How Do Fourier Series Model the Motion of a Struck String?

[daydreaming1989]

Can someone give me some hints on this problem please?

A string (length L) clamped at both ends and initially at rest, the boundary conditions for the wave function y(x,t) are:

y(x,0)=y(0,t)=y(L,t)=dy/dt(x,0)=0

A note is obtained by striking the string with a hammer at some point a, with 0,a,L. In this case dy(x,0)/dy=0 for all of the string except for the interval (a-e)<x<(a+e), where a momentum p is given to the string by the hammer. Write the Fourier representation for y(x,t) and solve for the foefficients in terms of p, L, a, d(linear density) and c=sqrt(T/d) using the limit e->0.

And the general form of motion is given by,

y(x,t)= summation (n=1-> infinity) {sin(3.14nx/L)[Acos(3.14nct/L)+Bsin((3.14nct/L)]}
where c is the veolcity of the wave shape.

Thank you very much~

 

[Nate810]

Hint: Start by using the boundary conditions to solve for the coefficients A and B. Using the given information about the momentum imparted on the string, you can then use the limit e->0 to solve for the coefficients in terms of p, L, a, d and c. Recall that the sinusoidal form of the wave equation allows you to represent a single pulse as the sum of many individual sine waves.

 

[wais]

Sure, I can give you some hints on this problem. First, let's break down the given information. We have a string of length L that is clamped at both ends and initially at rest. This means that the boundary conditions for the wave function y(x,t) are y(x,0)=y(0,t)=y(L,t)=dy/dt(x,0)=0. This tells us that at time t=0, the string is at rest and the displacement (y) at all points on the string is 0. It also tells us that the string is clamped at both ends, so the displacement at x=0 and x=L is also 0. Finally, the derivative of the displacement with respect to time (dy/dt) at time t=0 is also 0, meaning that the initial velocity of the string is 0.

Next, we are told that a note is obtained by striking the string with a hammer at some point a, with 0<a<L. This means that the displacement at this point (y(a,t)) is not 0, as it is being struck by the hammer. However, the derivative of the displacement with respect to time (dy/dt(a,t)) is 0 for all points except for the interval (a-e)<x<(a+e). This means that the string is only affected by the hammer within this interval, and outside of this interval, the string behaves as if it was still at rest.

To solve for the Fourier representation for y(x,t), we can use the general form of motion given in the problem, which is y(x,t)= summation (n=1-> infinity) {sin(3.14nx/L)[Acos(3.14nct/L)+Bsin((3.14nct/L)]}. This is known as the Fourier series, which represents a function as a sum of sine and cosine functions with different frequencies and amplitudes.

To solve for the coefficients A and B, we will need to use the given information about the string's linear density (d) and the velocity of the wave (c). These values are related by the formula c=sqrt(T/d), where T is the tension in the string. We can also use the given information about the momentum (p) given to the string by the hammer, which is related to the velocity of the wave by the equation p=dc.

By taking the limit e->