Equation of a Wave Homework: Intensity Variation w/ Phase Difference

[Unto]

Homework Statement

Equation of a wave is a = a°sin(ωt - kx + φ)

where φ is the phase of a wave. if 2 waves with phases φ1 and φ2 interfere, show how the intensity I =a² varies as a function of the phase difference φ1 - φ2. Use one of the trigonometric double angle forumula or otherwise to obtain your result.

Homework Equations

The double angle formulas

The Attempt at a Solution

Well am I supposed to map I =a² onto the equation?

If so then the only double angle formula is cos(2x) = cos²(x) - sin²(x)

But I get a really stupid answer when I square the wave equation..

What do I do?

 

[lanedance]

show your work

start by writing the sum of the amplitude of the 2 waves with same fequency but different phase & work from there...

also intuitively, what do you expect will happen?

 

[Unto]

I have 2 waves with a phase difference:

a1 = a°sin(ωt - kx + φ1)
a2 = a°sin(ωt - kx + φ2)

If the waves combine, then an interference occurrs...

a1 + a2 = a°sin(ωt - kx + φ1) + a°sin(ωt - kx + φ2)

K apparently, sin a + sin b = 2cos 0.5(a - b) sin 0.5(a + b)

So in relation:

a1 + a2 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)

a3 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)

Now I =a²

But I'm unsure of how to square this expression I have, assuming it is even right..

 

[tiny-tim]

Use one of the trigonometric double angle forumula or otherwise to obtain your result.
…
If so then the only double angle formula is cos(2x) = cos²(x) - sin²(x)

Hi Unto!

I think by a "double angle formula" they mean like sin(A + B) or (sinA + sinB) etc …

these are trigonometric identities which you must learn.

 

[lanedance]

using a few diffenrt trigonamteric identities you can show the identity you used
$$sin(a) + sin(b) = 2cos(\frac{a-b}{2})sin(\frac{a+b}{2})$$

a3 = 2a°cos 0.5(φ1-φ2) sin 0.5(2ωt - 2kx + φ1 + φ2)
QUOTE]
so you get the amplitude as
$$2cos(\frac{\phi_1-\phi_}{2})sin(\omega t - k x \frac{\phi_1+\phi_}{2})$$
which looks reasonable to me

before you even consider intensity, have a look at what you've got with the amplitude
- the sin term looks similar to the input waves just with a different phase factor
- now the cos term, what happens when phi_1 = phi_2? and for what z, does cos(z) = 0?
- then think about the periodicity of the functions