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Subject: A few questions about mechanical waves.
Mr Smith,
I have a few questions that shouldn't take too long to
answer.
1. If you give us a function in the form such as example:
y(x,t) = .83sin(770t-.63x). How are we supposed to
show whether it satisfies the wave equation and give the
velocity?
Look up the Wave Equation in your index and read the section carefully.
The Wave Equation relates the second partial derivative of y with respect to t to the second partial derivative of y with respect to x; the propagation velocity c is a ratio of these two second partials.
A partial derivative with respect to t is calculated by treating x as a constant; and a partial with respect to x treats t as a constant.
2. If you have a problem that designates 3 beads
separated along a string with different y position. Also a
velocity of bead 1 and the mass of the beads as well as
there separation. Also listed is the string tension. I was
wondering how you would go about finding
- the acceleration of the given bead
Provided y is small with respect to the interval between beads, the tension can be treated as constant.
Between any two beads the tension is a vector with an x component and a y component. If the y component of a vector is small compared to the x component, the x component will be very nearly equal to the magnitude of the vector. The magnitude of the y component will be very nearly equal to the product of the slope and the magnitude.
Every bead (except the ones on the ends) therefore experiences a net force due to the tensions in two string segments, one to its left and one to its right. The x components of these tensions will be equal and opposite, but unless the two segments have the same slope and lie along the same straight line, their y components will not be equal and opposite. The vector sum of the y components will be equal to the net force on the bead.
Knowing the mass of the bead, if you know its position you can then easily calculate is acceleration.
If you know the velocity of the bead, you can use this acceleration to estimate the change in its velocity over a given time interval. If the time interval is short enough this approximation will be reasonably accurate, and by choosing the a sufficiently short time interval the error in the approximation can be made as small as desired.
From the velocity at the beginning of the interval and the approximate velocity at the end you can determine the approximate average velocity and therefore the change in position over the time interval, and the new position of the bead.
Applying this procedure to every bead in turn allows us, given the initial positions and velocities, to approximate as accurately as we wish the new positions and velocities of the beads after a given time interval.
Iterating this procedure, using a short enough time interval we can predict the overall behavior of the system.
This process embodies the wave equation and illustrates its meaning. The second derivative of y with respect to x is the rate at which the slopes change, which is directly proportional to the difference in the slopes, used to calculate the net force on a bead. The net force is the second derivative of the y position with respect to clock time. The propagation velocity is a ratio of these two quantities, as dictated by the impulse momentum theorem.
- its approximate velocity at a later time(which
I know that would be easy once you find the
acceleration)
- the distance it will move in a given time interval
3. Also in the practice set, you said that total
internal reflection could not have n2/n1 greater then 1
because the sine of that function isn't possible; however,
in the solution n2 < n1 making it less then one. Could you
show me how to find the total internal reflection if n2/n1
is less than 1.
The sine of the angle of total internal reflection is (n2 / n1).
Good job. Let me know if you have questions.