Basic Facts Weeks 6

General Waves on a String

The general motion of a string with its ends fixed is given by a sum over all of the normal modes. On a string of length L with wave speed v we have

∞ ( ( ) ( ) ) y(x,t) = ∑ sin(n-πx) A cos nπvt + B sin nπvt n=1 L n L n L

where the An and Bn are constants determined by the initial conditions. In particular, the Anare determined by the shape of the string at some time, usually t = 0, while the Bn are determined by the velocity of each point on the string at t = 0.

This is another example of the general principle that the most general motion of a system is a linear sum (a mixture) of its normal modes.

Fourier Analysis

The idea that a periodic function, or simply a function on a finite domain, can be decomposed into a sum of sines and cosines with that period is called Fourier Analysis (after Jean Baptiste Joseph Fourier 1768-1830). It is an example of the decomposition of a function into a complete set of orthogonal functions. Fourier also showed how to compute the coefficients (Anand Bn above) for any such function.

Plucked Strings

A plucked string can be treated as starting from rest (Bn = 0) with an initial saw-tooth shape. The resulting set of normal modes, also called the Spectrum since it dictates the vibration frequencies, will have the following properties.

a) Generally decrease in amplitude as n increases.

b) Be missing any modes which have a node at the plucking point.

c) Emphasise any modes that have an anti-node at the plucking point.

Thus a string plucked at its center will vibrate only in modes 1,3,5,7,9,etc., a string plucked at L∕3 will vibrate only in modes 1,2,4,5,7,8,10,etc.

Useful Facts

General Solution of 1-D Wave Equation

The general solution can be written either as a sum of waves travelling from left to right and from right to left,in the form

y(x,t) = f(x- vt)+ g(x+ vt)

or as a sum of standing waves. For the string with ends fixed at x = 0 and x = L that has the form

∑∞ ( ) ( ) y(x,t) = An sin nπ-x cos nπvt+ ϕn n=1 L L

where the An and ϕm are constants that are determined by the way the wave was started.

Fourier Analysis

On the interval 0 - L the sine and cosine functions are orthogonal if we define the inner product (dot product) of two functions f(x) and g(x) to be

∫ L f ⋅g = 0 f(x)g(x )dx

so that

({L ∫ Lsin (nπx-)sin(m-πx) dx = -2 n = m 0 L L (0 n ⁄= m ({ L ∫ Lcos(nπx-)cos( mπx-)dx = 2- n = m 0 L L (0 n ⁄= m ∫ L (nπx-) (m-πx ) 0 sin L cos L dx = 0

For a wave on a string with fixed ends cosine terms spatial terms are excluded and the coefficients An and Bn in the topmost equation can be found from

An = 2L ∫0L y(x,0)sin(n-πLx) dx

and

2 ∫ L∂y (nπx ) Bn = nπv- 0 ∂t-(x,0)sin -L-- dx
.