Superposition and Interference

Superposition and Wave Interference

One of the most fundamental principles of wave motion is superposition. If two or more waves travel through the same medium, their displacements add algebraically. This gives rise to constructive and destructive interference:

• Constructive interference: When two waves are in phase, their amplitudes add, leading to a larger resultant wave.
• Destructive interference: When two waves are out of phase by $\pi$ radians (or $180^{\circ }$), they cancel out.

What do we observe?

• Optical and sound power
• Quantum probability

Adding Waves

• Linear superposition
• Interference
• Intensity and Probability

Where are we going with this?

• Understanding waves helps transition into quantum mechanics.
• Quantum physics of photons passing through two paths.
• Single photon experiments as tests of quantum mechanics.
• Interference of two waves at the same point.

Intensity, Probability, and Amplitude

• Energy of sinusoidal waves $\propto A^2$
• Examples:
  • String waves: $P_{\text{avg}}\propto A^2$
  • Sound waves: $I\propto A^2$
  • Electromagnetic waves: $I\propto E_{\max }^2$

Quantum Waves

• Probability of finding a particle: $|\Psi {|}^{2}dV$
• Probability current density: $J=\frac{\hslash k}{m}{A}^2$

Key Takeaways

• To determine probability or rate, square the amplitude of the wave.
• Though discussed using sound waves, concepts apply broadly.

Power and Wavefunctions

• Sound wave displacement:
  $y(x,t)=A\sin k(x-vt)$
• Instantaneous power per unit area:
  $\frac{P}{\text{area}}=\rho B{\omega }^2{y}^2(x,t)=\rho B{\omega }^2{A}^2{\sin }^2(kx-vt)$
• Time-averaged intensity:
  $⟨\frac{P}{\text{area}}⟩=\frac{1}{2}\rho B{\omega }^2{A}^2$

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Time Averages of Waves

• Mathematical definition:
  $⟨y(t)⟩=\frac{1}{T}{\int }_0^{T}y(t)dt$
• Sine and cosine functions:
  • Averages over a full cycle are zero.
  • Average of ${\sin }^2(\omega t)$ and ${\cos }^2(\omega t)$ is $1/2$.

Addition of Waves

• Linear Superposition:
  $y(x,t)=y_1(x,t)+y_2(x,t)$
• Interference:
  • Waves overlap in space.
  • Coherent waves (constant phase difference) show strong interference effects.
  • Resultant amplitude = sum of individual field amplitudes.

Adding Two Waves

• Equal amplitudes:
  $\sin a+\sin b=2\sin \left(\frac{a+b}{2}\right)\cos \left(\frac{a-b}{2}\right)$
• Unequal frequencies:
  $\sin ({\omega }_{1}t)+\sin ({\omega }_{2}t)=2\sin \left(\frac{{\omega }_1+{\omega }_2}{2}t\right)\cos \left(\frac{{\omega }_1-{\omega }_2}{2}t\right)$
• Beats:
  • Rapid oscillation at average frequency $\frac{{\omega }_1+{\omega }_2}{2}$.
  • Slow modulation at beat frequency $\frac{{\omega }_1-{\omega }_2}{2}$.

Interference of Waves

• Two waves with equal amplitudes and frequencies but different phase:
  $y_{\text{tot}}=y_1+y_2=2y_0\cos \left(\frac{\Delta \varphi }{2}\right)\sin \left(\omega t+\frac{{\varphi }_1+{\varphi }_2}{2}\right)$
• Interference Intensity:
  $I_R=4I_0{\cos }^2\left(\frac{\Delta \varphi }{2}\right)$

Controlling Phase Difference

• Methods:
  • Splitting a wave with a partially reflecting mirror.
  • Sending waves through different paths. See Mach-Zehnder Interferometer