Solving Superposition Problem: Find a3 and Bint

[FoolishMortal]

The superposition of two harmonic waves:
u1 = B * sin( a1(r) )
u2 = B * sin( a2(r) )
results in a sinusoidal wave of the form:
uint = Bint * sin( a3(r) )
Find a3 and Bint

I'm not sure what to do. I can't think of any way to get it into that "form". http://scienceworld.wolfram.com/physics/Interference.html (5) on that link gives the form I would expect.
I think the "int" subscript is talking about intensity, and I'm not sure but I don't think either u1 or u2 depend on time.
Can anyone help?

 

[berkeman]

Try looking up some trig identities for sine functions. Is there one involving the addition of two sine waves and getting another sine wave with a phase shift?

 

[piercas]

I would first clarify the problem by asking for more information. Is this a theoretical or experimental problem? Are we dealing with a specific physical system or is this a general case? Can we assume that the waves are propagating in the same medium and in the same direction? These details are important in order to provide an accurate and relevant solution.

Assuming that this is a theoretical problem and the waves are propagating in the same medium and direction, we can use the principle of superposition to solve for a3 and Bint. The principle states that when two or more waves overlap, the resulting wave is the algebraic sum of the individual waves.

In this case, we have two harmonic waves u1 and u2, with amplitudes B and frequencies a1 and a2, respectively. The resulting wave uint also has an amplitude Bint and frequency a3. Using the principle of superposition, we can write the following equation:

uint = u1 + u2

Substituting the given expressions for u1 and u2, we get:

Bint * sin(a3(r)) = B * sin(a1(r)) + B * sin(a2(r))

Now, using the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite the equation as:

Bint * sin(a3(r)) = 2B * sin[(a1(r) + a2(r))/2]cos[(a1(r) - a2(r))/2]

Comparing this with the form given in the link provided, we can see that:

a3(r) = (a1(r) + a2(r))/2
Bint = 2B * cos[(a1(r) - a2(r))/2]

Therefore, to solve for a3 and Bint, we need to know the values of a1 and a2 at a specific point in space, and the value of B. We can then use these equations to calculate a3 and Bint.

In conclusion, the superposition of two harmonic waves can be solved using the principle of superposition. By equating the resulting wave to the sum of the individual waves, we can obtain the values of a3 and Bint. However, the specific values of a1, a2, and B are needed to solve for these variables.