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06_Waves

1. Introduction to Waves

Three Main Types of Waves

  1. Mechanical Waves: Require a medium (e.g., water, sound, seismic waves).

  2. Electromagnetic Waves: Do not require a medium (e.g., visible light, radio waves).

  3. Matter Waves: The quantum mechanical view of fundamental particles.

Classification by Direction of Propagation (\(\vec{v}\))

  • Transverse Wave: The displacement is perpendicular to the direction of wave travel (\(\Delta\vec{y} \perp \vec{v}\)).

    • Example: A wave on a string attached to a wall.

  • Longitudinal Wave: The displacement is parallel to the direction of wave travel (\(\Delta\vec{y} \parallel \vec{v}\)).

    • Example: A stretched and compressed spring attached to a wall.

Requirements for Mechanical Waves

To generate and propagate a mechanical wave, three conditions must be met:

  1. A source of disturbance.

  2. A medium.

  3. A physical connection between adjacent portions of the medium.

Basic Variables of Periodic Waves

Similar to simple harmonic motion, periodic waves are described by:

  • Amplitude (\(A\))

  • Frequency (\(f\))

  • Angular Frequency (\(\omega\))

  • Period (\(T\))

  • Wavelength (\(\lambda\)): The distance between points whose oscillations differ by \(2\pi\).

2. 1D Travelling Waves and Superposition

Mathematical Description

For a 1D travelling wave, the displacement \(y\) is a function of both position \(x\) and time \(t\):

\[ y = f(x, t) \]

  • Right-moving pulse: \(y = f(x - vt)\)

  • Left-moving pulse: \(y = f(x + vt)\)

  • \(v\): Wave speed (波速 / 相位速度).

Principle of Superposition

Linear waves are waves that obey the superposition principle. When waves overlap, they add up algebraically:

\[ y'(x, t) = y_1(x, t) + y_2(x, t) \]

  • Interference: The phenomenon of combining waves.

    • Constructive Interference (相长干涉 / 建设性干涉): Pulses with the same sign add together to create a larger amplitude.

    • Destructive Interference (相消干涉 / 破坏性干涉): Pulses with opposite signs add together to momentarily cancel each other out.

3. Reflection and Transmission

Reflection (Upon Change of Medium)

  • Fixed Boundary: When a wave pulse hits a fixed wall, it exerts a force on the wall, and the wall exerts an equal and opposite downward force on the string (Newton's Third Law). As a result, the pulse inverts upon reflection.

图示：GP Waves1

  • schematic diagram from the hand-written notes of Mr. ztWang

  • Free Boundary: When a string is attached to a ring that slides freely on a frictionless vertical rod, the vertical force is zero. The pulse does not invert upon reflection.

图示：GP Waves2

Transmission Between Different Media

  • Light string to Heavy string: Behaves similarly to a fixed boundary. The transmitted pulse is upright (but slower/shorter), and the reflected pulse is inverted.

  • Heavy string to Light string: Behaves similarly to a free boundary. The transmitted pulse is upright (faster/longer), and the reflected pulse is upright.

图示：GP Waves3

4. The Linear Wave Equation

Derivation 1: Mechanical Waves in a Monoatomic Crystal (Phonon)

Consider atoms in a crystal with an equilibrium lattice spacing \(a\).

  • Equilibrium position: \(X_n = na\)

  • Deviation from equilibrium: \(u_n = x_n - X_n\)

Modeling nearest neighbor interaction (potential energy) with a spring constant \(K\):

\[ \phi(x_{n+1} - x_n) = \phi_0 + \frac{1}{2}K(x_{n+1} - x_n - a)^2 + \dots = \phi_0 + \frac{1}{2}K(u_{n+1} - u_n)^2 + \dots \]
\[ U^{total} \approx \frac{K}{2} \sum_n (u_{n+1} - u_n)^2 \]

Using Newton's Second Law for the \(n\)-th atom (\(F_n = m\ddot{u}_n\)):

\[ m\ddot{u}_n = K(u_{n+1} - u_n) - K(u_n - u_{n-1}) \]
\[ m\ddot{u}_n = Ka \frac{u_{n+1} - u_n}{a} - Ka \frac{u_n - u_{n-1}}{a} \]

Taking the continuum limit as \(a \to 0\):

\[ m\ddot{u}_n \approx Ka \left. \frac{\partial u}{\partial x} \right|_{x_n + a/2} - Ka \left. \frac{\partial u}{\partial x} \right|_{x_n - a/2} = Ka^2 \left. \frac{\partial^2 u}{\partial x^2} \right|_{x_n} \]

This yields the wave equation:

\[ \frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2} \quad \text{where} \quad v = a\sqrt{\frac{K}{m}} \]

Note on Continuum Limit: > The physical meaning of the limit \(a \to 0\) is \(\lambda \gg a\) (wavelength is much larger than the lattice spacing), or equivalently \(k \ll \frac{2\pi}{a}\).

Derivation 2: Wave on a String (Small Amplitude)

For a string with mass per unit length \(\mu\) and tension \(T\):

Applying Newton's second law to a small segment \(\Delta x\):

\[ \mu \Delta x \frac{\partial^2 y}{\partial t^2} \approx T \Delta x \left. \frac{\partial^2 y}{\partial x^2} \right|_x \]
\[ \frac{\partial^2 y}{\partial t^2} = \frac{T}{\mu} \frac{\partial^2 y}{\partial x^2} \equiv v^2 \frac{\partial^2 y}{\partial x^2} \quad \text{where} \quad v = \sqrt{\frac{T}{\mu}} \]

The General Linear Wave Equation

The generalized form applies to various physical systems:

\[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \]

  • Sound wave: \(y\) = displacement of air molecule.

  • Wave on a string: \(y\) = vertical displacement.

  • Electromagnetic wave: \(y\) = electric or magnetic field.

Verifying Solutions

The obvious solutions are \(y = f(x - vt)\) (right-moving) and \(y = f(x + vt)\) (left-moving).

To check \(f(x - vt)\):

  • Time derivatives: \(\frac{\partial y}{\partial t} = -vf'(x - vt) \implies \frac{\partial^2 y}{\partial t^2} = v^2 f''(x - vt)\)

  • Spatial derivatives: \(\frac{\partial y}{\partial x} = f'(x - vt) \implies \frac{\partial^2 y}{\partial x^2} = f''(x - vt)\)

    Substituting these back verifies the equation: \(\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}\).