How do I Find Phase Shifts for two Waves?

[He man]

Hey, I'm new here. Many of guys probably have this same issue, but my Physics teacher always puts more than he can handle on his plate, aka he prepares more information than he has time to teach. What does that leave his students with (all 400 in the lecture)? A PDF email of the power point presentation (oh, I forgot to mention he refuses to teach with a marker or chalk) and says we are responsible to learn it ourselves. This isn't homework, its questions from his previous tests that I am practicing on my own so I can pass his class. Anyway, here it is!

Homework Statement

Two traveling sinusoidal waves in a string are defined by the functions where y and x are in centimeters and t is seconds.
y1 = (2.20 m) sin (19.5x - 32.5t)
and
y2 = (2.20 m) sin (28.0x - 44.0t)


2. The Question
What is the phase difference between these two waves at the point x = 5.00 cm at t = 2.00 s?

What is the positive x value closest to the origin for which the two phases differ by π at t = 2.00 s? (This is where the two waves add to zero.)

The Attempt at a Solution

I can add both equations to find a resulting equation. But I have no idea how to find the phase shift between the two. They are of different k and w values and my textbook only covers superposition of the same variables. How would I go about finding a phase shift?

Oh I forgot to mention, he is teaching out of a different textbook, so I cannot see the examples in that text since I have the one that the school requires. I believe he is using a book by Young, 11th edition (I don't have the title)

 

[Gear300]

If you notice, both equations are periodic in respect to sine, so there is no explicit phase constant between the two. A phase difference would be a difference in value. So let us denote the values inside the sine function as N. At x = 5.00 cm and t = 2.00s, the value of N for y1 will differ with the value of N for y2. The difference is the phase difference at that point.

For the second question, since it is already specified that t = 2.00s, you simply have to find a value x in which the difference between N in y1 and N in y2 is pi.

 

[He man]

Okay, so if I get this correctly,

Since both equations are in respect with sine, we can say that there is no explicit phase difference, the only difference would be based on the frequency and wavelength of the of two waves (or in this case, the inside, Kx-wt term).

If we find the difference of this inside term, I find that the two waves are out of phase by -8.5x+11.5t, or 8.5-11.5t, which one would it be?

The answer is suppose to be in degrees though. I've only seen the phase shift in the form of

kx-wt +phi ( +/- for all values)

How Do I find the degree value based on that number?

For question 2, how did you get the difference of y1 and y2 as pi? I'm reading it as the first value of x, where the phase difference is n (how do i know what interger should the n be is my big question) at t =2.00s

Sorry if I'm being hard to deal with!

 

[Gear300]

Okay, so if I get this correctly,

Since both equations are in respect with sine, we can say that there is no explicit phase difference, the only difference would be based on the frequency and wavelength of the of two waves (or in this case, the inside, Kx-wt term).

If we find the difference of this inside term, I find that the two waves are out of phase by -8.5x+11.5t, or 8.5-11.5t, which one would it be?

The answer is suppose to be in degrees though. I've only seen the phase shift in the form of

kx-wt +phi ( +/- for all values)

How Do I find the degree value based on that number?

For question 2, how did you get the difference of y1 and y2 as pi? I'm reading it as the first value of x, where the phase difference is n (how do i know what interger should the n be is my big question) at t =2.00s

Sorry if I'm being hard to deal with!

You got the concept right. Between -8.5x+11.5t, or 8.5-11.5t, you'll notice that one expression is simply the negative of the other, so your answers will only differ by sign (you don't really need the sign, you just need the value). You need to get the value for the phase difference for x = 5.00 cm and t = 2.00 s. The number you'll get will be in radians (radians and regular numbers are no different, so if something is in a trigonometric function and if it isn't specified in degrees, its in radians); you just need to convert from radians to degrees.

Sorry about question 2...I thought it said pi...but if it is where the 2 waves cancel each other out, then the phase difference is a multiple of pi.

 

[He man]

Alright for the first answer, if i input the values as given in question a) i get 19.5 +/-, this is in radians, so after doing the conversion, I get 117.27 degrees. Aha! Thanks bro! Gotta write that down in my solutions notebook now lol.

For question 2.

What is the positive x value closest to the origin for which the two phases differ by π at t = 2.00 s? (This is where the two waves add to zero.)

If i in put t= 2s, i get,

y1= 2.2sin(19.5x-65)
y2= 2.2sin(28.0x-88)

I want to find x where there is deconstructive interference (aka what value of x where y1 and y2 are the same?). I'm stuck at this point (I don't know how i would find x when it is inside the sin) however, using MATLAB, or Ti-84, i can graph the equation. I assume we are finding the point where the waves crash (aka intersect). My calc says it is where x = 2.705.

Is this the right approach?

 

[Gear300]

You got the first question done well...if I'm right about things

If i in put t= 2s, i get,

y1= 2.2sin(19.5x-65)
y2= 2.2sin(28.0x-88)

I want to find x where there is deconstructive interference (aka what value of x where y1 and y2 are the same?). I'm stuck at this point (I don't know how i would find x when it is inside the sin) however, using MATLAB, or Ti-84, i can graph the equation. I assume we are finding the point where the waves crash (aka intersect). My calc says it is where x = 2.705.

Is this the right approach?

For the second one, you're almost there...its not where they intersect, its where y1 + y2 = 0.

 

[He man]

OH SNAP! lol right this is where there is no sound at all (aka a node), so i am looking for the zero point, this is x=3.3. Still no idea how i would solve it without a calculator! but that's not a physics issue that's just an arithmetic issue!

 

[Gear300]

Looks like you got the concept right. If you did the arithmetic right, those should be the values.

 

[He man]

Thanks man! I am going to log the answers and hunt down the answer sheet.

Big help man! I need an A in this class, never though id struggle with physics, but this guy makes it really hard. I've gotten A's in all my calc, physics nad chem classes pretty easily, but this one is just FRYING my brain for some reason.

Now I need to post my 2nd question. =D Thanks again man.

 

[Gear300]

no problem