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- Consider a string of mass per unit length μ and tension T. The y coordinate can be considered as a function of both x and t.
For a segment of length Δx (

*almost*horizontal, but blown up here for clarity):- F
_{y}= T sin θ ≈ T tan θ = T Δy/Δx. - F = ma takes the form
μ Δx d

and in the limit Δx → 0:^{2}y/dt^{2}= F_{y2}- F_{y1}= T ((Δy/Δx)|

_{x + Δx}- (Δy/Δx)|_{x})μ/T d

^{2}y/dt^{2}= d^{2}y/dx^{2}.

**wave equation**. Note that it is**linear**: if y_{1}and y_{2}are both solutions, so is y_{1}+ y_{2}(the**superposition**of the two solutions). - F
- The general solution is
y(x,t) = f(x - c t) + g(x + c t).

These are "right-movers" and "left-movers", respectively. With u = x - c t:μ/T d

Both "movers" have^{2}f(u)/dt^{2}= d^{2}f(u)/dx^{2}μ/T (-c) df'(u)/dt = df'(u)/dx

μ/T c

^{2}f''(u) = f''(u)**propagation**(translation) velocity c = √(T/μ), so the wave equation has the general formd

^{2}y/dt^{2}= c^{2}d^{2}y/dx^{2}. - Let f(x - c t) = A sin(k x - ω t + φ) (recall the simple harmonic oscillator).
Then
f(x - c t) = A sin(k (x - ω t / k) + φ)

andc = ω / k.

- From the periodicity of this function we see that k = 2 π / λ, where λ
is the
**wavelength**(the horizontal distance spanned by one cycle) and k is the**wave number**(the number of radians per unit length).The propagation velocity is then c = λ ν, where ν is the frequency.

See this applet for how changes in wave number and phase angle affect a wave.

- The preceding analysis of
**transverse waves**on a string (where the direction of wave displacement is normal to the direction of wave propagation) is much more general than it appears. It applies to any linear wave motion in a medium which has attributes corresponding to inertia and tension; in particular, to linear waves in fluid media (ie., air or water), to**longitudinal waves**(where the wave displacement is parallel to the direction of wave propagation), and as we shall see, to linear electromagnetic waves in any medium.It does

*not*, however, apply to nonlinear waves. - The power flowing through the point x is the product of the upward force (-T dy/dx) with the upward velocity (dy/dt). Integrating
this at x = 0 over one period, and dividing by the period, gives the average power delivered per cycle:
P = μ ω

^{2}A^{2}c / 2.

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©2011, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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