystyle \cos\left( \frac{\phi}{2} \right) = \pm 1\) or \(\displaystyle \Delta r = (2n)\cdot \frac{\lambda}{2}\), the waves are said to be everywhere in phase and thus interfere constructively.

When \(\displaystyle \cos\left( \frac{\phi}{2} \right) = 0\) or \(\displaystyle \Delta r = (2n+1)\cdot \frac{\lambda}{2}\), the resultant wave has zero amplitude everywhere, as a consequence of destructive interference.

Beating Case

Beating is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies.

Set

\[ \begin{gathered} y_1 = A\cos\omega_1t = A\cos 2\pi f_1t \\ y_2 = A\cos\omega_2t = A\cos 2\pi f_2t \end{gathered} \]

Then we can get

\[ \begin{aligned} y = y_1 + y_2 &= A(\cos 2\pi f_1t + \cos 2\pi f_2t) \\ &= \left[ 2A \cos 2\pi\left( \frac{f_1-f_2}{2} \right)t \right]\cos 2\pi\left( \frac{f_1+f_2}{2} \right)t \end{aligned} \]

Beat frequency: $f_b = |f_1 - f_2| $

Standing Wave

Set

\[ \begin{gathered} y_1 = A\sin(kx - \omega t) \\ y_2 = A\sin(kx + \omega t) \\ \end{gathered} \]

Then we can get

\[ \begin{aligned} y = y_1 + y_2 &= A\sin(kx - \omega t) + A\sin(kx + \omega t) \\ &= (2A\sin kx)\cos\omega t \end{aligned} \]

Nodes: \(kx = n\pi\)

Antiodes: \(\displaystyle kx = (n + \frac{1}{2})\pi\)

The wavelengths of standing waves for a string fixed at both ends are $\displaystyle \lambda_n = \frac{2L}{n} $

Then the frequencies are $\displaystyle f_n = \frac{v}{\lambda_n} = n\frac{v}{2L} $

This series is called harmonic series. The frequency with \(n=1\) is called fundamental frequency or the first harmonic. The frequency with \(n>1\) is called the \(n\)th harmonic, or the \((n-1)\)th overtones.

Sound Wave

• Sound waves in liquids and air are longitudinal waves.
• Sound waves in solids can be either longitudinal or transverse.

Pressure Fluctuation

For a sound wave confined to a tube, there are \(\(\Delta V = S[u(x + \Delta x, t) - u(x, t)]\)\)

So we can get

\[ \begin{aligned} \frac{\text{d}V}{V} &= \lim_{\Delta x \rightarrow 0} \frac{S[u(x + \Delta x, t) - u(x, t)]}{S\Delta x} \\ &= \frac{\partial u(x,t)}{\partial x} \end{aligned} \]

Hence \(\(\Delta p(x, t) = -B \frac{\text{d}V}{V} = -B \frac{\partial u(x,t)}{\partial x}\)\)

Speed of Sound in a Fluid

We consider an idealized case of a sound wave confined to a tube. Set \(A\) to be the area of cross section, \(v_y\) to be the initial speed the fluid is pushed as well as the speed of fluid, \(V\) to be the volume of fluid being pushed, \(\Delta V\) to be the reduced volume of the fluid, then we can get

\[ \begin{gathered} V = Avt \\\\ \Delta V = Av_yt \\\\ \Delta p = -B \frac{\Delta V}{V} = -B \frac{v_y}{v} \end{gathered} \]

By \(P = I\), we can get

\[ \begin{gathered} P = \rho V v_y = \rho (Av_yt)v \\\\ I = Ft = \Delta p At = B \frac{Av_yt}{v} \\\\ \Rightarrow v = \sqrt{\frac{B}{\rho}} \end{gathered} \]

Sound Intensity

Sound intensity \(I\) is defined as the power carried by sound waves per unit area in a direction perpendicular to that area, that is to say, \(\boldsymbol{I} = p\boldsymbol{v}\).

\[ \begin{aligned} I(x, t) &= \Delta p(x, t) v(x, t) \\ & = -B \frac{\partial u(x,t)}{\partial x} \frac{\partial u(x,t)}{\partial t} \\ &= kB\omega s_{max}^2 \cos^2(kx - \omega t + \phi) \end{aligned} \]

So the average is

\[ \begin{aligned} I = \frac{\mathscr{P}_{avg}}{A} &= \frac{1}{2} kB\omega s_{max}^2 \\ &= \frac{1}{2} k\rho v^2\omega s_{max}^2 \\ &= \frac{1}{2} \rho v(\omega s_{max})^2 \\ \end{aligned} \]

which means \(I \sim s^2\).

Sound intensity level is the logarithmic measure of the intensity of a sound relative to a reference value, that is

\[\beta = 10 \log\left( \frac{I}{I_0} \right) (\text{dB})\]

\[I_0 = 1.00\times 10^{-12} \text{ W}/\text{m}^2\]

Spherical Waves

• Wavefronts are surfaces over which the oscillations have the same value.
• Rays are directed lines perpendicular to the wavefronts that indicate the direction of travel of the wavefronts.

\[ \begin{gathered} I = \frac{\mathscr{P}_{avg}}{A} = \frac{\mathscr{P}_{avg}}{4\pi r^2} \\\\ \Rightarrow \phi(r, t) = \frac{s_0}{r}\sin(kr - \omega t) \end{gathered} \]

Doppler Effect

Moving Observer, Stationary Source Assume \(v_O\) is positive if the observer is moving from \(O\) to \(S\), then the speed of the sound relative to the observer become \(\(v' = v + v_O\)\)

However, the wavelength of the sound the observer receive is unchanged, so \(\(f' = \frac{v'}{\lambda} = \frac{v + v_O}{\lambda} = (1 + \frac{v_O}{v})f\)\)

Moving Source, Stationary Observer Assume \(v_S\) is positive if the source is moving from \(S\) to \(O\), then the wavelengh of the sound become \(\(\lambda' = \lambda - v_ST = \lambda - \frac{v_S}{f}\)\)

However, the speed of the sound the observer receive is unchanged, so \(\(f' = \frac{v}{\lambda'} = \frac{v}{\lambda - \dfrac{v_S}{f}} = \frac{v}{\dfrac{v}{f} - \dfrac{v_S}{f}} = \frac{1}{1 - \dfrac{v_S}{v}}f\)\)

Both Source and Observer in Motion The speed of the sound the observer receive is $\displaystyle v' = v + v_O $, the wavelengh of the sound the observer receive is $\displaystyle \lambda' = \lambda - \frac{v_S}{f} $, so we can get \(\(f' = \frac{v'}{\lambda'} = \frac{v + v_O}{v - v_S} f\)\)

Shock Waves

When \(v_S\) is higher than the wave velocity \(v\), the envelope surface of the wave surface is a cone, which is called Mach cone.

Set the conical angle to be \(\theta\), then we can get \(\(\sin\theta = \frac{vt}{v_St} = \frac{v}{v_S}\)\)

By this, we can set Mach number: \(v_S/v\)

At the shock front, which is the generatrix of the cone, it will form constructive interference and generate a wave with a very large amplitude.