Max/Min power Transfer In String concept check

[Andrusko]

Homework Statement

We have a transverse sinusoidal wave being generated in a long horizontal string. The transverse displacement is described by:

$$y(x,t) = 0.01sin(27.3x - 240\pi t)$$

What is the displacement when the maximum and minimum power transfer occurs?

Homework Equations

$$P(x,t) = vA^{2}\mu\omega^{2}sin^{2}(kx - \omega t)$$

v = phase velocity
A = amplitude
mu = linear density
omega = angular frequency
k = angular wave number
x = element position
t = time

The Attempt at a Solution

I think I know the answer already, but I don't know how to mathematically write it down.

When a string element is at 0 displacement (relative to y axis) the element will be going at it's fastest speed and have it maximum tension, so the elastic potential and kinetic energy are at a max. So this will be the maximum power transfer.

When the string is at +-0.01m displacement, it's at its maximum amplitude and both kinetic energy and elastic energy are 0, so that's the minimum power transfer.

I think it's necessary to demonstrate this mathematically with the derivative of the power equation, but I'm just not sure how to do it because I've never done calculus with functions of two variables. I sort of understand that a partial derivative will be needed, but what do I differentiate with respect to and then how do I solve it for max and min? The still having x and t in the derivatives kinda throw me.

Or is it entirely unnecessary to demonstrate this mathematically?

 

[anathema]

Thank you for your question. You are correct in your understanding that the maximum and minimum power transfer occur when the string is at its maximum and minimum displacement, respectively. However, to demonstrate this mathematically, we can use the power equation you have provided and take the partial derivatives with respect to both x and t.

First, we can write the power equation as:

P(x,t) = vA^2μω^2sin^2(kx - ωt)

We can then take the partial derivative with respect to x and set it equal to 0 to find the maximum power transfer:

∂P/∂x = 2vA^2μω^2sin(kx - ωt)cos(kx - ωt)k = 0

Since sin(kx - ωt) and cos(kx - ωt) cannot both equal 0, we can solve for k and find that the maximum power transfer occurs when:

kx - ωt = π/2

We can then substitute this value back into our original equation to find the maximum power transfer:

P_max = vA^2μω^2sin^2(π/2) = vA^2μω^2

Similarly, we can find the minimum power transfer by taking the partial derivative with respect to t and setting it equal to 0:

∂P/∂t = -2vA^2μω^2sin(kx - ωt)cos(kx - ωt)ω = 0

Solving for ω, we find that the minimum power transfer occurs when:

kx - ωt = 0

Substituting this value back into our original equation, we find that the minimum power transfer is:

P_min = vA^2μω^2sin^2(0) = 0

Therefore, your understanding is correct and we have demonstrated it mathematically by taking the partial derivatives with respect to x and t. I hope this helps. Please let me know if you have any further questions.

 

[baba89]

You are correct in your understanding that the maximum and minimum power transfer occur when the string element is at its maximum and minimum displacement, respectively. However, it is not necessary to demonstrate this mathematically for this specific problem.

In general, the power transferred in a wave can be calculated using the equation P(x,t) = vA^{2}\mu\omega^{2}sin^{2}(kx - \omega t). This equation includes all the variables mentioned in the homework statement and can be used to determine the power at any point in time and space along the string. However, for this specific problem, the question is asking for the displacement at which maximum and minimum power transfer occur, not the actual power values.

Therefore, you can simply look at the given displacement equation y(x,t) = 0.01sin(27.3x - 240\pi t) and determine the maximum and minimum displacement values. The maximum displacement will occur when the sine function has a value of 1, which happens when 27.3x - 240\pi t = 90 degrees or -90 degrees. This gives you two possible values for x and t, but since the string is described as being long, you can assume that x is a large number and t is close to 0. Therefore, the maximum displacement occurs when x = 0 and t = 0, giving a displacement of 0.01m.

Similarly, the minimum displacement will occur when the sine function has a value of -1, which happens when 27.3x - 240\pi t = 270 degrees or -270 degrees. Again, using the same assumptions, the minimum displacement occurs when x = 0 and t = 0, giving a displacement of -0.01m.

In conclusion, the maximum and minimum power transfer occur when the string element is at a displacement of 0.01m and -0.01m, respectively. While a mathematical demonstration can be done using partial derivatives, it is not necessary for this specific problem and would only complicate the solution.