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Phy 242
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Test 2 Question
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In terms of the bead model of wave propagation, explain why we would expect the second time derivative of the wave function to be proportional to the second derivative of this function with respect to x (those derivatives should really be called partial derivatives, but at this stage that's just a matter of wording).
The SHM of the left-hand end of a long string is given by y = .59 cm * sin ( ( 7 `pi rad/s) t ). This motion induces a traveling wave in the string. The string has tension 19 Newtons and mass per unit length is 12 grams / meter. Explain how we know there is energy in the wave, and find how much energy there is in 11.4 meters of this wave.
a=fnf(.5,2,.01) b=fnf(3,9,1) c=fnf(5,50,2) d=fnf(3,19,1) e=fnf(3,20,.1)
f=fnf(.03,.09,.001)
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These are two problems from practice tests for test number 2 that I don't know how to solve. If you could please explain them to me that'd be great. Also, in the second problem I don't understand what all the information at the bottom is (i.e. a=fnf(.5,2,.01))
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Don't worry about the first problem. The information you would need to solve that problem has inadvertently been left out. If that problem arises you can ignore it.
The second problem is directly from the Introductory Problem Sets. The codes you don't understand are just generators used to randomize the problem, and aren't relevant to its solution.
So as not to confuse the issue I'm going to direct you to the Introductory Problem Sets for an explanation of that problem. If you have questions about the given explanations, of course, I'll be glad to answer them.
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#$&*
Phy 242
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Practice Test 2 Question
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Show that the function y(x, t) = .6 e^-( 920 t - .9 x)^2 satisfies the wave equation, and give the frequency, wavelength and velocity of the wave. Sketch the wave from x = - 5 to x = 5 at t=0 and at t = .002178.
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I am working on the practice tests for test number two and came across this problem. I know I need to take the partial derivatives with respect to x and t.
Once I find the partial derivatives with respect to x and t I know that:
d^2y/dx^2= (1/v^2)*d^2y/dt^2
However, taking the partial derivative of this function is extremely in-depth. Do you expect us to do this on the test? If so, what is the easiest way to take the derivative of this equation?
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The partial derivatives are easily calculated using first-semester calculus.
The one thing you need to know is that when taking the partial derivative with respect to x, you treat t as a constant. And when taking the partial derivative with respect to t, you treat x as a constant.
Knowing this, the process is a straightforward application of the chain rule and the rules for the derivatives of elementary functions.
As a warmup I'll give you a few functions and their partial derivatives, which you should verify. I'll use y_x for the partial derivative with respect to x, and y_t for the partial derivative with respect to t.
If y = x^2 y^3 then
y_x = 2 x y^3 and
y_t = 3 x^2 y2.
Subsequent examples use the chain rule. Note that, if ' represents derivative with respect to t, the chain rule tells us that if y = e^(f(t)), y ' = f ' (t) e^(f(t)).
If y = e^(x t^2) then
y_x = t^2 e^(x t^2)
y_t = 2 x t e^(x t^2).
If y = cos( x^2 / t) then
y_x = -2 x sin(x^2 / t) and
y_t = x^2 / t^2 sin(x^2 / t).
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