Query 10

course Phy 232

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Question: Query introductory set 6, problems 1-10

explain how we know that the velocity of a periodic wave is equal to the product of its wavelength and frequency

Your Solution:Multiplying wavelength segements per segment by the length of each wavelength will give us the velocity.

confidence rating:

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Given Solution:

** we know how many wavelength segments will pass every second, and we know the length of each, so that multiplying the two gives us the velocity with which they must be passing **

Your Self-Critique:ok

Your Self-Critique Rating:ok

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Question: explain how we can reason out that the period of a periodic wave is equal to its wavelength divided by its velocity

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Your Solution:If we know the wavelength and the velocity we can divide the wavelength by the velocity which will give us the time between the waves which is the period.

confidence rating:

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Given Solution:

** If we know how far it is between peaks (wavelength) and how fast the wavetrain is passing (velocity) we can divide the distance between peaks by the velocity to see how much time passes between peaks at a given point. That is, period is wavelength / velocity. **

Your Self-Critique:ok

Your Self-Critique Rating:ok

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Question: explain why the equation of motion at a position x along a sinusoidal wave is A sin( `omega t - x / v) if the equation of motion at the x = 0 position is A sin(`omega t)

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Your Solution: I don’t really understand what the question is asking. I see the x=0 maybe that has something to do with the sine functions who’s values are -1,0, or 1.

confidence rating:

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Given Solution:

** the key is the time delay. Time for the disturbance to get from x = 0 to position x is x / v. What happens at the new position is delayed by time x/v, so what happens there at clock time t happened at x=0 when clock time was t = x/v.

In more detail: If x is the distance down the wave then x / v is the time it takes the wave to travel that distance. What happens at time t at position x is what happened at time t - x/v at position x=0.

That expression should be y = sin(`omega * (t - x / v)).

The sine function goes from -1 to 0 to 1 to 0 to -1 to 0 to 1 to 0 ..., one cycle after another. In harmonic waves the motion of a point on the wave (think of the motion of a black mark on a white rope with vertical pulses traveling down the rope) will go thru this sort of motion (down, middle, up, middle, down, etc.) as repeated pulses pass.

If I'm creating the pulses at my end, and that black mark is some distance x down in rope, then what you see at the black mark is what I did at time x/v earlier. **

STUDENT COMMENT (University Physics):

According to the Y&F book (p.553) we get the expression for a sinusoidal wave moving the the +x-direction with the equation:

Y(x,t) = A*cos[omega*(t-x/v)]

I am not sure where the sine came from in the equation in the question. The book uses the cosine function to represent the waves motion.

The choice of the cosine function is arbitrary. Either function, or a combination of both, can come out of the solution to the wave equation (that's the partial differential equation which relates the second derivative with respect to position to the second derivative with respect to time).

The sine and cosine functions differ only by a phase difference of 90 degrees, and either can be used to describe simple harmonic motion or the motion of harmonic waves. The choice simply depends on the initial conditions of the system.

We don't want to get into solving the wave equation here, but the point can be illustrated by considering simple harmonic motion, which is characterized by F_net = - k x (leading to m x '' = - k x or x '' = -k/m * x, where derivatives are with respect to time).

The general solution to the equation x '' = - k / m * x is x = B sin(omega t) + C cos(omega t), where B and C are arbitrary constants and omega = sqrt(k/m).

B sin(omega t) + C cos(omega t) = A sin(omega t + phi), where A and phi are determined by B, C and the choice to use the sine function on the right-hand side.

B sin(omega t) + C cos(omega t) = A cos(omega t + phi), where A and phi are determined by B, C and the choice to use the cosine function on the right-hand side. The value of A will be the same as if we had used the sine function on the right, and the value of phi will differ by 90 degrees or pi/2 radians.

Your Self-Critiqe:That’s pretty confusing, and I still don’t really see the answer.

You might need to write it out.

Try deconstructing the solution phrase-by-phrase and as soon as you get to something you're not sure of, insert a synopsis of what you do and do not know, and any questions, and submit it. With that information I can focus my response on what you need.

Your Self-Critique Rating: ok

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Question: Query introductory set six, problems 11-14

given the length of a string how do we determine the wavelengths of the first few harmonics?

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Your Solution:As the wavelength lessens the more half wave will fit onto the string. So you have 1 half wavelength =string length which would make lambda = 2L For 2 wavelengths you get L=lambda for 3 wavelengths you get lambda =2/3 L the wavelengths would continue to decrease.

confidence rating:

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Given Solution:

** As wavelength decreases you can fit more half-waves onto the string. You can fit one half-wave, or 2 half-waves, or 3, etc..

So you get

1 half-wavelength = string length, or wavelength = 2 * string length; using `lambda to stand for wavelength and L for string length this would be

1 * 1/2 `lambda = L so `lambda = 2 L.

For 2 wavelengths fit into the string you get

2 * 1/2 `lambda = L so `lambda = L.

For 3 wavelengths you get

3 * 1/2 `lambda = L so `lambda = 2/3 L; etc.

Your wavelengths are therefore 2L, L, 2/3 L, 1/2 L, etc.. **

Your Self-Critique:ok

Your Self-Critique Rating:ok

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Question: Given the wavelengths of the first few harmonics and the velocity of a wave disturbance in the string, how do we determine the frequencies of the first few harmonics?

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Your Solution:Frequency is the number of crests passing per unit of time. The frequency is equal to the wavelength is equal to the wave velocity/wavelength.

confidence rating:

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Given Solution:

** The frequency is the number of crests passing per unit of time.

We can imagine a 1-second chunk of the wave divided into segments each equal to the wavelength. The number of peaks is equal to the length of the entire chunk divided by the length of a 1-wavelength segment. This is the number of peaks passing per second.

So frequency is equal to the wave velocity divided by the wavelength. **

Your Self-Critique:ok

Your Self-Critique Rating:ok

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Question: Given the tension and mass density of a string how do we determine the velocity of the wave in the string?

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Your Solution:The equation would be v=sqrt(tension/mass per unit of length)

confidence rating:

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Given Solution:

** We divide tension by mass per unit length and take the square root:

v = sqrt ( tension / (mass/length) ). **

Your Self-Critique:ok

Your Self-Critique Rating:ok

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