Physics Projects

 

��������� The goal of these projects is to study standing waves and to relate the mathematics involved with concepts from linear algebra.� This will be done in 3 parts.

 

Project #1.� The Wave Equation�� due Monday November 17

 

1.       What is the 1-dimensional wave equation?� Include units and parameters in your answer

 

2.       What physical principle(s) is the wave equation based upon?

 

 

3.       What is the difference between transverse and longitudinal waves? What are examples

of each?

 

4.       What is �superposition� ? what does it have to do with �linearity�?

 

5.       Suppose for sake of discussion that the wave equation is� uxx = (1/c2) utt.� Show that solutions to

this are:

 

a)                  sin(x + ct) , sin(x-ct), cos(x + ct)� and cos(x �ct)

 

b)�������� sin(x)sin(ct),������ cos(x)cos(ct),� sin(x) cos(ct),�� and� cos(x)sin(ct)

 

c)�������� A sin(x)sin(ct)� +�� B cos(x)cos(ct)���� where A and B are arbitrary

 

(Note: showing that something is a �solution� is not the same as asking you to �solve�

it.� You simply have to verify that the given function really satisfies the equation.)

 

What does the answer to part c) have to do with superposition??

 

����������������������� 6) in the plots library of Maple, investigate how to use the animate function. Include an example.

 

Project #2�� Fourier Series and the Wave Equation���

 

From your first project, you have found the one dimensional wave equation to be

 

uxx = (1/c2)utt

 

where the subscripts denote partial derivatives with respect to position and time. I have used c2 for the constant to emphasize that it is positive and to make some of the later algebra simpler. Further we assume this applies to a string which is L units long thus limiting x to [0,L]

 

For a problem to be properly posed, there have to be additional conditions:

 

����������� u(0,t)= u(L,t) = 0����������� - fixed ends

 

����������� ut(x,0) = 0���� �������������� ����- no initial velocity

 

����������� u(x,0) =�� f(x) (given) � ����- initial displacement (more on that later).

 

This problem as a whole is called a boundary value problem.

 

Our starting point is� a general solution which we claim to be

 

(1)������������������� u(x,t) = (c1cos(lct) + c2 sin(lct) )(k1cos(lx) + k2sin(lx))

 

where the ci�s, k�s and l are arbitrary as of now.

 

problem 1: show this is a solution to the wave equation

 

Note that the solution,(1), is in the form of the product of a time varying function and a space varying function. A course in boundary value problems would show you the details in arriving at this. We just want to use it.

 

problem 2: apply boundary conditions and show that

 

����������������������� k1 = 0�� and� that sin(lL) = 0

 

which in turn means that�� lL = np�� where n=0,1,2,3,�.

 

This step is important � the rest of the work we do with the wave equation rests on it! It means there are infinitely many l that work and they are concisely given by a formula

 

����������������������������������� ln� = np/L

 

problem 3:� since there is no initial velocity, show that c2 = 0��������

 

At this point, the general solution has been refined considerably and we can say that it looks like any multiple of

 

(2) ��������������������cos(npct/L)sin(npx/L)���� n=0,1,2,3,�

 

These solutions are often called normal modes of vibration.� For n =1, the solution is often called the fundamental or first harmonic.

 

problem 4� deals with �standing waves�. To understand them, you simply need to remember from trigonometry that sinq = 0� whenever q is an integral multiple of p, a fact we already used above when we determined l.� At that step we wanted the solution to be 0 at the endpoints (x = 0 and L). It is also zero at other points in between, which is where the idea of standing waves comes from.

 

For n=2,� where is cos(npct/L)sin(npx/L) =0�� for� 0<x<L�� and all t ?

 

For n = 3, where is cos(npct/L)sin(npx/L) =0�� for� 0<x<L�� and all t ?

 

For n = 4, where is cos(npct/L)sin(npx/L) =0�� for� 0<x<L�� and all t ?

 

What we end up seeing is that our collection of solutions has points that never move, while the rest of the points move up and down with varying amplitude depending on x.

 

Can you summarize these things graphically for the n=2,3,4 cases? Comment on why the n=4 case looks like two solutions for L=L/2 pasted together.

 

problem 5:� in Project 1, you argued for the Principle of Superposition which says if you have two solutions, then their sum is also a solution as well as an arbitrary multiple. In mathematical terms this means the solutions form a vector space. Its basis is all solutions of the form (2) above, so it is infinite dimension.� From either point of view, this means we can form the general solution by adding up arbitrary multiplies:

 

u(x,t) = c1 cos(pct/L)sin(px/L) + c2 cos(2pct/L)sin(2px/L) + c3 cos(3pct/L)sin(3px/L) + . . . .

 

or� (3) u(x,t) = � cn cos(npct/L)sin(npx/L)�� (the sum being for n=1,2,3,�)

 

Note especially that this form of solution, as a sum, rests on the fact that the wave equation was a linear, homogeneous equation and therefore the Principle of Superposition applies.� This is a key step in the solution of the equation and also a key link between this problem and Linear Algebra.

 

Problem 6: We still have one condition to satisfy, which makes some sense since we still have a lot of arbitrary constants not determined (all the c�s in the sum).� That condition is the initial position.� We said that it was some given function� f(x) at the start of the project. We will now be more specific and say that the string has been taken in the middle, at x=L/2, and pulled up one unit.� Argue why this defines the function f as

 

f(x) =�� 2x/L���������� if� 0<x<L/2��

���������� -2x/L + 2�� if� L/2 < x < L

 

Problem 7: �Using equation 3 and setting t=0, we must have, since cos(0)=1, that

 

(4)   f(x) = � cn sin(npx/L)��

 

in other words, we have to express f(x) as a series of sin functions. This is a Fourier Sin Series! As discussed in class, while it looks imposing in that we have one equation and infinitely many unknowns (the cn �) we can use the orthogonality of trig functions and isolate the unknowns easily. Now to actually find the right dot product and finish the problem:

 

the dot product of any two functions� is defined here to be the integral of their product from x=0 to x=L.

 

a.� show the set of sin functions� { sin(npx/L)�� } is orthogonal� (suggestion: use Maple or use trig identities)

 

b. show that the dot product of sin(npx/L)�� with itself is� L/2

 

c. compute the dot product of f(x) (as above for this problem) with sin(npx/L)��

 

d. at this point, you should be able to write down a formula for the Fourier coefficients!

 

Problem 8

 

Now that you have the final solution to the wave equation for our particular problem, lets see how it looks. Essentially we want to see if it starts out ok � satisfying (4) for the initial condition. Plot the first 2 terms of the Fourier Series; then the first 3. Maple is a good tool for this. If your graphs are quite a bit of from the actual initial condition (a triangular shaped graph), something is off and you need to recheck things. Please see

me if you need help with this!! Don�t leave it until the last moment.