Conceptual question about longitudinal waves

[Saladsamurai]

So we are working on sound waves in my physics course now and I was doing some textbook reading. I have been following it pretty well, but I just came across a relationship that I am not quite following.

It is with reference to wave interference. Let us say that two sound waves are emitted from two different point sources $$S_1$$ and $$S_2$$. The waves have the same wavelength $$\lambda$$ and are in phase at their sources. They take paths of lengths $$L_1$$ and $$L_2$$ and pass through point P.

The text says that their phase difference $$\phi$$ is dependent on $$\Delta L=|L_1-L_2|$$

Thus to relate the variables $$\Delta L$$ and $$\phi$$ we can use the proportion: $$\frac{\phi}{2\pi}=\frac{\Delta L}{\lambda}$$

I know that I should see it, but I don't exactly follow this proportion.

Could somebody ellaborate on this a little for me? I sure would appreciate,
Casey

 

[learningphysics]

Suppose the equation of both waves is: y = Acos(kx) (going along the direction the wave is travelling)

The wavelength of this wave is 2*pi/k

So at the point of interest, suppose wave 1 has traveled L1, and wave 2 has traveled L2:

y1 = Acos(kL1)

y2 = Acos(kL2)

the phase of the first wave is kL1. the phase of the second is kL2.

phase difference is: kL1 - kL2 = [2*pi/wavelength]*(L1 - L2)

so from this we get the phase difference relationship.

 

[Saladsamurai]

Suppose the equation of both waves is: y = Acos(kx) (going along the direction the wave is travelling)

The wavelength of this wave is 2*pi/k

So at the point of interest, suppose wave 1 has traveled L1, and wave 2 has traveled L2:

y1 = Acos(kL1)

y2 = Acos(kL2)

the phase of the first wave is kL1. the phase of the second is kL2.

phase difference is: kL1 - kL2 = [2*pi/wavelength]*(L1 - L2)

so from this we get the phase difference relationship.

Ah. I see that now. Thanks LP. It makes even more sense now that I wrote out what you did^^^...the phase difference is $$\phi$$

Casey