Two identical traveling waves moving in the same direction are out of phase by pi/2 rad. What is the amplitude Y of the two combining waves.
I have no clue how to do this. The answer according to the back of the book is 1.4y max. If you could explain to me how they got that value I would appreciate it. Thank you.
Ok, at the are out of phase by 90 deg.
Calculus approach:
Y= A sinwT + A coswT (that is two sine waves added, one is out of phase by 90 deg)
dY/dt=Awcoswt - Aw sinwt
setting to zero,
The max occurs when sinwt= coswt or at 45 degrees (+n90)
at 45 deg, the value of sin, cos is .707A. So adding the two waves, the max is 1.414 A.
To understand how the amplitude of the combined waves is determined, let's break it down step by step.
First, we start with the equation Y = A sin(wt) + A cos(wt), where A represents the amplitude of each individual wave and wt represents the angular frequency.
Since the two traveling waves are out of phase by π/2 rad (or 90 degrees), we want to find the maximum amplitude when they combine.
To find the maximum, we need to take the derivative of Y with respect to time (t) and set it equal to zero. This helps us find the points where the waves reach their maximum or minimum values.
Taking the derivative of Y = A sin(wt) + A cos(wt), we get:
dY/dt = A w cos(wt) - A w sin(wt)
Setting this derivative equal to zero:
0 = A w cos(wt) - A w sin(wt)
Now, we want to find the values of t where the sine and cosine functions are equal. At these points, the waves will add constructively and give the maximum amplitude.
The sine and cosine functions are equal when sin(wt) = cos(wt). Solving this equation, we find that this occurs at 45 degrees (or π/4 rad) and 225 degrees (or 5π/4 rad), and so on.
At 45 degrees, the values of sin(45°) and cos(45°) are both approximately 0.707. Therefore, at these points, the amplitude of the combined waves is:
Amplitude = A sin(45°) + A cos(45°) = A(0.707) + A(0.707) = 1.414 A
So, the amplitude of the two combining waves when they are out of phase by π/2 rad is 1.414 times the individual amplitude A.
In the given problem, the amplitude Y is mentioned as 1.4y max. The book's answer probably uses "y max" to represent the individual amplitude A. Therefore, the book's answer of 1.4y max is equivalent to 1.414 A, which matches the explanation above.