The Wave Equation
Kinematic Derivation of the Wave Equation
http://prism.texarkanacollege.edu/physicsjournal/wave-e...
KINEMATIC DERIVATION OF THE HARMONIC WAVE EQUATION AND RELATED TOPICS An extremely important type of wave in physics is the harmonic wave. This is a wave consisting of propagating simple harmonic oscillations or linear combinations thereof. Attach a weight to a spring and hang the spring so the weight is free to move. Then lift the weight straight up and release it; it will oscillate up and down in a close approximation to simple harmonic motion. Now imagine that you have a whole array of such mass-spring systems, all oscillating independently. If you could add all their oscillations together, you would get linear combination of simple harmonic motions. Whether a harmonic wave involves propagating oscillations of a solid, liquid, gas, or electromagnetic field, the oscillations will consist, in general, of a linear combination of simple harmonic motions as exemplified by the sum of the oscillations of the multiple mass-spring system.
The One-Dimensional Wave Equation
Mathematically, a harmonic wave can represented by a sinusoidal function of time and position or a linear superposition of such functions (that is, you just add them together as in the case of sum of the oscillations of the spring and weight system described above), since a single such sinusoidal function describes the propagation of a single simple harmonic oscillation. The argument of each sinusoidal function is of the form x ± vt, where x is the position along the wave where the oscillation is observed, t the time variable for the oscillation, and v the speed of the wave, which is the same for all frequencies. The harmonic wave equation can be derived dynamically or kinematically. Dynamically, the derivation depends on the type of wave. For example, the wave equation for elastic media depends on the displacement of the medium through which the wave travels being proportional to the magnitude of the
http://prism.texarkanacollege.edu/physicsjournal/wave-e...
KINEMATIC DERIVATION OF THE HARMONIC WAVE EQUATION AND RELATED TOPICS An extremely important type of wave in physics is the harmonic wave. This is a wave consisting of propagating simple harmonic oscillations or linear combinations thereof. Attach a weight to a spring and hang the spring so the weight is free to move. Then lift the weight straight up and release it; it will oscillate up and down in a close approximation to simple harmonic motion. Now imagine that you have a whole array of such mass-spring systems, all oscillating independently. If you could add all their oscillations together, you would get linear combination of simple harmonic motions. Whether a harmonic wave involves propagating oscillations of a solid, liquid, gas, or electromagnetic field, the oscillations will consist, in general, of a linear combination of simple harmonic motions as exemplified by the sum of the oscillations of the multiple mass-spring system.
The One-Dimensional Wave Equation
Mathematically, a harmonic wave can represented by a sinusoidal function of time and position or a linear superposition of such functions (that is, you just add them together as in the case of sum of the oscillations of the spring and weight system described above), since a single such sinusoidal function describes the propagation of a single simple harmonic oscillation. The argument of each sinusoidal function is of the form x ± vt, where x is the position along the wave where the oscillation is observed, t the time variable for the oscillation, and v the speed of the wave, which is the same for all frequencies. The harmonic wave equation can be derived dynamically or kinematically. Dynamically, the derivation depends on the type of wave. For example, the wave equation for elastic media depends on the displacement of the medium through which the wave travels being proportional to the magnitude of the