Waves

Chem 3240 · Lecture 2.1

Davit Potoyan

What is a wave?

  • A wave is a self-propagating disturbance
  • Transfers energy, momentum, information
  • Does not transport matter
  • Governed by the wave equation, a PDE

Two key families: traveling waves (propagate) and standing waves (oscillate in place)

Kinds of waves

  • Medium disturbance: sound, strings, water
  • Quantum waves: complex wavefunctions for electrons, atoms
  • Electromagnetic waves: need no medium, travel in vacuum
  • Gravitational waves: ripples in spacetime

Transverse vs longitudinal

  • Transverse: disturbance is perpendicular to propagation
  • Longitudinal: disturbance is along propagation

Defining a wave mathematically

  • A disturbance \(u\) depends on space and time: \(u = f(x, t)\)
  • A surfer riding the wave sees it frozen at \(x'\)
  • A shore observer sees the front move: \(x = x' + vt\)

A right-moving wave of fixed shape:

\[u(x,t) = f(x-vt)\]

Set the front constant: \(x - vt = const \Rightarrow x = vt + const\) (moves right). The form \(f(x+vt)\) moves left.

Periodic traveling waves

A sine wave traveling along \(x\):

\[y(x,t)= A \sin(kx-\omega t+\phi)\]

  • Amplitude \(A\): max disturbance
  • Wave number \(k\): periodicity in space
  • Angular frequency \(\omega = kv\): periodicity in time
  • Phase \(\phi\): starting point

Complex representation

  • Easier to compute with complex exponentials, take real or imaginary part at the end

\[u(x,t) = Ae^{i(kx-\omega t)}\]

  • Real and imaginary parts are cosine and sine
  • The phasor rotates as \(t\) advances at fixed \(x\)

Periodicity in space and time

  • Repeats every wavelength \(\lambda\) in space: \(k\lambda = 2\pi\)

\[k=\frac{2\pi}{\lambda}\]

  • Repeats every period \(T\) in time, frequency \(\nu = 1/T\): \(\omega T = 2\pi\)

\[\omega=\frac{2\pi}{T} = 2\pi\nu\]

Wavelength, frequency, and speed are linked by \(\omega = kv\), i.e. \(\lambda \nu = v\)

The classical wave equation

  • Two \(x\)-derivatives: \(u_{xx}^{''} = -k^2 u\)
  • Two \(t\)-derivatives: \(u_{tt}^{''} = -\omega^2 u\)
  • Take the ratio to eliminate \(u\): \(\dfrac{k^2}{\omega^2} = \dfrac{1}{v^2}\)

\[\frac{\partial^2 u(x,t)}{\partial x^2 } = \frac{1}{v^2}\frac{\partial^2 u(x,t)}{\partial t^2}\]

Solutions are wave functions of space and time

Combining waves: interference

  • Superposition: if \(u_A\) and \(u_B\) solve the wave equation, so does \(u_C = u_A + u_B\)
  • Interference: combining waves yields greater, lower, or equal amplitude

Interference of two phases

Summing two waves differing only in phase:

\[\Psi_{\text{total}} = 2 \cos\left( \frac{\phi_1 - \phi_2}{2} \right) e^{i \left( kx - \omega t + \frac{\phi_1 + \phi_2}{2} \right)}\]

  • In phase (\(\phi=0\)): amplitude doubles
  • Out of phase (\(\phi=\pi\)): cancels

Takeaway

A wave is a disturbance described by \(u=f(x \pm vt)\) that obeys the classical wave equation \(u_{xx} = \frac{1}{v^2}u_{tt}\), with \(k=2\pi/\lambda\), \(\omega=2\pi\nu\), and \(\omega=kv\); superposing waves produces interference.