Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Differential Equations → Partial Differential Equations ↓


Wave Equation


The wave equation is a fundamental concept in the study of partial differential equations and has important applications in various fields such as physics, engineering and mathematics. In short, the wave equation models how waves propagate through various mediums - it can be sound waves traveling through air, light waves moving in space or sea waves in the ocean. Understanding the wave equation is important for understanding the behavior of these dynamic systems.

Introduction to the wave equation

At its core, the wave equation is a partial differential equation that describes the propagation of various types of waves and disturbances in a given medium. It is typically expressed in one spatial dimension as:

∂²u/∂t² = c² ∂²u/∂x²

where u(x, t) is a function representing the wave at position x and time t, and c is the speed at which the wave travels through the medium.

Components of the wave equation

  • Time derivative: ∂²u/∂t² represents the acceleration of the wave, or how quickly the displacement changes with time.
  • Space derivative: ∂²u/∂x² represents the curvature of the wave, which shows how the wave changes at different points in space.
  • Wave speed: c is the wave speed, which is determined by the properties of the medium through which the wave travels.

Visualization of wave equation

To better understand how the wave equation works, let's look at a simple example of a vibrating string. Imagine you pluck a guitar string. The string vibrates, creating a wave pattern that travels along the length of the string.

Definite end Definite end

In this case, the string vibrating at different points can be described by the wave equation. Each point on the string moves up and down, creating a traveling wave that goes back and forth between the fixed ends.

Deriving the wave equation

The derivation of the wave equation involves applying Newton's second law of motion to an element of the medium through which the wave propagates. For a small section of a string with length Δx, we can investigate the forces acting on it.

Consider a string under tension. The vertical displacement of the string at position x and time t is a function u(x, t). The tension in the string, T, is a constant that provides the force necessary for the oscillation.

The forces on the smaller section are due to tensions at both ends. The vertical components of these forces are given by:

F1 ≈ T (∂u/∂x)| at x F2 ≈ -T (∂u/∂x)| at x+Δx

Using Newton's second law (F = ma), the net force on the block is:

m ∂²u/∂t² = T [ (∂u/∂x)| at x - (∂u/∂x)| at x+Δx ]

Assuming m = ρ Δx, where ρ is the linear mass density of the string, and simplifying the Taylor series expansion, we arrive at our familiar wave equation:

∂²u/∂t² = (T/ρ) ∂²u/∂x²

Here, c² = T/ρ gives the speed of the wave.

Solutions of the wave equation

Solutions of the wave equation help us understand and predict wave behavior in practical scenarios. Generally, the wave equation allows different types of solutions depending on the initial and boundary conditions.

General solution

The general solution of the one-dimensional wave equation can be given in terms of two arbitrary functions f and g:

u(x, t) = f(x - ct) + g(x + ct)

This solution represents two traveling waves: one moving to the right with speed c, and the other moving to the left with speed c.

Example: Transverse waves on a string

As a practical example, consider a string fixed at both ends, one of which is displaced with an initial velocity of zero.

Initial conditions:

  • For initial displacement u(x, 0) = f(x)
  • for initial velocity ∂u/∂t(x, 0) = 0.

Boundary conditions:

  • For the fixed end at x = 0 u(0, t) = 0.
  • For the fixed end at x = L u(L, t) = 0.

The solution to this scenario is obtained by using standing waves:

u(x, t) = A sin(kx) cos(ωt)

where A is the amplitude, k is the wavenumber given by k = nπ/L, and ω is the angular frequency given by ω = ck.

Higher dimensional wave equations

The wave equation can also be expressed in more than one dimension. For example, in three-dimensional space, it can be written as:

∂²u/∂t² = c² (∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²)

Solutions in higher dimensions are more complex and usually require advanced techniques such as Fourier series or transforms to analyze wave patterns in different domains.

Properties of waves described by the wave equation

Waves that satisfy the wave equation have certain characteristics and phenomena that can be further analyzed:

1. Superposition principle

Since the wave equation is linear, the principle of superposition applies. This means that if u1(x, t) and u2(x, t) are solutions, then their sum u(x, t) = u1(x, t) + u2(x, t) is also a solution.

2. Reflection and transmission

When waves hit boundaries, they can be reflected or transmitted. For example, waves hitting a certain boundary are reflected back with phase reversal.

3. Standing waves

These arise from the superposition of two waves traveling in opposite directions. They do not propagate through the medium, but alternate in amplitude. Nodes are points of zero displacement, while antinodes are points of maximum displacement.

Example: harmonic standing waves

Node Antinode

In a harmonic standing wave, the displacement occurs at the antinodes while the nodes remain stationary.

Practical applications of the wave equation

The wave equation has wide application in real-world scenarios:

  • Acoustics: The wave equation models sound wave propagation in air or other mediums, important in designing concert halls and soundproofing.
  • Electromagnetism: In physics, the wave equation forms the basis for understanding light waves, radio waves, and other electromagnetic phenomena.
  • Seismology: The wave equation helps determine how seismic waves move through the Earth's layers during events such as earthquakes.
  • Engineering: Mechanical and civil engineers use wave equations to design structures that can withstand vibrations and wave-induced forces.

Conclusion

The wave equation plays a vital role in the mathematical understanding and solution of wave phenomena in a variety of fields. Despite being simple in its form, its applications and solutions reveal the complex interactions and behaviors of waves, making it an indispensable tool in scientific investigation and technological advancement.


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Wave Equation

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