Question

(The wave equation) Consider a string with fixed zero ends of length L with speed parameter c, with initial position -X u(x,0) = x € (0, L/2] c [L/2, L] C L and zero initial velocity. (a) Find the normal modes of the solution and specify the spatial and temporal frequencies for each. (You do not need to derive the general solution to the wave equation with fixed ends.) (b) Describe how the tension Th, density p and length L affect the first harmonic n1. How does changing each affect the perceived sound? (c) Compute the energy spectrum and determine which mode contributes most the the perceived sound.

Answer 1

The general wave equation is given by

2 at? u(x, t) = u(x, t)

c² is a proportionality constant. This constant is defined depending on the wave equation's applications. Here, particularly, since where are working with a string with certain mass, c² = (tension at everypoint of the string when the body is vibrating)/(lineal density of the system).

There are multiple ways to solve the above differential equation. However, given that we know the system is a stationary string, the solution would be along the form of vibration functions. Without solving the equation completely, we estimate the solution is of hte form

$\small u(x,t)=\sum_{k}A(k)\cos\left(\omega t + \phi \right )\sin\left(kx \right )$

(a)

The velocity condition renders $\small \phi = 0$ . That is

onu(x,t) =- A(k)y sin (wt + $) sin(kx) u(x,0) =- A(k)w sin(ø) sin(kx) = 0 a at

The only option we have for the last equation to be safistied is $\small \phi = 0$ . Otherwise, having other terms to vanish means an absurd result like there is no vibration.

Replacing the given solution function into the diffrential equation we obtained that

w=ck = T -K P

Now, we are going to use the conditon u(x,0) to determine k. when x=0 or x=L, u(0,0)=0 according to the initial condition. Using x=L, we can obtained the normal modes of the system.

$\small \begin{align*} u(x,0)&=\sum_{k}A(k)\sin\left(kx \right ) \\ u(L,0)&=\sum_{k}A(k)\sin\left(kL \right ) \\ &\Rightarrow kL=n\pi \end{align*}$

The second line implies that iether A(k) is zero or sine vanishes. We do not want A(k) to be zero otherwise, there is no solution to the system. The only alternative we have is the sine function to vanish. It does it if and only if its argument is certain interger of $\small \pi$ . Notice that the amplitude also depends on k values, the mentioned n should exclude those values that render zero the coefficient A(k).  Thus,

T T р 7T k=n

(b)

The first harmonic is described by n=1. Notice that modifying the tension T or lineal density rho only has an impact of the time frequency of the system. This means that the string would vibrate having the same number of nodes and antinodes while the frequency of vibration changes. That is if T increases and rho decreases, the string will emit a higher frequency note. However if the variation of both parameters is proportional to similar factor, there is no modification of the frequency.

(c)

A particle of mass m in the surrounding of the string will experience a change of its motion given by

E = 5mo(x, t) + mw?p(x, t)

The motion of this particle will be similar as the one given by the string. Replacing and averaging over time (that is from 0 to T) we obtained that

E mw2 A 2

1 ,27272 E= ma P L2

Here A is the amplitude of the string vibrational mode. If we compute the general solution of the system, we would obtain the complete mathematically form of how A depends of k. However, in this simple approach I would say that the modes that maximize the amplitude meaning the transmitted energy are the ones that contribute the most to the perceived sound.