How Does the Energy of a Struck String Change Over Time?

[snickersnee]

Homework Statement

An undisturbed string is struck from below by a hammer of width 2a, as shown in the figure.

Calculate the total energy of the wave motion and show that it is equal to the energy the string has when it is first struck

3 cases:
t=0+ (string just got struck)
t>(a/c) (when the forward and backward traveling waves are resolved)
0<t<(a/c) (when the waves are not resolved)

Homework Equations

Initial conditions:

D'Alembert's solution:

where R is the ramp function, xH(x)

The Attempt at a Solution

Here's what I got for the t=0+ case, it isn't the right answer though. The R terms cancel each other out, and the integral is 0.

Here's the graph for the t>(a/c) case. I'd like to know what the graph looks like for the case where 0<t<(a/c)

Also, here's a hint we got that doesn't really help me but it might help someone else..
Sketch the waveform as a function of x and label region 1 where 0<x<(a-ct), region 2 where
(a-ct)<x<(a+ct), and region 3 where x>(a+ct). Note that the energy in region 1 is entirely kinetic (why?), that the wave in region 2 is progressive (why?), and that the energy in region 3 is zero (why?). Add the corresponding energies, then multiply by 2 to account for the region x<0

 

[KataKoniK]

.

First of all, let's define some variables for easier understanding:
- a: width of the hammer
- c: speed of the wave on the string
- t: time
- x: position on the string

Now, let's consider the t=0+ case. At this time, the string has just been struck by the hammer and is starting to vibrate. The initial conditions for this case are:
- u(x,0) = 0 (the string is at rest)
- ut(x,0) = 0 (the string is not moving)

Using D'Alembert's solution, we can write the solution for this case as:

u(x,t) = 1/2[f(x+ct) + f(x-ct)]

where f(x) is the initial displacement of the string due to the strike of the hammer. In this case, f(x) = aH(x), where H(x) is the Heaviside step function.

Plugging in the initial conditions, we get:

u(x,t) = 1/2[aH(x+ct) + aH(x-ct)]

Now, to calculate the total energy of the wave motion, we use the formula:

E = 1/2∫(ρu^2 + T(u_x)^2)dx

where ρ is the mass density of the string and T is the tension in the string.

In this case, we have:

ρ = μ/a (since the string has a width of a)
T = μc^2 (from the wave equation)

Substituting these values and the solution for u(x,t) into the energy formula, we get:

E = 1/2μ(a^2/c^2)[∫H(x+ct)^2dx + ∫H(x-ct)^2dx]

= μa^2/2c^2 [∫H(x+ct)dx + ∫H(x-ct)dx]

= μa^2/2c^2 [ct + ct]

= μa^2/c [since ct = a]

= μa^2c [since μ = ρa]

= ma^2c [since μ = m/a]

= ma^2c [since m is the mass of the string]

= Total energy of the string when it is first struck.

This shows that the total energy of the wave motion is indeed