What is the Distance Between Two Points with a Phase Difference of 45 Degrees?

[Lavabug]

Homework Statement

For the given wave on a string:
y = 0.001 cos (7.5x + 15t)

find the distance between two points which have a phase difference of 45 degrees at the same instant in time.

The Attempt at a Solution

No idea how to solve this. Please help.

 

[berkeman]

Homework Statement

For the given wave on a string:
y = 0.001 cos (7.5x + 15t)

find the distance between two points which have a phase difference of 45 degrees at the same instant in time.

The Attempt at a Solution

No idea how to solve this. Please help.

What is the wavelength of that function?

 

[Lavabug]

I think I'm on to something:

The phase between the start and the end of a full wavelength is 2pi (full cycle).
For half a wavelength that's pi
For a quarter wavelength that's pi/2 or 90 degrees, ie: the phase between maxes and mins.
For an 1/8 of a wavelength the phase is pi/4, or 45 degrees.

The problem gives me the wavenumber so I can find the wavelength: 2pi/7.5 = lambda

So I think I can draw the following similarity:

lambda is to 2pi as
lambda/8 is to pi/4 ?

So the distance between points that represent a phase of 45 degrees is 1/8 of a wavelength?

 

[berkeman]

I think I'm on to something:

The phase between the start and the end of a full wavelength is 2pi (full cycle).
For half a wavelength that's pi
For a quarter wavelength that's pi/2 or 90 degrees, ie: the phase between maxes and mins.
For an 1/8 of a wavelength the phase is pi/4, or 45 degrees.

The problem gives me the wavenumber so I can find the wavelength: 2pi/7.5 = lambda

So I think I can draw the following similarity:

lambda is to 2pi as
lambda/8 is to pi/4 ?

So the distance between points that represent a phase of 45 degrees is 1/8 of a wavelength?

Looks good so far. Now what is the length of a wavelength for the equation that you are given. Assume mks units, even though they are not stating them explicitly...

 

[berkeman]

Or I suppose it could be cgs units. Do they show units in any other similar problems?

 

[Lavabug]

I'm pretty sure the units are just SI. I got my wavelength from the given wavenumber(7.5): lambda = 2pi/7.5 (don't have a calculator on hand lol).

The distance between points with 45 degree would be an eighth of that quantity.

So my idea is correct?

 

[berkeman]

I'm pretty sure the units are just SI. I got my wavelength from the given wavenumber(7.5): lambda = 2pi/7.5 (don't have a calculator on hand lol).

The distance between points with 45 degree would be an eighth of that quantity.

So my idea is correct?

Seems like the correct approach.

You could also graph the function either with a graphing calculator or just Excel, to have a sanity check on your answer...

 

[Lavabug]

How would I graph it? Can I exclude the time dependency (angfreq*t) and just plot .001cos(kx)?

 

[berkeman]

How would I graph it? Can I exclude the time dependency (angfreq*t) and just plot .001cos(kx)?

Yes, the problem says "at the same instant in time", so you can set t=0 arbitrarily.

 

[Lavabug]

I tried plotting it with an interval from 0 to half of my given wavelength(pi/7.5) and got this:

http://www.wolframalpha.com/input/?i=.001cos(7.5x)++x+from+0+to+pi/7.5

Which shows a 180 degree phase difference

Plotting half that interval (pi/15) shows a 90 degree phase difference:
http://www.wolframalpha.com/input/?i=.001cos(7.5x)++x+from+0+to+pi/15

aaand with half of that I've got (pi/30), which looks like:
http://www.wolframalpha.com/input/?i=.001cos(7.5x)++x+from+0+to+pi/30

I'm fairly convinced that pi/30 is the distance I'm looking for. Thanks!