1.5 Wave Equation and Wave Speed

Derivation of the Wave Equation

Fundamental Principles and Concepts

The wave equation is a second-order partial differential equation that describes how waves propagate through a medium. Its derivation rests on two foundational principles: Newton's Second Law and Hooke's Law, applied to a small element of the medium.

The one-dimensional wave equation takes the form:

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

where $y$ is the displacement of the medium, $x$ is position, $t$ is time, and $v$ is the wave speed. What this equation says, physically, is that the acceleration of any small piece of the medium (the left side) is proportional to how sharply the medium is curved at that point (the right side). A tightly curved region experiences a larger restoring force, producing greater acceleration.

This one-dimensional form applies to waves on a string. The equation extends to higher dimensions for more complex scenarios:

• Two dimensions for waves on surfaces like membranes or drumheads
• Three dimensions for sound waves in air or seismic waves through the Earth

Mathematical Derivation Steps

For a string under tension, the derivation proceeds as follows:

1. Consider a small segment of string with length $\Delta x$ and linear mass density $\mu$.

2. Apply Newton's Second Law ($F = ma$) to that segment. The mass of the segment is $\mu \, \Delta x$, and its acceleration is $\frac{\partial^2 y}{\partial t^2}$.

3. The net transverse force on the segment comes from the difference in the vertical components of tension at each end. For small displacements, this net force is proportional to the change in slope across the segment: $F_{\text{net}} = T \left(\frac{\partial y}{\partial x}\bigg|_{x+\Delta x} - \frac{\partial y}{\partial x}\bigg|_{x}\right)$

4. Set the net force equal to mass times acceleration: $T \left(\frac{\partial y}{\partial x}\bigg|_{x+\Delta x} - \frac{\partial y}{\partial x}\bigg|_{x}\right) = \mu \, \Delta x \, \frac{\partial^2 y}{\partial t^2}$

5. Divide both sides by $\Delta x$ and take the limit as $\Delta x \to 0$. The left side becomes $T \frac{\partial^2 y}{\partial x^2}$.

6. Rearrange to get $\frac{\partial^2 y}{\partial t^2} = \frac{T}{\mu} \frac{\partial^2 y}{\partial x^2}$.

7. Identify the wave speed as $v = \sqrt{\frac{T}{\mu}}$, which recovers the standard wave equation.

Wave Speed in Different Media

Mechanical Waves

Wave speed is not a property of the wave itself; it's determined by the medium. The general pattern for mechanical waves is:

$v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}}$

The elastic property measures how strongly the medium resists deformation (the restoring force). The inertial property measures how much mass resists acceleration. A stiffer medium with less inertia produces faster waves.

Specific formulas for different media:

• Strings or wires: $v = \sqrt{\frac{T}{\mu}}$ where $T$ is tension and $\mu$ is linear mass density (kg/m)
• Sound in fluids: $v = \sqrt{\frac{B}{\rho}}$ where $B$ is the bulk modulus and $\rho$ is the fluid density
• Longitudinal waves in solids: $v = \sqrt{\frac{Y}{\rho}}$ where $Y$ is Young's modulus and $\rho$ is density

Some representative wave speeds to build intuition:

• A nylon guitar string under medium tension: ~500 m/s
• Sound in air at room temperature (20°C): ~343 m/s
• Seismic P-waves in granite: ~5000 m/s

Notice the trend: stiffer, denser solids generally carry waves faster than gases because the increase in elastic stiffness outweighs the increase in density.

Electromagnetic Waves

Electromagnetic waves don't need a medium, but their speed changes when passing through materials. The speed in a medium is:

$v = \frac{c}{n}$

where $c = 299{,}792{,}458$ m/s is the speed of light in vacuum and $n$ is the refractive index of the medium (always $\geq 1$, so light always slows down in a material).

• Light in water ($n \approx 1.33$): ~2.25 × 10⁸ m/s
• Light in diamond ($n \approx 2.42$): ~1.24 × 10⁸ m/s

Diamond's high refractive index is what gives it that distinctive sparkle: light bends sharply at the surface because of the large speed change.

Factors Affecting Wave Speed

Several environmental factors alter wave speed:

• Temperature: Sound travels faster in warmer air because air molecules move faster and transmit pressure disturbances more quickly. At 0°C, sound speed is about 331 m/s; at 20°C, it's about 343 m/s.
• Pressure: Higher pressure increases the speed of sound in gases, and affects wave speed in compressed solids as well.
• Composition: Salinity raises the speed of sound in seawater (important for sonar), and humidity slightly increases the speed of sound in air because water vapor is less dense than nitrogen and oxygen.

Solving the Wave Equation

Boundary Conditions and Their Effects

The wave equation alone doesn't fully determine a wave's behavior. You also need boundary conditions, which specify what happens at the edges of the medium.

Common types:

• Fixed ends (e.g., a guitar string clamped at both ends): The displacement must be zero at the boundaries. This forces standing waves with nodes at each end, restricting the string to specific wavelengths and frequencies (harmonics).
• Free ends (e.g., an open-ended organ pipe): The medium is free to move at the boundary, producing antinodes there. The allowed frequencies differ from the fixed-end case.
• Periodic boundaries (e.g., vibrations around a circular drumhead): The wave pattern must repeat smoothly as you go around the boundary.

The boundary conditions are what select the specific standing wave patterns you actually observe. A guitar string can only vibrate at frequencies where a whole number of half-wavelengths fit between the two fixed ends.

Solution Methods and Applications

The standard technique for solving the wave equation with boundary conditions is separation of variables:

1. Assume the solution can be written as a product of a spatial function and a time function: $y(x, t) = X(x) \cdot T(t)$.
2. Substitute into the wave equation and separate terms so that one side depends only on $x$ and the other only on $t$.
3. Since a function of $x$ can only equal a function of $t$ if both equal a constant, you get two ordinary differential equations (much easier to solve than the original PDE).
4. Apply boundary conditions to the spatial equation to find the allowed wavelengths.
5. The resulting solutions are the normal modes of the system, each with its own natural frequency (resonant frequency).

These techniques apply broadly:

• Calculating the harmonic series of musical instruments
• Designing concert halls and acoustic spaces for optimal sound
• Analyzing structural vibrations in bridges and buildings to avoid resonance failures

Dispersion Relation for Waves

Concept and Mathematical Representation

The dispersion relation connects a wave's angular frequency $\omega$ to its wavenumber $k$:

$\omega = \omega(k)$

This relationship tells you everything about how waves of different frequencies travel through a given medium.

In a non-dispersive medium, the relationship is linear:

$\omega = vk$

Here $v$ is constant, meaning all frequencies travel at the same speed. Waves on a taut string (in the ideal case) and electromagnetic waves in vacuum are non-dispersive.

In a dispersive medium, different frequencies travel at different speeds. Two distinct velocities become important:

• Phase velocity: $v_p = \frac{\omega}{k}$ describes how fast individual wave crests move.
• Group velocity: $v_g = \frac{d\omega}{dk}$ describes how fast the overall envelope of a wave packet moves. This is typically the velocity at which energy and information travel.

When $v_p \neq v_g$, the shape of a wave packet changes as it propagates because its component frequencies drift apart.

Examples and Phenomena

Deep-water waves are a classic example of dispersion. Their dispersion relation is:

$\omega = \sqrt{gk}$

where $g$ is gravitational acceleration. From this, the phase velocity is $v_p = \sqrt{g/k}$, so longer wavelengths (smaller $k$) travel faster. That's why after a distant storm, long-period swells arrive at shore before shorter choppier waves.

Electromagnetic waves in certain materials also show dispersion. Light of different colors (frequencies) travels at slightly different speeds through glass, which is why a prism separates white light into a spectrum. This same effect causes chromatic aberration in lenses, where different colors focus at slightly different points.

In dispersive media, a localized wave packet gradually spreads out over time as its component frequencies separate. Near absorption frequencies of a material, anomalous dispersion can occur, where the refractive index decreases with frequency, and the group velocity can exceed the phase velocity or even appear to exceed $c$ (though no energy or information actually travels faster than light).