Standing Waves

Standing WavesThis presentation will explicate the key principals behind standing waves.

Mathematical Definition of a Standing WaveConsider two waves moving in the opposite directions with the same wavelength, amplitude, and frequency. This is the conceptual foundation for a stand wave.Now for rough mathematical bits:Wave 1: D1(x,t) = A sin(kx - t) moving in direction of increasing xWave 2:D2(x,t) = A sin(kx + t) moving in direction of decreasing xNow we add the waves together. D(x,t) = D1(x,t) + D2(x,t)= A sin(kx - t) + A sin(kx + t) = A [sin(kx - wt) + sin(kx - t)] --Factoring out A = 2A sin(kx) cos (t) --Using trigonometric identities

Interpretation of the MathWhen two waves of opposite direction but with equal amplitude, frequency, and wavelength combine along a string, the result is the string oscillating in Simple Harmonic Motion.Visually, the result is something as such:

The result is something where the string is moving only up and down. Moreover, the above image also shows the points where the amplitude are zero, the nodes, and where its at a maximum at 2A, the anti-node. All other points have amplitude between zero and 2A. If we consider wavelength,, to be distance between two nodes, we can therefore assume that the position of node is 0, /2, , and so forth. Similarly, the anti-nodes occur at /4, 3/4, 5/4, and so forth.Since the nodes are stationary, no energy flows through a standing wave. Similarly to SHM, the energy continuously transforms between potential and kinetic energy.

Application One: Standing Waves on A String

Application One: Standing Waves on A String Cont.

Application Two: Stringed Instruments