The study of waves has a long history, dating back to ancient times with philosophers like Pythagoras exploring the relationship between pitch and string length in musical instruments. The relationship between pitch and frequency was discovered by Giovanni Benedetti, Isaac Beeckman, and Galileo, leading to the science of acoustics. Joseph Sauveur coined the term "acoustics" and showed that strings can vibrate at fundamental and harmonic frequencies. Isaac Newton calculated the speed of sound, while Laplace resolved discrepancies by including adiabatic effects. Brook Taylor provided the first analytical solution for a vibrating string, and Daniel Bernoulli, Leonard Euler, and Jean d'Alembert found solutions to the linear wave equation. Joseph Fourier conjectured that arbitrary functions can be represented by an infinite sum of sines and cosines, a theory later proven by Dirichlet. John William Strutt (Lord Rayleigh) laid the foundation for classical acoustics in his treatise "Theory of Sound." Hydrostatics and hydrodynamics were studied alongside acoustics, with Archimedes making significant contributions to hydrostatics. Fluid dynamics, including vortices and water waves, was advanced by mathematicians like Siméon-Denis Poisson, Claude Louis Marie Henri Navier, Augustin Louis Cauchy, and George Gabriel Stokes. Electromagnetism was initiated by William Gilbert, with contributions from Henry Cavendish, Charles-Augustin de Coulomb, and Alessandro Volta. James Clerk Maxwell unified electricity and magnetism in his "Treatise on Electricity and Magnetism," predicting the speed of light. The 20th century saw the discovery of general relativity and quantum mechanics, leading to new physical phenomena. This article focuses on classical wave phenomena, discussing linear and nonlinear waves, their applications, and solution methods. Linear waves are described by linear equations, allowing for superposition and Fourier analysis. Nonlinear waves, on the other hand, do not follow the superposition principle and are more challenging to analyze mathematically. The Korteweg-de Vries (KdV) equation, a famous solitary wave equation, describes small-amplitude, shallow-water waves and admits solitary wave solutions. Numerical methods for solving linear and nonlinear wave problems are also discussed, including finite difference, finite volume, and spectral methods.The study of waves has a long history, dating back to ancient times with philosophers like Pythagoras exploring the relationship between pitch and string length in musical instruments. The relationship between pitch and frequency was discovered by Giovanni Benedetti, Isaac Beeckman, and Galileo, leading to the science of acoustics. Joseph Sauveur coined the term "acoustics" and showed that strings can vibrate at fundamental and harmonic frequencies. Isaac Newton calculated the speed of sound, while Laplace resolved discrepancies by including adiabatic effects. Brook Taylor provided the first analytical solution for a vibrating string, and Daniel Bernoulli, Leonard Euler, and Jean d'Alembert found solutions to the linear wave equation. Joseph Fourier conjectured that arbitrary functions can be represented by an infinite sum of sines and cosines, a theory later proven by Dirichlet. John William Strutt (Lord Rayleigh) laid the foundation for classical acoustics in his treatise "Theory of Sound." Hydrostatics and hydrodynamics were studied alongside acoustics, with Archimedes making significant contributions to hydrostatics. Fluid dynamics, including vortices and water waves, was advanced by mathematicians like Siméon-Denis Poisson, Claude Louis Marie Henri Navier, Augustin Louis Cauchy, and George Gabriel Stokes. Electromagnetism was initiated by William Gilbert, with contributions from Henry Cavendish, Charles-Augustin de Coulomb, and Alessandro Volta. James Clerk Maxwell unified electricity and magnetism in his "Treatise on Electricity and Magnetism," predicting the speed of light. The 20th century saw the discovery of general relativity and quantum mechanics, leading to new physical phenomena. This article focuses on classical wave phenomena, discussing linear and nonlinear waves, their applications, and solution methods. Linear waves are described by linear equations, allowing for superposition and Fourier analysis. Nonlinear waves, on the other hand, do not follow the superposition principle and are more challenging to analyze mathematically. The Korteweg-de Vries (KdV) equation, a famous solitary wave equation, describes small-amplitude, shallow-water waves and admits solitary wave solutions. Numerical methods for solving linear and nonlinear wave problems are also discussed, including finite difference, finite volume, and spectral methods.