two sinusoidal waves traveling in the same direction having same frequency but different amplitudes and maintaining a phase difference derive an expression for intensity of maxima and minima.
the waves are
a sin(x) and b sin(x-k)
add them up and you have
a sinx + b(sinx cosk - cosx sink)
= (a+b*cosk)sinx - (b*sink)cosx
This is basically just
Asinx + Bcosx
which can be reduced to a single sine wave of amplitude √(A^2+B^2)
See a nice discussion at
https://en.wikibooks.org/wiki/Trigonometry/Simplifying_a_sin(x)_%2B_b_cos(x)
To derive the expression for the intensity of maxima and minima when two sinusoidal waves with the same frequency, different amplitudes, and a phase difference are traveling in the same direction, we can use the concept of superposition of waves.
Let's consider two waves with the following equations:
Wave 1: A1 * sin(ωt)
Wave 2: A2 * sin(ωt + φ)
Where:
A1 and A2 are the amplitudes of the waves,
ω is the angular frequency (equal for both waves),
t is the time variable,
φ is the phase difference between the waves.
To find the resultant wave, we add the two waves together:
Resultant wave: A1 * sin(ωt) + A2 * sin(ωt + φ)
Now, let's find the maximum and minimum values of the resultant wave by considering two cases: the maximum and minimum constructive interference.
Case 1: Maximum Constructive Interference (Maxima):
In this case, the waves are perfectly in phase, so the phase difference (φ) is zero. Therefore, the equation becomes:
Resultant wave (Maxima): A1 * sin(ωt) + A2 * sin(ωt)
Since sin(ωt) + sin(ωt) = 2 * sin(ωt), the equation reduces to:
Resultant wave (Maxima): (A1 + A2) * sin(ωt)
Thus, the intensity of the maxima is proportional to the square of the sum of the amplitudes:
Intensity (Maxima) ∝ (A1 + A2)^2
Case 2: Minimum Constructive Interference (Minima):
In this case, the waves are perfectly out of phase, so the phase difference (φ) is π. Therefore, the equation becomes:
Resultant wave (Minima): A1 * sin(ωt) + A2 * sin(ωt + π)
Since sin(ωt) + sin(ωt + π) = 0, the equation reduces to:
Resultant wave (Minima): 0
Hence, the intensity of the minima is zero.
In summary:
Intensity (Maxima) ∝ (A1 + A2)^2
Intensity (Minima) = 0
This expression gives an idea about the intensity of the resultant wave at the maximum and minimum points when two sinusoidal waves with the same frequency, different amplitudes, and a phase difference are superimposed.